chapter 2: reasoning and proof - clarkwork.com

Reasoning and ProofChapter Overview and PacingReasoning and ProofChapter Overview and Pacing

PACING (days)Regular Block

Basic/ Basic/ Average Advanced Average Advanced

Inductive Reasoning and Conjecture (pp. 62–66) 1 1 0.5 0.5• Make conjectures based on inductive reasoning.• Find counterexamples.

Logic (pp. 67–74) 2 1 1 0.5• Determine truth values of conjunctions and disjunctions.• Construct truth tables.

Conditional Statements (pp. 75–80) 2 1 1 0.5 • Analyze statements in if-then form. (with 2-3 (with 2-3 (with 2-3 (with 2-3• Write the converse, inverse, and contrapositive of if-then statements. Follow-Up) Follow-Up) Follow-Up) Follow-Up)

Deductive Reasoning (pp. 82–88) 1 1 0.5 0.5• Use the Law of Detachment. (with 2-4 (with 2-4 (with 2-3 (with 2-3• Use the Law of Syllogism. Follow-Up) Follow-Up) Follow-Up) Follow-Up)Follow-Up: Use a table and deductive reasoning to solve a logic problem.

Postulates and Paragraph Proofs (pp. 89–93) 1.5 1 1 0.5• Identify and use basic postulates about points, lines, and planes.• Write paragraph proofs.

Algebraic Proof (pp. 94–100) 1.5 1 1 0.5• Use algebra to write two-column proofs.• Use properties of equality in geometry proofs.

Proving Segment Relationships (pp. 101–106) 2 2 1 1• Write proofs involving segment addition.• Write proofs involving segment congruence.

Proving Angle Relationships (pp. 107–114) 2 2 1 1• Write proofs involving supplementary and complementary angles.• Write proofs involving congruent and right angles.

Study Guide and Practice Test (pp. 115–121) 1 1 0.5 0.5Standardized Test Practice (pp. 122–123)

Chapter Assessment 1 1 0.5 0.5

TOTAL 15 12 8 6

LESSON OBJECTIVES

60A Chapter 2 Reasoning and Proof

An electronic version of this chapter is available on StudentWorksTM. This backpack solution CD-ROMallows students instant access to the Student Edition, lesson worksheet pages, and web resources.

Year-long pacing: pages T20–T21.

*Key to Abbreviations: GCC � Graphing Calculator and Computer MastersSC � School-to-Career Masters

Chapter 2 Reasoning and Proof 60B

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75–76 77–78 79 80 119, 121 SC 3 2-4 2-4

81–82 83–84 85 86 2-5 2-5

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93–94 95–96 97 98 89–90, SC 4 2-7 2-7101–104

99–100 101–102 103 104 120 81–82, 2-8 2-8 patty paper, protractor, paper85–86

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60C Chapter 2 Reasoning and Proof

Mathematical Connections and BackgroundMathematical Connections and Background

Inductive Reasoning andConjectureA conjecture is an educated guess based on known

information. Examining several specific situations toarrive at a conjecture is called inductive reasoning.Meteorologists use inductive reasoning to predictweather conditions.

Just because a conjecture is true in most circum-stances does not make it a true conjecture. If just oneexample contradicts the conjecture, the conjecture is nottrue. The false example is called a counterexample.

LogicA statement is any sentence that is either true or

false, but not both. The truth or falsity of a statement iscalled its truth value. The negation of a statement hasthe opposite meaning as well as an opposite truth value.This means that if a statement is represented by p, thennot p is the negation of the statement. You could alsowrite not p as �p.

Two or more statements can be joined to form acompound statement. A conjunction is a compoundstatement formed by joining two or more statementswith the word and. The symbol � can be used instead ofand. Two or more statements can also be joined to forma disjunction. A disjunction is a compound statementformed by joining two or more statements with theword or. You might see the symbol � instead of theword or in a disjunction.

Conjunctions and disjunctions can be illustratedwith Venn diagrams. Truth tables can also be helpful inevaluating the truth values of statements. A truth tablewill show that a conjunction is true only when bothstatements are true. A disjunction, on the other hand, istrue unless both statements are false.

Conditional StatementsA conditional statement is a statement that can be

written in if-then form: if p, then q. The phrase immedi-ately following the word if is called the hypothesis. Thephrase immediately following the word then is calledthe conclusion. An arrow pointing to the right is writtenbetween p and q to symbolize an if-then situation. Aconditional statement is true in all cases except wherethe hypothesis is true and the conclusion is false.

Related conditionals are statements constructedfrom an if-then statement. A converse statement isformed by exchanging the hypothesis and the conclusion:if q, then p. An inverse statement is formed by negatingboth the hypothesis and the conclusion of the original

Prior KnowledgePrior KnowledgeIn algebra, students learned to solve for avariable. In Chapter 1, students learnedabout points, lines and planes. They wereintroduced to adjacent and vertical angles,complementary and supplementary angles,and right angles. They can identify

congruent segments and angles as well as perpendicular

lines.

This Chapter

Future ConnectionsFuture ConnectionsIn Chapter 4, students will build on theirknowledge of proofs when they investigatetriangles. Students will also write proofs inChapter 7. Thinking logically is a crucial skill fordaily living. Logic is used to make informedchoices and to examine a statement for truth.

Continuity of InstructionContinuity of Instruction

This ChapterIn this chapter, students explore methods ofreasoning and learn to apply those methods togeometry. They make conjectures, determinethe truth values of compound statements,and construct truth tables. They also analyzeconditional statements and write relatedconditionals. The terms postulate and theorem are introduced. Algebraic propertiesof equality are applied to geometry, enablingstudents to write formal and informal proofsproving segment and angle relationships.

Chapter 2 Reasoning and Proof 60D

statement: if �p, then �q. A contrapositive is formedby negating both the hypothesis and the conclusionof the converse statement: if �q, then �p.

Deductive ReasoningDeductive reasoning uses facts, rules, defini-

tions, or properties to reach logical conclusions. Aform of deductive reasoning that is used to draw con-clusions from true conditional statements is called theLaw of Detachment. This law states that if p → q istrue and p is true, then q is also true.

The Law of Syllogism is another law of logic.It states that if p → q and q → r are true, then p → r isalso true. You may see the similarity between this lawand the Transitive Property of Equality from algebra.

Postulates and Paragraph ProofsIn geometry, a postulate is a statement that

describes a fundamental relationship between thebasic terms of geometry. Postulates are accepted astrue without proof. Several postulates based on therelationship among points, lines, and planes wereintroduced in Chapter 1, but were not labeled aspostulates.

In this course, you will learn to use variousmethods to justify the truth of a statement or conjec-ture. Once a statement or conjecture has been shownto be true, it is called a theorem. A theorem can beused like a definition or postulate to justify that otherstatements are true.

A proof is a logical argument in which eachstatement you make is supported by a statement thatis accepted as true. One type of proof is called a para-graph or informal proof. It is a written explanation ofwhy a conjecture for a given situation is true. A goodproof states the theorem or conjecture to be proven. Itlists the given information and, if possible, supplies adiagram to illustrate the given information. The proofstates what is to be proved and develops a system ofdeductive reasoning.

Algebraic ProofIn algebra, you learned to use properties of

equality to solve algebraic equations and to verifyrelationships. These properties can be used to justifyeach step when solving an equation. A group of alge-braic steps used to solve problems form a deductiveargument. This argument can be demonstrated by

writing the solution to the equation in the first columnand listing the property justifying each step in thesecond column.

In geometry, a similar format is used to proveconjectures and theorems. A two-column, or formal,proof contains statements and reasons organized intwo columns. Each step is called a statement, and theproperties that justify each step are called reasons.

Proving SegmentRelationshipsAs you learned in Chapter 1, a segment can be

measured, and measures can be used in calculationsbecause they are real numbers. One postulate aboutsegments is called the Ruler Postulate. It states that thepoints on any line or line segment can be paired withreal numbers so that, given any two points A and Bon a line, A corresponds to 0, and B corresponds to apositive real number. That number is the length ofthe segment. Another postulate states that if point Blies between points A and C on the same line, AB � BC � AC. The converse statement holds true as well.

The Reflexive, Symmetric, and TransitiveProperties of Equality can be used to write proofsabout segment congruence. The theorem resultingfrom the proofs states that congruence of segments isreflexive, symmetric, and transitive.

Proving Angle RelationshipsThis lesson introduces postulates and theorems

about angle relationships. The Protractor Postulate states, “Given AB�� and a number r between 0 and 180,there is exactly one ray with endpoint A, extending on either side of AB��, such that the measure of theangle formed is r.” The Angle Addition Postulatestates that if R is in the interior of �PQS, thenm�PQR � m�RQS � m�PQS. If m�PQR � m�RQS �m�PQS, then R is in the interior of �PQS. This postulate can be used with other angle relationshipsto prove other theorems relating to angles.

Some of these theorems relate to supplementaryand complementary angles. Another theorem extendsthe Reflexive, Transitive, and Symmetric properties toangle congruence. There is also a series of theoremsabout perpendicular lines and right angles.

60E Chapter 2 Reasoning and Proof

Key to Abbreviations:TWE = Teacher Wraparound Edition; CRM = Chapter Resource Masters

Ongoing Prerequisite Skills, pp. 61, 80,87, 93, 100, 106

Practice Quiz 1, p. 80Practice Quiz 2, p. 100

GeomPASS: Tutorial Plus,Lesson 6

www.geometryonline.com/self_check_quiz

www.geometryonline.com/extra_examples

5-Minute Check TransparenciesPrerequisite Skills Workbook, pp. 41–44, 81–86,

89–90, 93–94, 101–104Quizzes, CRM pp. 119–120Mid-Chapter Test, CRM p. 121Study Guide and Intervention, CRM pp. 57–58, 63–64,

69–70, 75–76, 81–82, 87–88, 93–94, 99–100

MixedReview

Cumulative Review, CRM p. 122 pp. 66, 74, 80, 93, 100, 106, 114

ErrorAnalysis

Find the Error, TWE pp. 84, 111Unlocking Misconceptions, TWE p. 91Tips for New Teachers, TWE p. 70

Find the Error, pp. 84, 111Common Misconceptions, p. 76

StandardizedTest Practice

TWE pp. 122–123Standardized Test Practice, CRM pp. 123–124

Standardized Test PracticeCD-ROM

www.geometryonline.com/standardized_test

pp. 66, 74, 80, 86, 87, 93, 96,97, 99, 106, 114, 121, 122

Open-EndedAssessment

Modeling: TWE pp. 74, 87, 106Speaking: TWE pp. 80, 93Writing: TWE pp. 66, 100, 114Open-Ended Assessment, CRM p. 117

Writing in Math, pp. 66, 74, 79,86, 93, 99, 106, 114, 123

Open Ended, pp. 63, 71, 78, 84,91, 97, 103, 111

Standardized Test, p. 123

ChapterAssessment

Multiple-Choice Tests (Forms 1, 2A, 2B), CRM pp. 105–110

Free-Response Tests (Forms 2C, 2D, 3), CRM pp. 111–116

Vocabulary Test/Review, CRM p. 118

ExamView® Pro (see below)MindJogger Videoquizzes www.geometryonline.com/

vocabulary_reviewwww.geometryonline.com/

chapter_test

Study Guide, pp. 115–120Practice Test, p. 121

and Assessmentand AssessmentA

SSES

SMEN

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Type Student Edition Teacher Resources Technology/Internet

Geometry Lesson Yearly ProgressPro Skill Lesson

2-1 Inductive Reasoning and Conjecture2-2 Logic2-3 Conditional Statements2-4 Deductive Reasoning2-5 Postulates and Paragraph Proofs2-6 Algebraic Proof2-7 Proving Segment Relationships2-8 Proving Angle Relationships

ExamView® ProUse the networkable ExamView® Pro to:• Create multiple versions of tests.• Create modified tests for Inclusion students.• Edit existing questions and add your own questions.• Use built-in state curriculum correlations to create

tests aligned with state standards.• Apply art to your test from a program bank of artwork.

For more information on Yearly ProgressPro, see p. 2.

For more information on Intervention and Assessment, see pp. T8–T11.

http://www.geometryonline.com/self_check_quiz

http://www.geometryonline.com/extra_examples

http://www.geometryonline.com/standardized_test

http://www.geometryonline.com/vocabulary_review

http://www.geometryonline.com/chapter_test

Chapter 2 Reasoning and Proof 60F

Reading and Writing in MathematicsReading and Writing in Mathematics

Student Edition

• Foldables Study Organizer, p. 61• Concept Check questions require students to verbalize

and write about what they have learned in the lesson.(pp. 63, 71, 78, 84, 91, 97, 103, 111)

• Reading Mathematics, p. 81 • Writing in Math questions in every lesson, pp. 66, 74,

79, 86, 93, 99, 106, 114• Reading Study Tip, p. 75• WebQuest, p. 65

Teacher Wraparound Edition

• Foldables Study Organizer, pp. 61, 115• Study Notebook suggestions, pp. 64, 72, 78, 81, 84, 88,

91, 97, 104, 111 • Modeling activities, pp. 74, 87, 106• Speaking activities, pp. 80, 93• Writing activities, pp. 66, 100, 114• Differentiated Instruction (Verbal/Linguistic), p. 83• Resources, pp. 60, 65, 73, 79, 81, 83, 86, 92, 99,

105, 113, 115ELL

Glencoe Geometry provides numerous opportunities to incorporate reading and writinginto the mathematics classroom.

Additional Resources

• Vocabulary Builder worksheets require students todefine and give examples for key vocabulary terms asthey progress through the chapter. (Chapter 2 ResourceMasters, pp. vii-viii)

• Proof Builder helps students learn and understand theorems and postulates from the chapter. (Chapter 2Resource Masters, pp. ix–x)

• Reading to Learn Mathematics master for each lesson(Chapter 2 Resource Masters, pp. 61, 67, 73, 79, 85, 91,97, 103)

• Vocabulary PuzzleMaker software creates crossword,jumble, and word search puzzles using vocabulary liststhat you can customize.

• Teaching Mathematics with Foldables provides suggestions for promoting cognition and language.

• Reading Strategies for the Mathematics Classroom• WebQuest and Project Resources

Many of the vocabulary terms introduced inChapter 2 can be represented by symbols. Three-column notes can be a helpful way for students to organize new vocabulary terms. To reinforceunderstanding, students can write an explanationof each term in their own words and provide theappropriate symbol. The table at the right showsnotes for Lesson 2-2. Students can add on to thissample with other terms from Chapter 2.

Term Explanation Symbol

negation the opposite of the given �statement

conjunction a compound statement formed p � qwith the word “and”

disjunction a compound statement formed p � qwith the word “or”

For more information on Reading and Writing in Mathematics, see pp. T6–T7.

Have students read over the listof objectives and make a list ofany words with which they arenot familiar.

Point out to students that this isonly one of many reasons whyeach objective is important.Others are provided in theintroduction to each lesson.

60 Chapter 2 Reasoning and Proof

Reasoning and Proof

• inductive reasoning (p. 62)• deductive reasoning (p. 82)• postulate (p. 89)• theorem (p. 90)• proof (p. 90)

Key Vocabulary

B. Busco/Getty Images

Logic and reasoning are used throughout geometry to solveproblems and reach conclusions. There are many professionsthat rely on reasoning in a variety of situations. Doctors, for example, use reasoning to diagnose and treat patients.You will investigate how doctors use reasoning in Lesson 2-4.

• Lessons 2-1 through 2-3 Make conjectures,determine whether a statement is true or false, and find counterexamples for statements.

• Lesson 2-4 Use deductive reasoning to reach valid conclusions.

• Lessons 2-5 and 2-6 Verify algebraic and geometric conjectures using informal and formal proof.

• Lessons 2-7 and 2-8 Write proofs involvingsegment and angle theorems.


NotesNotes

NCTM LocalLesson Standards Objectives

2-1 1, 6, 7, 8, 9, 10

2-2 6, 7, 8, 9, 10

2-3 3, 6, 7, 8, 9, 10

2-4 3, 6, 7, 8, 9, 10

2-4 6, 7Follow-Up

2-5 3, 6, 7, 8, 9, 10

2-6 2, 3, 6, 7, 8, 9, 10

2-7 3, 6, 7, 8, 9, 10

2-8 3, 6, 7, 8, 9, 10

Key to NCTM Standards: 1=Number & Operations, 2=Algebra,3=Geometry, 4=Measurement, 5=Data Analysis & Probability, 6=ProblemSolving, 7=Reasoning & Proof,8=Communication, 9=Connections,10=Representation

Vocabulary BuilderThe Key Vocabulary list introduces students to some of the main vocabulary termsincluded in this chapter. For a more thorough vocabulary list with pronunciations ofnew words, give students the Vocabulary Builder worksheets found on pages vii andviii of the Chapter 2 Resource Masters. Encourage them to complete the definition of each term as they progress through the chapter. You may suggest that they addthese sheets to their study notebooks for future reference when studying for theChapter 2 test.

ELL

This section provides a review ofthe basic concepts needed beforebeginning Chapter 2. Pagereferences are included foradditional student help.Additional review is provided inthe Prerequisite Skills Workbook,pages 41–44, 81–86, 89–90, 93–94,101–104.

Prerequisite Skills in the GettingReady for the Next Lesson sectionat the end of each exercise setreview a skill needed in the nextlesson.

Chapter 2 Reasoning and Proof 61

Prerequisite Skills To be successful in this chapter, you’ll need to masterthese skills and be able to apply them in problem-solving situations. Reviewthese skills before beginning Chapter 2.

For Lesson 2-1 Evaluate Expressions

Evaluate each expression for the given value of n. (For review, see page 736.)

1. 3n � 2; n � 4 10 2. (n � 1) � n; n � 6 13 3. n2 � 3n; n � 3 0

4. 180(n � 2); n � 5 540 5. n��n2

��; n � 10 50 6. �n(n

2� 3)�; n � 8 20

For Lessons 2-6 through 2-8 Solve Equations

Solve each equation. (For review, see pages 737 and 738.)

7. 6x � 42 � 4x 21 8. 8 � 3n � �2 � 2n 2 9. 3(y � 2) � �12 � y �9

10. 12 � 7x � x � 18 �5 11. 3x � 4 � �12

�x � 5 ��158� 12. 2 � 2x � �

23

�x � 2 �32

�

For Lesson 2-8 Adjacent and Vertical Angles

For Exercises 13–14, refer to the figure at the right. (For review, see Lesson 1-5.)

13. If m�AGB � 4x � 7 and m�EGD � 71, find x. 1614. If m�BGC � 45, m�CGD � 8x � 4, and m�DGE � 15x � 7,

find x. 6

A B

CF

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G

Reasoning and Proof Make this Foldable to help you organize your notes. Beginwith eight sheets of 8 ” by 11” grid paper.1

�2

Reasoning & Proof

2-2

2-3

2-4

2-5

2-6

2-7

2-8

2-1

Reading and Writing As you read and study each lesson, use the corresponding page to write proofs andrecord examples of when you used logical reasoning in your daily life.

Staple

Label

Cut Tabs

Stack and staple the eightsheets together to form abooklet.

Cut the bottom of eachsheet to form a tabbedbook.

Label each of the tabs witha lesson number. Add thechapter title to the first tab.

Chapter 2 Reasoning and Proof 61

For PrerequisiteLesson Skill

2-3 Evaluating AlgebraicExpressions, p. 74

2-4 Solving Equations, p. 80

2-5 Information from Figures, p. 87

2-6 Solving Equations, p. 93

2-7 Segment Measures, p. 100

2-8 Complementary andSupplementary Angles, p. 106

Organization of Data Use this Foldable for student writing aboutreasoning and proofs. After students make their Foldable, have themlabel the tabs to correspond to the eight lessons in this chapter.Students use their Foldable to take notes, define terms, recordconcepts, write statements in if-then form, and write paragraphproofs. On the back of the Foldable, have students record examplesof ways in which they use reasoning and proofs in their daily lives.Note how columnists and authors present their reasoning and ways in which they try to prove or disprove their points of view.

TM

For more informationabout Foldables, seeTeaching Mathematicswith Foldables.

5-Minute CheckTransparency 2-1 Use as a

quiz or review of Chapter 1.

Mathematical Background notesare available for this lesson on p. 60C.

can inductive reasoninghelp predict weather

conditions?Ask students:• What are normal temperatures

for the month of January?Sample answer: The temperaturesin January are usually in the 30s or40s. (Answers will vary in differentparts of the country.)

• How do people benefit fromthe inductive reasoningtechniques of meteorologists?Sample answers: People can planfor outdoor events a few days inadvance; they can dressappropriately for daily weatherconditions and carry weather-related items, such as umbrellas,sunglasses, and so on.

MAKE CONJECTURES A is an educated guess based on knowninformation. Examining several specific situations to arrive at a conjecture is calledinductive reasoning. is reasoning that uses a number of specificexamples to arrive at a plausible generalization or prediction.

Inductive reasoning

conjecture

In Chapter 1, you learned some basic geometric concepts. These concepts can beused to make conjectures in geometry.

Vocabulary• conjecture• inductive reasoning• counterexample

Inductive Reasoning and Conjecture


• Make conjectures based on inductive reasoning.

• Find counterexamples.

can inductive reasoning helppredict weather conditions?can inductive reasoning helppredict weather conditions?

Patterns and ConjectureThe numbers represented below are called triangular numbers. Make aconjecture about the next triangular number based on the pattern.

Observe: Each triangle is formed by adding another row of dots.

Find a Pattern: 1 3 6 10 15

�2 �3 �4 �5

The numbers increase by 2, 3, 4, and 5.

Conjecture: The next number will increase by 6. So, it will be 15 � 6 or 21.

1 3 6 10 15

Example 1Example 1

Bob Daemmrich/Stock Boston

Meteorologists use science and weatherpatterns to make predictions about futureweather conditions. They are able to makeaccurate educated guesses based on pastweather patterns.

ConjecturesList your observations andidentify patterns beforeyou make a conjecture.

Study Tip

LessonNotes

1 Focus1 Focus

Chapter 2 Resource Masters• Study Guide and Intervention, pp. 57–58• Skills Practice, p. 59• Practice, p. 60• Reading to Learn Mathematics, p. 61• Enrichment, p. 62

Graphing Calculator and Computer Masters, p. 19

5-Minute Check Transparency 2-1Answer Key Transparencies

TechnologyInteractive Chalkboard

Workbook and Reproducible Masters

Resource ManagerResource Manager

Transparencies

33

In-Class ExampleIn-Class Example PowerPoint®

11

22

In-Class ExamplesIn-Class Examples PowerPoint®

MAKE CONJECTURES

Teaching Tip Tell students totest all fundamental operations,including powers and roots,when they are looking forpatterns in a series of numbers.Advise students that sometimestwo operations can be used.

Make a conjecture about thenext number based on thepattern. 2, 4, 12, 48, 240 1440

For points L, M, and N, LM � 20, MN � 6, and LN � 14. Make a conjectureand draw a figure toillustrate your conjecture.

Conjecture: L, M, and N arecollinear.

FIND COUNTEREXAMPLES

UNEMPLOYMENT Based onthe table showingunemployment rates forvarious cities in Kansas, finda counterexample for thefollowing statement.The unemployment rate is highestin the cities with the most people.

Source: Labor Market Information Services—KansasDepartment of Human Resources

Osage has only 10,182 people onits civilian labor force, and it hasa higher rate of unemploymentthan Shawnee, which has 90,254people on its civilian labor force.

CivilianCounty

Labor ForceRate

Shawnee 90,254 3.1%Jefferson 9,937 3.0%Jackson 8,915 2.8%Douglas 55,730 3.2%Osage 10,182 4.0%Wabaunsee 3,575 3.0%Pottawatomie 11,025 2.1%

L614

MN

20

Lesson 2-1 Inductive Reasoning and Conjecture 63

FIND COUNTEREXAMPLES A conjecture based on several observations maybe true in most circumstances, but false in others. It takes only one false example toshow that a conjecture is not true. The false example is called a .counterexample

1. Write an example of a conjecture you have made outside of school.

2. Determine whether the following conjecture is always, sometimes, or never truebased on the given information.Given: collinear points D, E, and FConjecture: DE � EF � DF

3. OPEN ENDED Write a statement. Then find a counterexample for the statement.

Geometric ConjectureFor points P, Q, and R, PQ � 9, QR � 15, and PR � 12. Make a conjecture anddraw a figure to illustrate your conjecture.Given: points P, Q, and R; PQ � 9, QR � 15, and PR � 12

Examine the measures of the segments. Since PQ � PR � QR, the points cannot be collinear.

Conjecture: P, Q, and R are noncollinear. Q

P R

15

12

9

Example 2Example 2

Find a CounterexampleFINANCE Find a counterexamplefor the following statement basedon the graph.The rates for CDs are at least 1.5%less than the rates a year ago.Examine the graph. The statement is true for 6-month, 1-year, and 2�

12

�-year CDs. However, the difference in the rate for a 5-year CD is 0.74% less, which is less than1.5%. The statement is false for a 5-year certificate of deposit. Thus,the change in the 5-year rate is acounterexample to the originalstatement.

Example 3Example 3

Concept Check1–3. See p. 123A.

• Updated data• More on finding

counterexamples

Log on for:Log on for:

www.geometryonline.com/usa_today

Latest CD rates

USA TODAY Snapshots®

USA TODAY

Average certificate of deposit rates as of Wednesday:

Source: Bank Rate Monitor, 800-327-7717, www.bankrate.com

This weekLast week

Year ago6-month

1-year

21⁄2-year

5-year

This weekLast week

Year ago

This weekLast week

Year ago

This weekLast week

Year ago

1.80%1.80%

4.55%

2.12%2.11%

4.64%

2.96%2.96%

4.74%

4.22%4.23%

4.96%



2 Teach2 Teach

InteractiveChalkboard

PowerPoint®

Presentations

This CD-ROM is a customizable Microsoft® PowerPoint®presentation that includes:• Step-by-step, dynamic solutions of each In-Class Example

from the Teacher Wraparound Edition• Additional, Try These exercises for each example• The 5-Minute Check Transparencies• Hot links to Glencoe Online Study Tools


http://www.geometryonline.com/usa_today

3 Practice/Apply3 Practice/Apply

Study NotebookStudy NotebookHave students—• add the definitions/examples of

the vocabulary terms to theirVocabulary Builder worksheets forChapter 2.

• include any other item(s) that theyfind helpful in mastering the skillsin this lesson.

Make a conjecture about the next item in each sequence.4.

5. �8, �5, �2, 1, 4 7

Make a conjecture based on the given information. Draw a figure to illustrateyour conjecture. 6–7. See margin.6. PQ � RS and RS � TU 7. AB�� and CD�� intersect at P.

Determine whether each conjecture is true or false. Give a counterexample for anyfalse conjecture.8. Given: x is an integer.

Conjecture: �x is negative. False; if x � �2, then �x � �(�2) or 2.

9. Given: WXYZ is a rectangle.Conjecture: WX � YZ and WZ � XY true

10. HOUSES Most homes in the northern United States have roofs made withsteep angles. In the warmer areas of the southern states, homes often have flatroofs. Make a conjecture about why the roofs are different. Sample answer:Snow will not stick on a roof with a steep angle.


Guided Practice

Application

Practice and ApplyPractice and Applyindicates increased difficulty★

Make a conjecture about the next item in each sequence.11.

12.

13. 1, 2, 4, 8, 16 32 14. 4, 6, 9, 13, 18 24 15. �13

�, 1, �53

�, �73

�, 3 �131�

16. 1, �12

�, �14

�, �18

�, �116� �

312� 17. 2, �6, 18, �54 162 18. �5, 25, �125, 625

�3125

Make a conjecture about the number of blocks in the next item of each sequence.19. 20.

19–20. See p. 123A.

Make a conjecture based on the given information. Draw a figure to illustrateyour conjecture. 21–28. See p. 123A for figures.21. Lines � and m are perpendicular. 22. A(�2, �11), B(2, 1), C(5, 10)

23. �3 and �4 are a linear pair. 24. BD�� is an angle bisector of �ABC.

25. P(�1, 7), Q(6, �2), R(6, 5) 26. HIJK is a square.

27. PQRS is a rectangle. 28. �B is a right angle in �ABC.PQ � SR, QR � PS (AB )2 � (BC )2 � (AC )2

ForExercises

11–2021–2829–36

SeeExamples

123

Extra Practice See page 756.


21. Lines � and mform four right angles.22. A, B, and C arecollinear.23. �3 and �4 aresupplementary.24. �ABD ��DBC.25. ∆PQR is ascalene triangle.26. HI � IJ � JK �KH

GUIDED PRACTICE KEYExercises Examples

4–5 16–7 28–9 3


About the Exercises…Organization by Objective• Make Conjectures: 11–28• Find Counterexamples:

29–36

Odd/Even AssignmentsExercises 11–36 are structuredso that students practice thesame concepts whether theyare assigned odd or evenproblems.

Assignment GuideBasic: 11–41 odd, 43–67Average: 11–41 odd, 43–67Advanced: 12–40 even, 41–64(optional: 65–67)

Answers

6. PQ � TU

7. A, B, C, and D are noncollinear.

PA

C

D

B

P Q

R S

T U

Naturalist Students can practice brainstorming conjectures and findingcounterexamples in nature. For example have students consider thestatement, “If plants don’t receive water daily, they will not survive.” Acounterexample would be a cactus, which can go weeks without water.Nature topics could include plants, animals, predator/food supplyrelationships, insects, weather, and so on.

Differentiated Instruction

Study Guide and InterventionInductive Reasoning and Conjecture

NAME ______________________________________________ DATE ____________ PERIOD _____

2-12-1

Gl /M G Hill 57 Gl G

Less

on

2-1

Make Conjectures A conjecture is a guess based on analyzing information orobserving a pattern. Making a conjecture after looking at several situations is calledinductive reasoning.

Make a conjecture aboutthe next number in the sequence 1, 3, 9,27, 81.Analyze the numbers:Notice that each number is a power of 3.

1 3 9 27 8130 31 32 33 34

Conjecture: The next number will be 35 or 243.

Make a conjecture about the number of smallsquares in the next figure.Observe a pattern: The sides of the squareshave measures 1, 2, and 3 units.Conjecture: For the next figure, the side ofthe square will be 4 units, so the figurewill have 16 small squares.

Example 1Example 1 Example 2Example 2

ExercisesExercises

Describe the pattern. Then make a conjecture about the next number in thesequence.

1. �5, 10, �20, 40 Pattern: Each number is �2 times the previous number.Conjecture: The next number is �80.

2. 1, 10, 100, 1000 Pattern: Each number is 10 times the previous number.Conjecture: The next number is 10,000.

3. 1, �65�, �

75�, �

85� Pattern: Each number is �

15

� more than the previous number.

Conjecture: The next number is �95

�.

Make a conjecture based on the given information. Draw a figure to illustrateyour conjecture. 4–7. Sample answers are given.

4. A(�1, �1), B(2, 2), C(4, 4) 5. �1 and �2 form a right angle.Points A, B, and C are collinear. �1 and �2 are complementary.

6. �ABC and �DBE are vertical angles. 7. �E and �F are right angles.�ABC and �DBE are congruent. �E and �F are congruent.

FERT

QP

DC

EBA

T W

12

P

R

x

y

OA(–1, –1)

B(2, 2)

C(4, 4)

Study Guide and Intervention, p. 57 (shown) and p. 58


Make a conjecture about the next item in each sequence.

1.

2. 5, �10, 15, �20 25 3. �2, 1, ��12�, �

14�, ��

18� �

116� 4. 12, 6, 3, 1.5, 0.75 0.375

Make a conjecture based on the given information. Draw a figure to illustrateyour conjecture. 5–8. Sample answers are given.

5. �ABC is a right angle. 6. Point S is between R and T.

BA�� ⊥ BC�� RS � ST � RT

7. P, Q, R, and S are noncollinear 8. ABCD is a parallelogram.and P�Q� � Q�R� � R�S� � S�P�.

The segments form a square. AB � CD and BC � AD.

Determine whether each conjecture is true or false. Give a counterexample forany false conjecture.

9. Given: S, T, and U are collinear and ST � TU.Conjecture: T is the midpoint of S�U�.

true

10. Given: �1 and �2 are adjacent angles.Conjecture: �1 and �2 form a linear pair.

False; �1 and �2 could each measure 60°.

11. Given: G�H� and J�K� form a right angle and intersect at P.Conjecture: G�H� ⊥ J�K�true

12. ALLERGIES Each spring, Rachel starts sneezing when the pear trees on her street blossom.She reasons that she is allergic to pear trees. Find a counterexample to Rachel’s conjecture.Sample answer: Rachel could be allergic to other types of plants thatblossom when the pear trees blossom.

D

A

C

B

S

P

R

Q

TSRA

CB

Practice (Average)

Inductive Reasoning and Conjecture

NAME ______________________________________________ DATE ____________ PERIOD _____

2-12-1Skills Practice, p. 59 and Practice, p. 60 (shown)

Reading to Learn MathematicsInductive Reasoning and Conjecture

NAME ______________________________________________ DATE ____________ PERIOD _____

2-12-1

Less

on

2-1

Pre-Activity How can inductive reasoning help predict weather conditions?

Read the introduction to Lesson 2-1 at the top of page 62 in your textbook.

• What kind of weather patterns do you think meteorologists look at tohelp predict the weather? Sample answer: patterns of high andlow temperatures, including heat spells and cold spells;patterns of precipitation, including wet spells and dry spells

• What is a factor that might contribute to long-term changes in theweather? Sample answer: global warming due to high usageof fossil fuels

Reading the Lesson1. Explain in your own words the relationship between a conjecture, a counterexample, and

inductive reasoning.Sample answer: A conjecture is an educated guess based on specificexamples or information. A counterexample is an example that showsthat a conjecture is false. Inductive reasoning is the process of making aconjecture based on specific examples or information.

2. Make a conjecture about the next item in each sequence.

a. 5, 9, 13, 17 21 b. 1, �13�, �

19�, �2

17�

�811�

c. 0, 1, 3, 6, 10 15 d. 8, 3, �2, �7 �12e. 1, 8, 27, 64 125 f. 1, �2, 4, �8 16g. h.

3. State whether each conjecture is true or false. If the conjecture is false, give acounterexample.

a. The sum of two odd integers is even.

trueb. The product of an odd integer and an even integer is odd.

False; sample answer: 5 � 8 � 40, which is even.c. The opposite of an integer is a negative integer. False; sample answer: The

opposite of the integer �5 is 5, which is a positive integer.d. The perfect squares (squares of whole numbers) alternate between odd and even.

true

Helping You Remember4. Write a short sentence that can help you remember why it only takes one counterexample

to prove that a conjecture is false.Sample answer: True means always true.

Reading to Learn Mathematics, p. 61

CounterexamplesWhen you make a conclusion after examining several specificcases, you have used inductive reasoning. However, you must becautious when using this form of reasoning. By finding only onecounterexample, you disprove the conclusion.

Is the statement �1x� � 1 true when you replace x with

1, 2, and 3? Is the statement true for all reals? If possible, find a counterexample.

�11

� � 1, �12

� � 1, and �13

� � 1. But when x � �12

�, then �1x

� � 2. This counterexample

shows that the statement is not always true.

Answer each question.

1. The coldest day of the year in Chicago 2. Suppose John misses the school busoccurred in January for five straight four Tuesdays in a row. Can youyears. Is it safe to conclude that the safely conclude that John misses the coldest day in Chicago is always in school bus every Tuesday? no

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

2-12-1

ExampleExample

Enrichment, p. 62


Determine whether each conjecture is true or false. Give a counterexample for anyfalse conjecture.29. Given: �1 and �2 are complementary angles.

Conjecture: �1 and �2 form a right angle.

30. Given: m � y � 10, y � 4Conjecture: m � 6 False; if y � 7 and m � 5, then 7 � 5 � 10

31. Given: points W, X, Y, and ZConjecture: W, X, Y, and Z are noncollinear.

32. Given: A(�4, 8), B(3, 8), C(3, 5)Conjecture: � ABC is a right triangle. true

33. Given: n is a real number.Conjecture: n2 is a nonnegative number. true

34. Given: DE � EFConjecture: E is the midpoint of D�F�.

35. Given: JK � KL � LM � MJConjecture: JKLM forms a square. False; JKLM may not have a right angle.

36. Given: noncollinear points R, S, and TConjecture: R�S�, S�T�, and R�T� form a triangle. true

37. MUSIC Many people learn to play the piano by ear. This means that they firstlearned how to play without reading music. What process did they use? trialand error, a process of inductive reasoning

CHEMISTRY For Exercises 38–40, use the following information.Hydrocarbons are molecules composed of only carbon (C) and hydrogen (H) atoms. The simplest hydrocarbons are called alkanes. The first three alkanes areshown below.

38. Make a conjecture about butane, which is the next compound in the group.Write its structural formula.

39. Write the chemical formula for the 7th compound in the group. C7H1640. Develop a rule you could use to find the chemical formula of the nth substance

in the alkane group. CnH2n � 2

41. CRITICAL THINKING The expression n2 � n � 41 has a prime value for n � 1, n � 2, and n � 3. Based on this pattern, you might conjecture that thisexpression always generates a prime number for any positive integral value of n. Try different values of n to test the conjecture. Answer true if you think the conjecture is always true. Answer false and give a counterexample if youthink the conjecture is false. false; n � 41

False; D, E, and F do not have to becollinear.

False; see margin forcounterexample.

False; see margin forcounterexample.

Mary Kate Denny/PhotoEdit

MusicThe average medium-sized piano has about230 strings. Each stringhas about 165 poundsof tension. That’s acombined tension ofabout 18 tons.Source: www.pianoworld.com

38. Butane will have 4 carbon atoms and 10 hydrogen atoms.

H

H

CH C

H

H

C

H

H

C

H

H

H

★

Methane Ethane Propane

CH4 C2H6 C3H8

H

H

CH C

H

H

C

H

H

H

H

H

CH C

H

H

H

H

H

HH C

Alkanes

Compound Name

Chemical Formula

Structural Formula

You can use scatter plots to makeconjectures about therelationships betweenlatitude, longitude,degree distance, andthe monthly hightemperature. Visit www.geometryonline.com/WebQuestto continue work onyour WebQuest project.

and 5 � 4, but 7 � 6.



ELL

Answers

29.

31. W X Y Z

12


http://www.geometryonline.com/webquest

Open-Ended AssessmentWriting Ask students to writefive conjectures about schoolrules or activities. Then havestudents swap papers with apartner and try to come up witha counterexample for eachconjecture. An example statementcould be: Students must attendschool Monday through Friday. Acounterexample for this wouldbe a holiday or a snow day.

Getting Ready forLesson 2-2Basic Skill Students will learnabout logic statements in Lesson2-2. They will determine thetruth value of various situations.Use Exercises 65–67 to determineyour students’ familiarity withdetermining which elementsmake a statement true.

Answer

42. Sample answer: By pastexperience, when dark cloudsappear, there is a chance of rain.Answers should include thefollowing.• When there is precipitation in

the summer, it is usually rainbecause the temperature isabove freezing. When thetemperature is below freezing, asin the winter, ice or snow forms.

• See students’ work.

42. Answer the question that was posed at the beginning ofthe lesson. See margin.

How can inductive reasoning help predict weather conditions?

Include the following in your answer:• an explanation as to how a conjecture about a weather pattern in the summer

might be different from a similar weather pattern in the winter, and• a conjecture about tomorrow’s weather based on your local weather over the

past several days.

43. What is the next term in the sequence 1, 1, 2, 3, 5, 8? C11 12 13 14

44. ALGEBRA If the average of six numbers is 18 and the average of three of thenumbers is 15, then what is the sum of the remaining three numbers? D

21 45 53 63DCBA

DCBA

WRITING IN MATH


48. Yes; the symboldenotes that �KJN isa right angle.49. No; we do notknow anything aboutthe angle measures.50. No; we do notknow whether �MNPis a right angle.51. Yes; they form alinear pair.52. Yes; since theother three angles inrectangle KLPJ areright angles, �KLPmust also be a rightangle.

Maintain Your SkillsMaintain Your Skills

Mixed Review

Getting Ready forthe Next Lesson

Name each polygon by its number of sides and then classify it as convex orconcave and regular or irregular. (Lesson 1-6)

45. 46. 47.

Determine whether each statement can be assumed from the figure. Explain. (Lesson 1-5)

48. �KJN is a right angle.

49. �PLN �NLM

50. �PNL and �MNL are complementary.

51. �KLN and �MLN are supplementary.

52. �KLP is a right angle.

Find the coordinates of the midpoint of a segment having the given endpoints.(Lesson 1-3)

53. A�B� for A(�1, 3), B(5, �5) (2, �1) 54. C�D� for C(4, 1), D(�3, 7) (0.5, 4)55. F�G� for F(4, �9), G(�2, �15) (1, �12) 56. H�J� for H(�5, �2), J(7, 4) (1, 1)57. K�L� for K(8, �1.8), L(3, 6.2) (5.5, 2.2) 58. M�N� for M(�1.5, �6), N(�4, 3)

(�2.75, �1.5)Find the value of the variable and MP, if P is between M and N. (Lesson 1-2)

59. MP � 7x, PN � 3x, PN � 24 8; 56 60. MP � 2c, PN � 9c, PN � 63 7; 1461. MP � 4x, PN � 5x, MN � 36 4; 16 62. MP � 6q, PN � 6q, MN � 60 5; 3063. MP � 4y � 3, PN � 2y, MN � 63 64. MP � 2b � 7, PN � 8b, MN � 43

10; 43 5; 3BASIC SKILL Determine which values in the given replacement set make theinequality true.65. x � 2 5 4, 5 66. 12 � x � 0 13, 14 67. 5x � 1 25 5, 6, 7

{2, 3, 4, 5} {11, 12, 13, 14} {4, 5, 6, 7}

K L M

J P N

hexagon, convex, irregular

pentagon,convex, regular

heptagon, concave, irregular



4 Assess4 Assess


quiz or review of Lesson 2-1.


does logic apply toschool?

Ask students:• Determine whether the

following statement is true orfalse: “South Carolina bordersNorth Carolina, Georgia, andTennessee.” false

• Locate Wilmington on the mapof North Carolina. Is this acoastal or inland city? Make aconjecture about whether youmight find ocean or lakes inWilmington given its locationon the map. Coastal; sampleanswer: You would find ocean inWilmington, North Carolina.

Negation

Logic

Lesson 2-2 Logic 67

Vocabulary• statement• truth value• negation• compound statement• conjunction• disjunction• truth table

does logic apply to school?

• Determine truth values of conjunctions and disjunctions.

• Construct truth tables.

When you answer true-false questions on a test, you are using a basic principle of logic. For example, refer to the map, and answer true or false.

Raleigh is a city in North Carolina.

You know that there is only one correct answer, either true or false.

VIRGINIA

SOUTHCAROLINA

Charlotte

RaleighNORTH CAROLINA

Willmington

Ashville

Gastonia

• Words If a statement is represented by p, then not p is the negation of thestatement.

• Symbols ~p, read not p

DETERMINE TRUTH VALUES A , like the true-false exampleabove, is any sentence that is either true or false, but not both. Unlike a conjecture,we know that a statement is either true or false. The truth or falsity of a statement is called its .

Statements are often represented using a letter such as p or q. The statement abovecan be represented by p.

p: Raleigh is a city in North Carolina. This statement is true.

The of a statement has the opposite meaning as well as an oppositetruth value. For example, the negation of the statement above is not p.

not p: Raleigh is not a city in North Carolina. In this case, the statement is false.

negation

truth value

statement

Two or more statements can be joined to form a . Considerthe following two statements.

p: Raleigh is a city in North Carolina.q: Raleigh is the capital of North Carolina.

The two statements can be joined by the word and.

p and q: Raleigh is a city in North Carolina, and Raleigh is the capital of North Carolina.

compound statement

StatementsA mathematical statementwith one or morevariables is called an opensentence. The truth valueof an open sentencecannot be determineduntil values are assignedto the variables. Astatement with onlynumeric values is a closedsentence.

Study Tip

Lesson x-x Lesson Title 67

Chapter 2 Resource Masters• Study Guide and Intervention, pp. 63–64• Skills Practice, p. 65• Practice, p. 66• Reading to Learn Mathematics, p. 67• Enrichment, p. 68• Assessment, p. 119

Graphing Calculator and Computer Masters, p. 20

Teaching Geometry With ManipulativesMasters, p. 16





Transparencies

LessonNotes

1 Focus1 Focus

11


DETERMINE TRUTHVALUES

Use the following statementsto write a compoundstatement for eachconjunction. Then find itstruth value.p: One foot is 14 inches.q: September has 30 days.r: A plane is defined by three

noncollinear points.

a. p and qOne foot is 14 inches, andSeptember has 30 days; false.

b. r � pA plane is defined by threenoncollinear points, and one footis 14 inches; false.

c. �q � rSeptember does not have 30 days, and a plane is defined bythree noncollinear points; false.

d. �p � rA foot is not 14 inches, and aplane is defined by threenoncollinear points; true.

Conjunction

The statement formed by joining p and q is an example of a conjunction.

Truth Values of ConjunctionsUse the following statements to write a compound statement for eachconjunction. Then find its truth value.

p: January 1 is the first day of the year.q: �5 � 11 � �6r: A triangle has three sides.

a. p and qJanuary is the first day of the year, and �5 � 11 � �6.p and q is false, because p is true and q is false.

b. r � pA triangle has three sides, and January 1 is the first day of the year.r � p is true, because r is true and p is true.

c. p and not rJanuary 1 is the first day of the year, and a triangle does not have three sides.p and not r is false, because p is true and not r is false.

d. ~q � r�5 � 11 � �6, and a triangle has three sides�q � r is true because �q is true and r is true.

Example 1Example 1

• Words A is a compound statement formed by joining two ormore statements with the word and.

• Symbols p � q, read p and q

conjunction

NegationsThe negation of astatement is notnecessarily false. It has the opposite truthvalue of the originalstatement.

Study Tip

Disjunction• Words A is a compound statement formed by joining two or

more statements with the word or.

• Symbols p � q, read p or q

disjunction


A conjunction is true only when both statements in it are true. Since it is true thatRaleigh is in North Carolina and it is the capital, the conjunction is also true.

Statements can also be joined by the word or. This type of statement is adisjunction. Consider the following statements.

p: Ahmed studies chemistry.

q: Ahmed studies literature.

p or q: Ahmed studies chemistry, or Ahmed studies literature.


2 Teach2 Teach

22


Use the following statementsto write a compoundstatement for eachdisjunction. Then find itstruth value.p: A�B� is proper notation for

“line AB.”q: Centimeters are metric units.r: 9 is a prime number.

a. p or qA�B� is proper notation for “lineAB,” or centimeters are metricunits; true.

b. q � rCentimeters are metric units, or9 is a prime number; true.

Conjunctions can be illustrated with Venn diagrams. Refer to the statement at the beginning of the lesson. The Venn diagram at the right shows that Raleigh (R) is represented by the intersection of the set of cities in North Carolina and the set of state capitals. In other words, Raleigh must be in the set containing cities in North Carolina and in the set of state capitals.

A disjunction can also be illustrated with a Venn diagram. Consider the following statements.

p: Jerrica lives in a U.S. state capital.

q: Jerrica lives in a North Carolina city.

p � q: Jerrica lives in a U.S. state capital, or Jerrica lives in a North Carolina city.

In the Venn diagrams, the disjunction is represented by the union of the two sets.The union includes all U.S. capitals and all cities in North Carolina. The city inwhich Jerrica lives could be located in any of the three regions of the union.

The three regions represent

A U.S. state capitals excluding the capital of North Carolina,

B cities in North Carolina excluding the state capital, and

C the capital of North Carolina, which is Raleigh.

Lesson 2-2 Logic 69

A disjunction is true if at least one of the statements is true. In the case of p or qabove, the disjunction is true if Ahmed either studies chemistry or literature or both.The disjunction is false only if Ahmed studies neither chemistry nor literature.

Truth Values of DisjunctionsUse the following statements to write a compound statement for eachdisjunction. Then find its truth value.

p: 100 � 5 � 20

q: The length of a radius of a circle is twice the length of its diameter.

r: The sum of the measures of the legs of a right triangle equals the measure of the hypotenuse.

a. p or q100 5 � 20, or the length of a radius of a circle is twice the length of itsdiameter.p or q is true because p is true. It does not matter that q is false.

b. q � rThe length of a radius of a circle is twice the length of its diameter, or the sumof the measures of the legs of a right triangle equals the measure of thehypotenuse.q � r is false since neither statement is true.

Example 2Example 2

Venn DiagramsThe size of theoverlapping region in aVenn Diagram does notindicate how many itemsfall into that category.

Study TipAll U.S. Cities

U.S.State

Capitals

Citiesin

NorthCarolina

R

All U.S. Cities

A B

C

U.S.State

Capitals

Citiesin

NorthCarolina


Lesson 2-2 Logic 69


33


DANCING The Venn diagramshows the number ofstudents enrolled inMonique’s Dance School fortap, jazz, and ballet classes.

a. How many students areenrolled in all three classes? 9

b. How many students areenrolled in tap or ballet? 121

c. How many students areenrolled in jazz and balletand not tap? 25

TRUTH TABLES

Tap28

Jazz43

29Ballet

13

17 259

Venn diagrams can be used to solve real-world problems involving conjunctionsand disjunctions.


Use Venn DiagramsRECYCLING The Venn diagram shows the number of neighborhoods that havea curbside recycling program for paper or aluminum.

a. How many neighborhoods recycle both paper and aluminum?The neighborhoods that have paper and aluminum recycling are represented by the intersection of the sets. There are 46 neighborhoods that have paper andaluminum recycling.

b. How many neighborhoods recycle paper or aluminum?The neighborhoods that have paper or aluminum recycling are represented by the union of the sets. There are 12 � 46 � 20 or 78 neighborhoods that havepaper or aluminum recycling.

c. How many neighborhoods recycle paper and not aluminum?The neighborhoods that have paper and not aluminum recycling arerepresented by the nonintersecting portion of the paper region. There are 12 neighborhoods that have paper and not aluminum recycling.

Curbside Recycling

12 2046

Paper Aluminum

Example 3Example 3

TRUTH TABLES A convenient method for organizing the truth values ofstatements is to use a .

If p is a true statement, then �p is a false statement.If p is a false statement, then �p is a true statement.

Truth tables can also be used to determine truth values of compound statements.

You can use the truth values for negation, conjunction, and disjunction toconstruct truth tables for more complex compound statements.

truth table

Negation

p �p

T F

F T

Conjunction

p q p � q

T T T

T F F

F T F

F F F

Disjunction

p q p � q

T T T

T F T

F T T

F F F

A conjunction istrue only whenboth statementsare true.

A disjunction isfalse only whenboth statementsare false.

TautologyA compound sentence is atautology if its truth valueis always true. Forexample, “It is snowing orit is not snowing” is atautology.

Study Tip


InterventionTell studentsthat truth tablesmust displayall combinations

of Ts and Fs to exhaust allpossible outcomes, so for eachstatement, p, q, and r, they willneed to mix the occurrences oftrue and false. First, they shoulddetermine the number of rowsthey need. Then fill the top halfof the p column with Ts and theother half with Fs. For the qcolumn, they can alternate Tand F the whole way down. Ifthey need an r column, they canalternate sets of 2 Ts and 2 Fsall the way down, and so on.Assure students that as longas they initially set up thebasic structure correctly, theyshould be able to produce therest of the table quite easily.

New

Answers

1. The conjunction (p and q) is represented by the intersection of the two circles.2a. Sample answer: October has 31 days or �5 � 3 � �8.2b. Sample answer: A square has five right angles and the Postal Service does not deliver

mail on Sundays.2c. Sample answer: July 5th is not a national holiday.3. A conjunction is a compound statement using the word and, while a disjunction is a

compound statement using the word or.

44


Teaching Tip Tell studentsthat they could potentiallyinterchange the columns for p,q, and r in 4c, and as long asthey correctly fill in the columnsfor p � q and (p � q) � r, theoverall outcome would be thesame. They would still end upwith 5 Ts and 3 Fs, but the Tsand Fs would be in a differentorder.

Construct a truth table foreach compound statement.

a. �p � q

b. p � (�q � r)

c. (p � q) � �r

p q r �r p � q (p � q) � �rT T T F T FT F T F T FT T F T T TT F F T T TF T T F T FF F T F F FF T F T T TF F F T F F

p q r �q �q � r p � (�q � r)T T T F F TT F T T T TT T F F F TT F F T F TF T T F F FF F T T T TF T F F F FF F F T F F

p q �p �p � qT T F TT F F FF T T TF F T T

Construct Truth TablesConstruct a truth table for each compound statement.a. p � �q

Step 1 Make columns with the headings p, q, �q, and p � � q.Step 2 List the possible combinations of truth values for p and q.Step 3 Use the truth values of q to determine the truth values of �q.Step 4 Use the truth values for p and �q to write the truth values for p � �q.

Step 1

Step 2 Step 3 Step 4b. �p � �q

c. (p � q) � rMake columns for p, q, p � q, r, and (p � q) � r.

Lesson 2-2 Logic 71

Truth TablesUse the FundamentalCounting Principle todetermine the number of rows necessary.

Study Tip

Concept Check1–3. See margin.

Example 4Example 4

p q �q p � �q

T T F F

T F T T

F T F F

F F T F

p q p � q r (p � q) � r

T T T T T

T F F T T

T T T F T

T F F F F

F T F T T

F F F T T

F T F F F

F F F F F

1. Describe how to interpret the Venn diagram for p � q.

2. OPEN ENDED Write a compound statement for each condition.a. a true disjunctionb. a false conjunctionc. a true statement that includes a negation

3. Explain the difference between a conjunction and a disjunction.

p � q

p q

p q �p �q �p � �q

T T F F F

T F F T T

F T T F T

F F T T T

Lesson 2-2 Logic 71

Logical/Mathematical Have students examine the relationship betweenthe number of simple statements (p, q, and r) and the number of rowsnecessary to exhaust all possible combinations in a truth table. Point outthat for Example 4a, there are 2 statements and 4 rows; for Example 4c,there are 3 statements and 8 rows. Ask students to form a conjectureabout how many rows would be needed for 4, 5, and n statements.Similarly, students can examine the relationship between the number ofcircles and the number of intersecting areas of a Venn diagram.





• include an example of a Venndiagram and an example of atruth table.



Guided Practice Use the following statements to write a compound statement for each conjunctionand disjunction. Then find its truth value. 4–9. See margin for statements.p: 9 � 5 � 14q: February has 30 days.r: A square has four sides.

4. p and q false 5. p and r true 6. q � r false7. p or ~q true 8. q � r true 9. ~p � ~r false

10. Copy and complete the truth table.

Construct a truth table for each compound statement. 11–14. See p. 123A.11. p � q 12. q � r 13. ~p � r 14. (p � q) � r

AGRICULTURE For Exercises 15–17, refer to the Venn diagram that represents the states producing more than 100 million bushels of corn or wheat per year.15. How many states produce more than 100 million

bushels of corn? 1416. How many states produce more than 100 million

bushels of wheat? 717. How many states produce more than 100 million

bushels of corn and wheat? 3


4–6 17–9 2

10–14 315–17 4

Application


p q �q p � �q

T T F F

T F T TF T F FF F T F

Corn Wheat

IA PA IL ND

WA

MTID

NE INMO CO

WI OHMI KY

MNSDKS

Grain Production

Source: U.S. Department of Agriculture

Use the following statements to write a compound statement for each conjunctionand disjunction. Then find its truth value. 18–29. See p. 123A for statements.

p: ��64� � 8q: An equilateral triangle has three congruent sides. r: 0 0s: An obtuse angle measures greater than 90° and less than 180°.

18. p and q false 19. p or q true 20. p and r false21. r and s false 22. q or r true 23. q and s true24. p � s false 25. q � r false 26. r � p false27. s � q true 28. (p � q) � s true 29. s � (q and r) true

Copy and complete each truth table.30. 31.

ForExercises

18–2930–4142–48

SeeExamples

1, 243



p q �p �p � q

T T F TT F F FF T T TF F T T

p q �p �q �p � �q

T T F F FT F F T FF T T F FF F T T T


About the Exercises…Organization by Objective• Determine Truth Values:

18–29, 42–48• Truth Tables: 30–41

Odd/Even AssignmentsExercises 18–40 are structuredso that students practice thesame concepts whether theyare assigned odd or evenproblems.Alert! Exercises 48–50 requirethe Internet or other researchmaterials.

Assignment GuideBasic: 19–37 odd, 41–51 odd,52–73Average: 19–51 odd, 52–73Advanced: 18–50 even, 51–52,54–69 (optional: 70–73)

Answers

4. 9 � 5 � 14 and February has 30 days.

5. 9 � 5 � 14 and a square has foursides.

6. February has 30 days and asquare has four sides.

7. 9 � 5 � 14 or February does nothave 30 days.

8. February has 30 days ora square has four sides.

9. 9 � 5 14 or a squaredoes not have foursides.

45.

AcademicClubs

60

Sports95 20

Level of ParticipationAmong 310 Students

Study Guide and InterventionLogic

NAME ______________________________________________ DATE ____________ PERIOD _____

2-22-2


Less

on

2-2

Determine Truth Values A statement is any sentence that is either true or false. Thetruth or falsity of a statement is its truth value. A statement can be represented by using aletter. For example,

Statement p: Chicago is a city in Illinois. The truth value of statement p is true.

Several statements can be joined in a compound statement.

Statement p and statement q joined Statement p and statement q joined Negation: not p is the negation ofby the word and is a conjunction. by the word or is a disjunction. the statement p.

Symbols: p � q (Read: p and q ) Symbols: p � q (Read: p or q) Symbols: �p (Read: not p)

The conjunction p � q is true only The disjunction p � q is true if p is The statements p and �p have when both p and q are true. true, if q is true, or if both are true. opposite truth values.

Write a compoundstatement for each conjunction. Thenfind its truth value.p: An elephant is a mammal.q: A square has four right angles.

a. p � qJoin the statements with and: An elephantis a mammal and a square has four rightangles. Both parts of the statement aretrue so the compound statement is true.

b. �p � q�p is the statement “An elephant is not amammal.” Join �p and q with the wordand: An elephant is not a mammal and asquare has four right angles. The firstpart of the compound statement, �p, isfalse. Therefore the compound statementis false.

Write a compoundstatement for each disjunction. Thenfind its truth value.p: A diameter of a circle is twice the radius.q: A rectangle has four equal sides.

a. p � qJoin the statements p and q with theword or: A diameter of a circle is twicethe radius or a rectangle has four equalsides. The first part of the compoundstatement, p, is true, so the compoundstatement is true.

b. �p � qJoin �p and q with the word or: Adiameter of a circle is not twice theradius or a rectangle has four equalsides. Neither part of the disjunction istrue, so the compound statement is false.


ExercisesExercises

Write a compound statement for each conjunction and disjunction.Then find its truth value.p: 10 � 8 � 18 q: September has 30 days. r: A rectangle has four sides.

1. p and q 10 � 8 � 18 and September has 30 days; true.

2. p or r 10 � 8 � 18 or a rectangle has four sides; true.

3. q or r September has 30 days or a rectangle has four sides; true.

4. q and �r September has 30 days and a rectangle does not have foursides; false.



Use the following statements to write a compound statement for each conjunctionand disjunction. Then find its truth value.p: 60 seconds � 1 minuteq: Congruent supplementary angles each have a measure of 90.r : �12 � 11 �1

1. p � q 60 seconds � 1 minute and congruent supplementary angles eachhave a measure of 90; true.

2. q � r Congruent supplementary angles each have a measure of 90 or�12 � 11 �1; true.

3. �p � q 60 seconds 1 minute or congruent supplementary angles eachhave a measure of 90; true.

4. �p � �r 60 seconds 1 minute and �12 � 11 � �1; false.

Copy and complete each truth table.

5. 6.

Construct a truth table for each compound statement.

7. q � (p � �q) 8. �q � (�p � q)

SCHOOL For Exercises 9 and 10, use the following information.The Venn diagram shows the number of students in the band who work after school or on the weekends.

9. How many students work after school and on weekends? 3

10. How many students work after school or on weekends? 25

WorkWeekends

17

WorkAfter

School5

3

p q �p �q �p � q �q � (�p � q)

T T F F T F

T F F T F F

F T T F T F

F F T T T T

p q �q p � �q q � (p � �q)

T T F F T

T F T T T

F T F F T

F F T F F

p q �p �p � q p � (�p � q)

T T F T TT F F F FF T T T FF F T T F

p q �p �q �p � �q

T T F F FT F F T TF T T F TF F T T T

Practice (Average)

Logic

NAME ______________________________________________ DATE ____________ PERIOD _____


Reading to Learn MathematicsLogic

NAME ______________________________________________ DATE ____________ PERIOD _____

2-22-2

Less

on

2-2

Pre-Activity How does logic apply to school?


How can you use logic to help you answer a multiple-choice question on astandardized test if you are not sure of the correct answer? Sample answer:Eliminate the choices that you know are wrong.Then choose theone you think is most likely correct from the ones that are left.

Reading the Lesson1. Supply one or two words to complete each sentence.

a. Two or more statements can be joined to form a statement.b. A statement that is formed by joining two statements with the word or is called a

.c. The truth or falsity of a statement is called its .d. A statement that is formed by joining two statements with the word and is called a

.e. A statement that has the opposite truth value and the opposite meaning from a given

statement is called the of the statement.

2. Use true or false to complete each sentence.a. If a statement is true, then its negation is .b. If a statement is false, then its negation is .c. If two statements are both true, then their conjunction is and

their disjunction is .d. If two statements are both false, then their conjunction is and

their disjunction is .e. If one statement is true and another is false, then their conjunction is

and their disjunction is .

3. Consider the following statements:p: Chicago is the capital of Illinois. q: Sacramento is the capital of California.Write each statement symbolically and then find its truth value.a. Sacramento is not the capital of California. �q; falseb. Sacramento is the capital of California and Chicago is not the capital of Illinois.

q � �p; true

Helping You Remember4. Prefixes can often help you to remember the meaning of words or to distinguish between

similar words. Use your dictionary to find the meanings of the prefixes con and dis andexplain how these meanings can help you remember the difference between aconjunction and a disjunction. Sample answer: Con means together and dismeans apart, so a conjunction is an and (or both together) statement anda disjunction is an or statement.

truefalse

falsefalse

truetrue

truefalse

negation

conjunction

truth valuedisjunction

compound


Letter PuzzlesAn alphametic is a computation puzzle using letters instead ofdigits. Each letter represents one of the digits 0–9, and twodifferent letters cannot represent the same digit. Some alphameticpuzzles have more than one answer.

Solve the alphametic puzzle at the right.

Since R � E � E, the value of R must be 0. Notice that thethousands digit must be the same in the first addend and thesum. Since the value of I is 9 or less, O must be 4 or less. Use trial and error to find values that work.

F � 8, O � 3, U � 1, R � 0

N � 4, E � 7, I � 6, and V � 5.

Can you find other solutions to this puzzle?

Find a value for each letter in each alphametic. Sample answers are shown

1. 2. 734TWO9703HALF

8310� 347��

8657

FOUR� ONE��

F I VE

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

2-22-2

ExampleExample

Enrichment, p. 68

SchoolNationwide, approximately80% of high school seniors participate inextracurricular activities.Athletics, performing arts,and clubs are the mostpopular.Source: National Center forEducation Statistics

Lesson 2-2 Logic 73

32. Copy and complete the truth table.

Construct a truth table for each compound statement. 33– 40. See pp. 123A–123B.33. q and r 34. p or q 35. p or r 36. p and q37. q � �r 38. �p � �q ★ 39. �p � (q � �r) ★ 40. p � (�q � �r)

MUSIC For Exercises 41–44, use the following information.A group of 400 teens were asked what type of music they listened to. They could choose among pop, rap, and country. The results are shown in the Venn diagram.41. How many teens said that they listened to none of

these types of music? 4242. How many said that they listened to all three types

of music? 743. How many said that they listened to only pop and

rap music? 2544. How many teens said that they listened to pop, rap, or country music? 358

SCHOOL For Exercises 45–47, use the following information.In a school of 310 students, 80 participate in academic clubs, 115 participate insports, and 20 students participate in both.45. Make a Venn diagram of the data. See margin.46. How many students participate in either clubs or sports? 17547. How many students do not participate in either clubs or sports? 135

RESEARCH For Exercises 48–50, use the Internet or another resource to determinewhether each statement about cities in New York is true or false.48. Albany is not located on the Hudson river. false49. Either Rochester or Syracuse is located on Lake Ontario. true50. It is false that Buffalo is located on Lake Erie. false

CRITICAL THINKING For Exercises 51 and 52, use the following information.All members of Team A also belong to Team B, but only some members of Team B also belong to Team C. Teams A and C have no members in common.

51. Draw a Venn diagram to illustrate the situation. See margin.52. Which of the following statements is true? b

a. If a person is a member of Team C, then the person is not a member of Team A.

b. If a person is not a member of Team B, then the person is not a member of Team A.

c. No person that is a member of Team A can be a member of Team C.

Bill Bachmann/PhotoEdit

★

Pop175

Country45

Rap62

25 10

42

7

34

Music Preference

p q r p � q (p � q) � r

T T T T TT T F T FT F T T TT F F T FF T T T TF T F T FF F T F FF F F F F


Lesson 2-2 Logic 73

ELL

51.

C

B

A


Open-Ended AssessmentModeling Have students modela Venn diagram and a truth tablewith buttons or chips. For theVenn diagram, students candraw two large overlappingcircles on a piece of paper andlabel them Science and English.Then they can place buttons onthe diagram to represent thenumber of students in the classwho like one, the other, or bothsubjects. Students can also drawa grid and use white buttons fortrue and black buttons for false tomodel one of the truth tables inthe lesson.

Getting Ready forLesson 2-3Prerequisite Skill Students willlearn about conditional statementsin Lesson 2-3. They will substitutethe hypothesis and conclusion forthe if and then parts of statements.Use Exercises 70–73 to determineyour students’ familiarity withsubstituting numbers for variablesin algebraic expressions.

Assessment OptionsQuiz (Lessons 2-1 and 2-2) isavailable on p. 119 of the Chapter 2Resource Masters.

Answer

53. Sample answer: Logic can beused to eliminate false choices ona multiple choice test. Answersshould include the following.• Math is my favorite subject and

drama club is my favoriteactivity.

• See students’ work.

53. Answer the question that was posed at the beginning of the lesson. See margin.

How does logic apply to school?

Include the following in your answer:• an example of a conjunction using statements about your favorite subject

and your favorite extracurricular activity, and• a Venn diagram showing various characteristics of the members of your

geometry class (for example, male/female, grade in school, and so on).

54. Which statement about �ABC has the same truth value as AB � BC? A

m�A � m�C m�A � m�BAC � BC AB � AC

55. ALGEBRA If the sum of two consecutive even integers is 78, which number is the greater of the two integers? C

36 3840 42DC

BA

DC

BA

WRITING IN MATH


B

A C


Mixed Review


Make a conjecture about the next item in each sequence. (Lesson 2-1)

56. 3, 5, 7, 9 11 57. 1, 3, 9, 27 81 58. 6, 3, �32

�, �34

� �38

�

59. 17, 13, 9, 5 1 60. 64, 16, 4, 1 �14

� 61. 5, 15, 45, 135 405

COORDINATE GEOMETRY Find the perimeter of each polygon. Round answersto the nearest tenth. (Lesson 1-6)

62. triangle ABC with vertices A(�6, 7), B(1, 3), and C(�2, �7) 33.163. square DEFG with vertices D(�10, �9), E(�5, �2), F(2, �7), and G(�3, �14)64. quadrilateral HIJK with vertices H(5, �10), I(�8, �9), J(�5, �5), and K(�2, �4)65. hexagon LMNPQR with vertices L(2, 1), M(4, 5), N(6, 4), P(7, �4), Q(5, �8),

and R(3, �7) 29.5

Measure each angle and classify it as right,acute, or obtuse. (Lesson 1-4)

66. �ABC 145°, obtuse67. �DBC 55°, acute68. �ABD 90°, right

69. FENCING Michelle wanted to put a fence around her rectangular garden. Thefront and back measured 35 feet each, and the sides measured 75 feet each. Ifshe wanted to make sure that she had enough feet of fencing, how much shouldshe buy? (Lesson 1-2) 222 ft

PREREQUISITE SKILL Evaluate each expression for the given values.(To review evaluating algebraic expressions, see page 736.)

70. 5a � 2b if a � 4 and b � 3 14 71. 4cd � 2d if c � 5 and d � 2 4472. 4e � 3f if e � �1 and f � �2 �10 73. 3g2 � h if g � 8 and h � �8 184

A B

C

D

63. 34.464. 30.4



4 Assess4 Assess




are conditionalstatements used in

advertisements?Ask students:• Use the advertisements to

answer the following questions:What happens if you buy a newcar? How can you get a freephone? You get $1500 cash back;enroll in phone service for one year.

• How effective are these typesof advertisements? Why? Veryeffective; people like to getsomething for free, even if theyhave to pay for something else.

Conditional Statements

Lesson 2-3 Conditional Statements 75

Vocabulary• conditional statement• if-then statement• hypothesis• conclusion• related conditionals• converse• inverse• contrapositive• logically equivalent

• Analyze statements in if-then form.

• Write the converse, inverse, and contrapositive of if-then statements.

Advertisers often lure consumers into purchasing expensive items by convincingthem that they are getting something for free in addition to their purchase.

If-Then Statement• Words An is written in the form if p, then q. The phrase

immediately following the word if is called the , and thephrase immediately following the word then is called the .

• Symbols p → q, read if p then q, or p implies q.

conclusionhypothesis

if-then statement

Reading MathThe word if is not part ofthe hypothesis. The wordthen is not part of theconclusion.

Study Tip

Identify Hypothesis and ConclusionIdentify the hypothesis and conclusion of each statement.a. If points A, B, and C lie on line �, then they are collinear.

If points A, B, and C lie on line �, then they are collinear.

hypothesis conclusion

Hypothesis: points A, B, and C lie on line �Conclusion: they are collinear

b. The Tigers will play in the tournament if they win their next game.Hypothesis: the Tigers win their next gameConclusion: they will play in the tournament

Example 1Example 1

Identifying the hypothesis and conclusion of a statement is helpful whenwriting statements in if-then form.

are conditional statementsused in advertisements?are conditional statementsused in advertisements?

-

IF-THEN STATEMENTS The statements above are examples of conditionalstatements. A is a statement that can be written in if-thenform. The first example above can be rewritten to illustrate this.

If you buy a car, then you get $1500 cash back.

conditional statement




5-Minute Check Transparency 2-3Real-World Transparency 2Answer Key Transparencies

TechnologyGeomPASS: Tutorial Plus, Lesson 6Interactive Chalkboard



Transparencies

LessonNotes

1 Focus1 Focus

11

22

33


IF-THEN STATEMENTS

Identify the hypothesis andconclusion of each statement.

a. If a polygon has 6 sides, thenit is a hexagon. Hypothesis: apolygon has 6 sides; Conclusion:it is a hexagon

b. Tamika will advance to thenext level of play if shecompletes the maze in hercomputer game. Hypothesis:Tamika completes the maze inher computer game; Conclusion:she will advance to the next level of play.

Identify the hypothesis andconclusion of each statement.Then write each statement inthe if-then form.

a. Distance is positive.Hypothesis: a distance isdetermined; Conclusion: it ispositive; If a distance isdetermined, then it is positive.

b. A five-sided polygon is apentagon. Hypothesis: a polygonhas five sides; Conclusion: it is apentagon; If a polygon has fivesides, then it is a pentagon.

Teaching Tip Tell students touse parentheses to identify thehypothesis and conclusion ineach situation. Explain that ifthe hypothesis in the situationmatches the hypothesis in theoriginal statement, students canmark a T over the parentheses; ifnot, they can mark an F. They cando the same for the conclusions.

Determine the truth value ofthe following statement foreach set of conditions. If Yukonrests for 10 days, his ankle willheal.

a. Yukon rests for 10 days, andhe still has a hurt ankle. false

b. Yukon rests for 3 days, andhe still has a hurt ankle. true

c. Yukon rests for 10 days, andhe does not have a hurt ankleanymore. true

d. Yukon rests for 7 days, andhe does not have a hurt ankleanymore. true

Recall that the truth value of a statement is either true or false. The hypothesisand conclusion of a conditional statement, as well as the conditional statement itself,can also be true or false.


Write a Conditional in If-Then FormIdentify the hypothesis and conclusion of each statement. Then write eachstatement in if-then form.a. An angle with a measure greater than 90 is an obtuse angle.

Hypothesis: an angle has a measure greater than 90Conclusion: it is an obtuse angleIf an angle has a measure greater than 90, then it is an obtuse angle.

b. Perpendicular lines intersect.Sometimes you must add information to a statement. In this case, it is necessaryto know that perpendicular lines come in pairs.Hypothesis: two lines are perpendicularConclusion: they intersectIf two lines are perpendicular, then they intersect.

Example 2Example 2

If-Then StatementsWhen you write a statementin if-then form, identify the condition that causes theresult as the hypothesis. The result is the conclusion.

Study Tip

CommonMisconceptionA true hypothesis does notnecessarily mean that aconditional is true. Likewise,a false conclusion does notguarantee that a conditionalis false.

Study Tip

Truth Values of ConditionalsSCHOOL Determine the truth value of the following statement for each set ofconditions.If you get 100% on your test, then your teacher will give you an A.a. You get 100%; your teacher gives you an A.

The hypothesis is true since you got 100%, and the conclusion is true because the teacher gave you an A. Since what the teacher promised is true, the conditional statement is true.

b. You get 100%; your teacher gives you a B.The hypothesis is true, but the conclusion is false. Because the result is not whatwas promised, the conditional statement is false.

c. You get 98%; your teacher gives you an A.The hypothesis is false, and the conclusion is true. The statement does not saywhat happens if you do not get 100% on the test. You could still get an A. It isalso possible that you get a B. In this case, we cannot say that the statement isfalse. Thus, the statement is true.

d. You get 85%; your teacher gives you a B.As in part c, we cannot say that the statement is false. Therefore, the conditionalstatement is true.

Example 3Example 3

The resulting truth values in Example 3 can be usedto create a truth table for conditional statements. Noticethat a conditional statement is true in all cases exceptwhere the hypothesis is true and the conclusion is false.

p q p → q

T T T

T F F

F T T

F F T


Nancy Lee Keen Martinsville High School, Martinsville, IN

To develop the concept of conditional statements, I made posters of each of thefour related conditionals. I wrote the hypotheses on yellow poster board, theconclusions on blue poster board, and NOT on red poster board. As weintroduced each type of conditional, we placed the posters in the correct order.

Teacher to TeacherTeacher to Teacher

44


CONVERSE, INVERSE, ANDCONTRAPOSITIVE

Write the converse, inverse,and contrapositive of thestatement All squares arerectangles. Determine whethereach statement is true or false.If a statement is false, give acounterexample.Conditional: If a shape is asquare, then it is a rectangle.Converse: If a shape is arectangle, then it is a square.False; a rectangle with � � 2 andw � 4 is not a square. Inverse: Ifa shape is not a square, then it isnot a rectangle. False; a 4-sidedpolygon with side lengths 2, 2, 4,and 4 is not a square.Contrapositive: If a shape is not a rectangle, then it is not asquare. true

Concept CheckIn Lesson 2-2, p and qrepresented simple statements,not necessarily related to oneanother. In this lesson, theybecome the hypothesis andconclusion of a conditionalstatement. Make sure studentsknow that separately, p and q arestill simple statements, but theynow have an interdependentrelationship. Before moving on,students should feel verycomfortable identifying thehypothesis and conclusion,determining the truth value ofeach one separately, anddetermining their combinedtruth value in various forms ofconditional statements.


Related ConditionalsWrite the converse, inverse, and contrapositive of the statement Linear pairs ofangles are supplementary. Determine whether each statement is true or false. If astatement is false, give a counterexample.First, write the conditional in if-then form.

Conditional: If two angles form a linear pair, then they are supplementary.The conditional statement is true.

Write the converse by switching the hypothesis and conclusion of the conditional.

Converse: If two angles are supplementary, then they form a linear pair. The converse is false. �ABC and �PQR are supplementary, but are not a linear pair.

Inverse: If two angles do not form a linear pair, then they are not supplementary. Theinverse is false. �ABC and �PQR do not form a linear pair, but they are supplementary.

The contrapositive is the negation of the hypothesis and conclusion of the converse.

Contrapositive: If two angles are not supplementary, then they do not form alinear pair. The contrapositive is true.

Example 4Example 4

Related ConditionalsStatement Formed by Symbols Examples

given hypothesis and conclusion p → q If two angles have the same measure,then they are congruent.

exchanging the hypothesis and q → p If two angles are congruent,conclusion of the conditional then they have the same measure.

negating both the hypothesis and �p → �q If two angles do not have the sameconclusion of the conditional measure, then they are not congruent.

negating both the hypothesis and �q → �p If two angles are not congruent, thenconclusion of the converse statement they do not have the same measure.

Contrapositive

Inverse

Converse

Conditional

CONVERSE, INVERSE, AND CONTRAPOSITIVE Other statements basedon a given conditional statement are known as .related conditionals

If a given conditional is true, the converse and inverse are not necessarily true. However, the contrapositive of a true conditional is always true, and thecontrapositive of a false conditional is always false. Likewise, the converse andinverse of a conditional are either both true or both false.

Statements with the same truth values are said to be . So, aconditional and its contrapositive are logically equivalent as are the converse andinverse of a conditional. These relationships are summarized below.

logically equivalent

Conditional Converse Inverse Contrapositivep qp → q q → p �p → �q �q → �p

T T T T T T

T F F T T F

F T T F F T

F F T T T T

ContrapositiveThe relationship of thetruth values of aconditional and itscontrapositive is known asthe Law of Contrapositive.

Study Tip

A B

CP

Q R70°

110°



Kinesthetic Provide index cards for each student labeled “Hypothesis,”“Conclusion,” and “Implies” (or an arrow pointing to the right). Give eachstudent two cards labeled “Not” in red ink. Ask students to use the cardsto form a conditional, a converse, an inverse, and a contrapositive.Students should respond by placing the cards in the correct position andorder to reflect the requests. Students can also use the cards to worksome examples or exercises in this lesson by writing the parts ofconditional statements on corresponding cards.






• include a simplified version of theRelated Conditionals chart andthe truth table on page 77.



Concept Check

Guided Practice

Application

1–3. See margin.


4–6 17–9, 15 210–12 313, 14 4

Practice and ApplyPractice and Apply

Identify the hypothesis and conclusion of each statement. 16 –21. See p. 123B.16. If 2x � 6 � 10, then x � 2.17. If you are a teenager, then you are at least 13 years old.18. If you have a driver’s license, then you are at least 16 years old.19. If three points lie on a line, then they are collinear.20. “If a man hasn’t discovered something that he will die for, he isn’t fit to live.”

(Martin Luther King, Jr., 1963)21. If the measure of an angle is between 0 and 90, then the angle is acute.

Write each statement in if-then form. 22–27. See p. 123B.22. Get a free visit with a one-year fitness plan.23. Math teachers love to solve problems.24. “I think, therefore I am.” (Descartes)25. Adjacent angles have a common side.26. Vertical angles are congruent.27. Equiangular triangles are equilateral.

ForExercises

16–2122–2728–3940–45

SeeExamples

1234



1. Explain why writing a conditional statement in if-then form is helpful.2. OPEN ENDED Write an example of a conditional statement.3. Compare and contrast the inverse and contrapositive of a conditional.

Identify the hypothesis and conclusion of each statement. 4– 6. See margin.4. If it rains on Monday, then I will stay home.5. If x � 3 � 7, then x � 10.6. If a polygon has six sides, then it is a hexagon.

Write each statement in if-then form.7. A 32-ounce pitcher holds a quart of liquid.8. The sum of the measures of supplementary angles is 180.9. An angle formed by perpendicular lines is a right angle.

Determine the truth value of the following statement for each set of conditions.If you drive faster than 65 miles per hour on the interstate, then you will receive a speeding ticket.10. You drive 70 miles per hour, and you receive a speeding ticket. true11. You drive 62 miles per hour, and you do not receive a speeding ticket. true12. You drive 68 miles per hour, and you do not receive a speeding ticket. false

Write the converse, inverse, and contrapositive of each conditional statement.Determine whether each related conditional is true or false. If a statement is false,find a counterexample. 13 –14. See margin.13. If plants have water, then they will grow.14. Flying in an airplane is safer than riding in a car.

15. FORESTRY In different regions of the country, different variations of treesdominate the landscape. In Colorado, aspen trees cover high areas of themountains. In Florida, cypress trees rise from swamps. In Vermont, maple treesare prevalent. Write these conditionals in if-then form. See p. 123B.

7. If a pitcher is a 32-ounce pitcher, then it holds a quart of liquid.8. If two angles aresupplementary, thenthe sum of the measures of theangles is 180.9. If an angle is formedby perpendicular lines,then it is a right angle.


About the Exercises…Organization by Objective• If-Then Statements: 16–39• Converse, Inverse, and

Contrapositive: 40–45


Assignment GuideBasic: 17–47 odd, 48–68Average: 17–47 odd, 48–68Advanced: 16–48 even, 50–65(optional: 66–68)All: Quiz 1 (1–5)

Answers

1. Writing a conditional in if-thenform is helpful so that the hypoth-esis and conclusion are easilyrecognizable.

2. Sample answer: If you eat yourpeas, then you will have dessert.

3. In the inverse, you negate boththe hypothesis and the conclusionof the conditional. In the contra-positive, you negate the hypothesisand the conclusion of the converse.

4. H: it rains on Monday; C: I willstay home

5. H: x � 3 � 7; C: x � 10

6. H: a polygon has six sides; C: it isa hexagon

13. Converse: If plants grow, then theyhave water; true. Inverse: If plantsdo not have water, then they willnot grow; true. Contrapositive: Ifplants do not grow, then they donot have water. False; they mayhave been killed by overwatering.

14. Converse: If you are safer than riding in a car,then you are flying in an airplane. False; thereare other places that are safer than riding in acar. Inverse: If you are not flying in an airplane,then you are not safer than riding in a car. False;there are other places that are safer than ridingin a car. Contrapositive: If you are not safer thanriding in a car, then you are not flying in anairplane; true.

Study Guide and InterventionConditional Statements

NAME ______________________________________________ DATE ____________ PERIOD _____

2-32-3


Less

on

2-3

If-then Statements An if-then statement is a statement such as “If you are readingthis page, then you are studying math.” A statement that can be written in if-then form iscalled a conditional statement. The phrase immediately following the word if is thehypothesis. The phrase immediately following the word then is the conclusion.

A conditional statement can be represented in symbols as p → q, which is read “p implies q”or “if p, then q.”

Identify the hypothesis and conclusion of the statement.

If �X � �R and �R � �S, then �X � �S.hypothesis conclusion

Identify the hypothesis and conclusion.Write the statement in if-then form.

You receive a free pizza with 12 coupons.

If you have 12 coupons, then you receive a free pizza.hypothesis conclusion

Example 1Example 1

Example 2Example 2

ExercisesExercises

Identify the hypothesis and conclusion of each statement.

1. If it is Saturday, then there is no school. H: it is Saturday; C: there is no school

2. If x � 8 � 32, then x � 40. H: x � 8 � 32; C: x � 40

3. If a polygon has four right angles, then the polygon is a rectangle.H: a polygon has four right angles; C: the polygon is a rectangle

Write each statement in if-then form.

4. All apes love bananas.If an animal is an ape, then it loves bananas.

5. The sum of the measures of complementary angles is 90. If two anglesare complementary, then the sum of their measures is 90.

6. Collinear points lie on the same line.If points are collinear, then they lie on the same line.

Determine the truth value of the following statement for each set of conditions.If it does not rain this Saturday, we will have a picnic.

7. It rains this Saturday, and we have a picnic. true

8. It rains this Saturday, and we don’t have a picnic. true

9. It doesn’t rain this Saturday, and we have a picnic. true

10. It doesn’t rain this Saturday, and we don’t have a picnic. false



Identify the hypothesis and conclusion of each statement.

1. If 3x � 4 � �5, then x � �3.H: 3x � 4 � �5; C: x � �3

2. If you take a class in television broadcasting, then you will film a sporting event.H: you take a class in television broadcasting;C: you will film a sporting event

Write each statement in if-then form.

3. “Those who do not remember the past are condemned to repeat it.” (George Santayana)If you do not remember the past, then you are condemned to repeat it.

4. Adjacent angles share a common vertex and a common side.If two angles are adjacent, then they share a common vertex and acommon side.

Determine the truth value of the following statement for each set of conditions.If DVD players are on sale for less than $100, then you buy one.

5. DVD players are on sale for $95 and you buy one. true

6. DVD players are on sale for $100 and you do not buy one. true

7. DVD players are not on sale for under $100 and you do not buy one. true

8. Write the converse, inverse, and contrapositive of the conditional statement. Determinewhether each statement is true or false. If a statement is false, find a counterexample.If (�8)2 � 0, then �8 � 0.Converse: If �8 � 0, then (�8)2 � 0; true.Inverse: If (�8)2 � 0, then �8 � 0; true.Contrapositive: If �8 � 0, then (�8)2 � 0; false.

SUMMER CAMP For Exercises 9 and 10, use the following information.Older campers who attend Woodland Falls Camp are expected to work. Campers who arejuniors wait on tables.

9. Write a conditional statement in if-then form.Sample answer: If you are a junior, then you wait on tables.

10. Write the converse of your conditional statement.If you wait on tables, then you are a junior.

Practice (Average)

Conditional Statements

NAME ______________________________________________ DATE ____________ PERIOD _____


Reading to Learn MathematicsConditional Statements

NAME ______________________________________________ DATE ____________ PERIOD _____

2-32-3

Less

on

2-3

Pre-Activity How are conditional statements used in advertisements?


Does the second advertising statement in the introduction mean that youwill not get a free phone if you sign a contract for only six months ofservice? Explain your answer. No; it only tells you what happens ifyou sign up for one year.

Reading the Lesson1. Identify the hypothesis and conclusion of each statement.

a. If you are a registered voter, then you are at least 18 years old. Hypothesis: youare a registered voter; Conclusion: you are at least 18 years old

b. If two integers are even, their product is even. Hypothesis: two integers areeven; Conclusion: their product is even

2. Complete each sentence.a. The statement that is formed by replacing both the hypothesis and the conclusion of a

conditional with their negations is the .b. The statement that is formed by exchanging the hypothesis and conclusion of a

conditional is the .

3. Consider the following statement:You live in North America if you live in the United States.a. Write this conditional statement in if-then form and give its truth value. If the

statement is false, give a counterexample. If you live in the United States, thenyou live in North America; false: You live in Hawaii.

b. Write the inverse of the given conditional statement in if-then form and give its truthvalue. If the statement is false, give a counterexample. If you do not live in theUnited States, then you do not live in North America; false; sampleanswer: You live in Mexico.

c. Write the contrapositive of the given conditional statement in if-then form and giveits truth value. If the statement is false, give a counterexample. If you do not livein North America, then you do not live in the United States; false: Youlive in Hawaii.

d. Write the converse of the given conditional statement in if-then form and give itstruth value. If the statement is false, give a counterexample. If you live in NorthAmerica, then you live in the United States; false; sample answer: Youlive in Canada.

Helping You Remember4. When working with a conditional statement and its three related conditionals, what is

an easy way to remember which statements are logically equivalent to each other?Sample answer: The two statements whose names contain verse (theconverse and the inverse) are a logically equivalent pair. The other two(the original conditional and the contrapositive) are the other logicallyequivalent pair.

converse

inverse


Venn Diagrams

A type of drawing called a Venn diagram can be useful in explaining conditionalstatements. A Venn diagram uses circles to represent sets of objects.

Consider the statement “All rabbits have long ears.” To make a Venn diagram for thisstatement, a large circle is drawn to represent all animals with long ears. Then asmaller circle is drawn inside the first to represent all rabbits. The Venn diagramshows that every rabbit is included in the group of long-eared animals.

The set of rabbits is called a subset of the set of long-eared animals.

The Venn diagram can also explain how to write thestatement, “All rabbits have long ears,” in if-then form. Everyrabbit is in the group of long-eared animals, so if an animal isa rabbit, then it has long ears.

For each statement, draw a Venn diagram. Then write the sentence in if-then form.

1 Every dog has long hair 2 All rational numbers are real

animals withlong ears

rabbits

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

2-32-3Enrichment, p. 74


Determine the truth value of the following statement for each set of conditions.If you are over 18 years old, then you vote in all elections.28. You are 19 years old and you vote. true29. You are 16 years old and you vote. true30. You are 21 years old and do not vote. false31. You are 17 years old and do not vote. true32. Your sister is 21 years old and votes. true33. Your dad is 45 years old and does not vote. false

In the figure, P, Q, and R are collinear, P and A lie in plane M , and Q and B lie in plane N . Determine the truth value of each statement.34. P, Q, and R lie in plane M . true35. QB�� lies in plane N . true36. Q lies in plane M . true37. P, Q, A, and B are coplanar. false38. AP�� contains Q. false39. Planes M and N intersect at RQ��. true

Write the converse, inverse, and contrapositive of each conditional statement.Determine whether each related conditional is true or false. If a statement is false,find a counterexample. 40–45. See p. 123B.40. If you live in Dallas, then you live in Texas.

41. If you exercise regularly, then you are in good shape.

42. The sum of two complementary angles is 90.

43. All rectangles are quadrilaterals.

44. All right angles measure 90.

45. Acute angles have measures less than 90.

SEASONS For Exercises 46 and 47, use the following information.Due to the movement of Earth around the sun, summer days in Alaska have morehours of daylight than darkness, and winter days have more hours of darkness thandaylight.46. Write two true conditional statements in if-then form for summer days and

winter days in Alaska.

47. Write the converse of the two true conditional statements. State whether each istrue or false. If a statement is false, find a counterexample. See p. 123B.

48. CRITICAL THINKING Write a false conditional statement. Is it possible to insertthe word not into your conditional to make it true? If so, write the trueconditional.


How are conditional statements used in advertisements?

Include the following in your answer:• an example of a conditional statement in if-then form, and• an example of a conditional statement that is not in if-then form.

WRITING IN MATH

N

MA

P Q R

B

Galen Rowell/CORBIS

46. Sample answer: In Alaska, if it issummer, then thereare more hours ofdaylight than dark-ness. In Alaska, if it iswinter, then there aremore hours of dark-ness than daylight.

SeasonsAt the poles, sunlight mayshine continuously for sixmonths during spring andsummer, but never risesmore than 23.5° above thehorizon. During the othersix months of the year, thepoles are in darkness.Source: U.S. Geological Survey



ELL

Answer

49. Conditional statements can be used todescribe how to get a discount, rebate, orrefund. Sample answers should include thefollowing.• If you are not 100% satisfied, then return

the product for a full refund.• Wearing a seatbelt reduces the risk of

injuries.


Open-Ended AssessmentSpeaking Students can practicetheir speaking skills by identifyingparts of statements and translatingstatements into the converse,inverse, and contrapositive aloud.

Getting Ready forLesson 2-4Prerequisite Skill Students willlearn about deductive reasoningin Lesson 2-4. They will applyconcepts of solving equations todeductive-reasoning techniques.Use Exercises 66–68 to determineyour students’ familiarity withsolving equations.

Assessment OptionsPractice Quiz 1 The quizprovides students with a briefreview of the concepts and skillsin Lessons 2-1 through 2-3.Lesson numbers are given to theright of the exercises orinstruction lines so students canreview concepts not yetmastered.

Answers

52. George Washington was the firstpresident of the United States anda hexagon has 5 sides.

53. A hexagon has five sides or 60 � 3 � 18.

54. George Washington was the firstpresident of the United States or ahexagon has five sides.

55. A hexagon doesn’t have five sidesor 60 � 3 � 18.

56. George Washington was the firstpresident of the United States anda hexagon doesn’t have five sides.

57. George Washington was not thefirst president of the United Statesand 60 � 3 18.

58. D

A

C

B



50. Which statement has the same truth value as the following statement? CIf Ava and Willow are classmates, then they go to the same school.

If Ava and Willow go to the same school, then they are classmates.If Ava and Willow are not classmates, then they do not go to the same school.If Ava and Willow do not go to the same school, then they are not classmates.If Ava and Willow go to the same school, then they are not classmates.

51. ALGEBRA In a history class with 32 students, the ratio of girls to boys is 5 to 3.How many more girls are there than boys? B

2 8 12 20

Use the following statements to write a compound statement for each conjunctionand disjunction. Then find its truth value. (Lesson 2-2) 52–57. See margin.p: George Washington was the first president of the United States.q: A hexagon has five sides.r: 60 � 3 � 1852. p � q false 53. q � r false 54. p � q true55. ~q � r true 56. p � ~q true 57. ~p � ~r false

Make a conjecture based on the given information. Draw a figure to illustrateyour conjecture. (Lesson 2-1) 58–61. See margin for sample figures.58. ABCD is a rectangle. 59. In �FGH, m�F � 45, m�G � 67, m�H � 68.60. J(�3, 2), K(1, 8), L(5, 2) 61. In �PQR, m�PQR � 90 �PQR is a right angle.

Use the Distance Formula to find the distance between each pair of points. (Lesson 1-3)

62. C(�2, �1), D(0, 3) �20� � 4.5 63. J(�3, 5), K(1, 0) �41� � 6.464. P(�3, �1), Q(2, �3) �29� � 5.4 65. R(1, �7), S(�4, 3) �125� � 11.2

PREREQUISITE SKILL Identify the operation used to change Equation (1) toEquation (2). (To review solving equations, see pages 737 and 738.) 66–68. See margin.66. (1) 3x � 4 � 5x � 8 67. (1) �

12

�(a � 5) � 12 68. (1) 8p � 24(2) 3x � 5x � 12 (2) a � 5 � 24 (2) p � 3

DCBA

D

C

B

A

Practice Quiz 1Practice Quiz 1

Determine whether each conjecture is true or false. Give a counterexample for any false conjecture.(Lesson 2-1)

1. Given: WX � XY 2. Given: �1 and �2 are complementary. Conjecture: W, X, and Y are collinear. �2 and �3 are complementary.False; see p. 123B for counterexample. Conjecture: m�1 � m�3 true

Construct a truth table for each compound statement. (Lesson 2-2) 3–4. See p. 123B.3. ~p � q 4. p � (q � r)

5. Write the converse, inverse, and contrapositive of the following conditional statement. Determine whether each related conditional is true or false. If a statement is false, find a counterexample. (Lesson 2-3)

If two angles are adjacent, then the angles have a common vertex. See p. 123C.

Lessons 2-1 through 2-3

Mixed Review

58. AB � CD ; AD � BC59. The sum of themeasures of theangles in a triangle is 180.60. �JKL has twosides congruent.




4 Assess4 Assess

59. 60.

x

y

O

J L

KG

F

H

45

67

68

61.

66. Subtract 4 from each side.67. Multiply each side by 2.68. Divide each side by 8.

Q

P

R

ReadingMathematics

Getting StartedGetting Started

TeachTeach

AssessAssess

Study NotebookStudy Notebook

Explain that true biconditionalstatements are extremely helpfulfor writing proofs because theycan be used forwards orbackwards. Tell students,however, that they are not ascommon as regular conditionals,and students will need to bewary and thorough when theyare determining whether abiconditional is true or false.

Biconditional StatementsStudents can also get some extrapractice writing the inverse andcontrapositive of eachbiconditional statement.

Ask students to summarize whatthey have learned aboutbiconditional statements.

Biconditional Statements

Reading Mathematics Biconditional Statements 81

Ashley began a new summer job, earning $10 an hour. If she works over 40 hours a week, she earns time and a half, or $15 an hour. If she earns $15 an hour, she hasworked over 40 hours a week.

p: Ashley earns $15 an hourq: Ashley works over 40 hours a week

p → q: If Ashley earns $15 an hour, she has worked over 40 hours a week.q → p: If Ashley works over 40 hours a week, she earns $15 an hour.

In this case, both the conditional and its converse are true. The conjunction of thetwo statements is called a .

So, the biconditional statement is as follows.

p ↔ q: Ashley earns $15 an hour if and only if she works over 40 hours a week.

ExamplesWrite each biconditional as a conditional and its converse. Then determinewhether the biconditional is true or false. If false, give a counterexample.

a. Two angle measures are complements if and only if their sum is 90.Conditional: If two angle measures are complements, then their sum is 90.Converse: If the sum of two angle measures is 90, then they are complements.Both the conditional and the converse are true, so the biconditional is true.

b. x � 9 iff x � 0Conditional: If x 9, then x 0.Converse: If x 0, then x 9.The conditional is true, but the converse is not. Let x � 2. Then 2 0 but 2 � 9.So, the biconditional is false.

Reading to Learn 1–5. See margin.Write each biconditional as a conditional and its converse. Then determinewhether the biconditional is true or false. If false, give a counterexample.

1. A calculator will run if and only if it has batteries.

2. Two lines intersect if and only if they are not vertical.

3. Two angles are congruent if and only if they have the same measure.

4. 3x � 4 � 20 iff x � 7.

5. A line is a segment bisector if and only if it intersects the segment at its midpoint.

biconditional

Biconditional Statement• Words A biconditional statement is the conjunction of a conditional

and its converse.

• Symbols (p → q) � (q → p) is written (p ↔ q) and read p if and only if q.

If and only if can be abbreviated iff.

Reading Mathematics Biconditional Statements 81

Answers

English LanguageLearners may benefit fromwriting key concepts from thisactivity in their Study Notebooksin their native language and thenin English.

ELL

1. Conditional: If a calculator runs, then it hasbatteries. Converse: If a calculator hasbatteries, then it will run. False; a calculatormay be solar powered.

2. Conditional: If two lines intersect, then theyare not vertical. Converse: If two lines arenot vertical, then they intersect. False; twoparallel horizontal lines will not intersect.

5. Conditional: If a line is a segmentbisector, then it intersects thesegment at its midpoint. Converse:If a line intersects a segment atits midpoint, then it is a segmentbisector. true

3. Conditional: If two angles are congruent,then they have the same measure.Converse: If two angles have the samemeasure, then they are congruent. true

4. Conditional: If 3x � 4 � 20, then x � 7.Converse: If x � 7, then 3x � 4 � 20.False; 3x � 4 � 17 when x � 7.



Mathematical Background notesare available for this lesson on p. 60D.

does deductivereasoning apply

to health?Ask students:• If you have a mass of 57.8 kg,

what dose will a doctor giveyou based on the chart? 350 mg

• What might happen if a patientused inductive reasoning to forma conjecture about the dose ofan antidepressant based on thechart above? Is this a safemethod for health situations?Sample answer: The dose of theantidepressant might be much lessthan that of the antibiotic for thesame weight, so the patient couldpotentially overdose on theantidepressant; no.

Law of Detachment

LAW OF DETACHMENT The process that doctors use to determine theamount of medicine a patient should take is called . Unlikeinductive reasoning, which uses examples to make a conjecture, deductive reasoninguses facts, rules, definitions, or properties to reach logical conclusions.

A form of deductive reasoning that is used to draw conclusions from trueconditional statements is called the .Law of Detachment

deductive reasoning

Vocabulary• deductive reasoning• Law of Detachment• Law of Syllogism

Deductive Reasoning


• Use the Law of Detachment.

• Use the Law of Syllogism.

When you are ill, your doctor may prescribe anantibiotic to help you get better. Doctors may usea dose chart like the one shown to determine thecorrect amount of medicine you should take.

Determine Valid ConclusionsThe following is a true conditional. Determine whether each conclusion is validbased on the given information. Explain your reasoning.If a ray is an angle bisector, then it divides the angle into two congruent angles.

a. Given: BD�� bisects �ABC.Conclusion: �ABD � �CBD

The hypothesis states that BD�� is the bisector of �ABC. Since the conditional is true and the hypothesis is true, the conclusion is valid.

b. Given: �PQT � �RQS

Conclusion: QS�� and QT�� are angle bisectors.

Knowing that a conditional statement and its conclusion are true does not make the hypothesis true. An angle bisector divides an angle into two separate congruent angles. In this case, the given angles are not separated by one ray. Instead, they overlap. The conclusion is not valid.

Q

P

S

T

R

A

D

C

B

Example 1Example 1

ValidityWhen you apply the Lawof Detachment, make surethat the conditional is truebefore you test the validityof the conclusion.

Study Tip

does deductive reasoningapply to health?does deductive reasoningapply to health?

• Words If p → q is true and p is true, then q is also true.

• Symbols [(p → q) � p] → q

Weight(kg)

10–2020–30

30–4040–50

50–6060–70

Dose(mg)150

200250

300350

400

LessonNotes

1 Focus1 Focus

Chapter 2 Resource Masters• Study Guide and Intervention, pp. 75–76• Skills Practice, p. 77• Practice, p. 78• Reading to Learn Mathematics, p. 79• Enrichment, p. 80• Assessment, pp. 119, 121

School-to-Career Masters, p. 3Teaching Geometry With Manipulatives

Masters, p. 47





Transparencies

22


11


LAW OF DETACHMENT

The following is a trueconditional. Determinewhether each conclusion isvalid based on the giveninformation. Explain yourreasoning.If two segments are congruentand the second segment iscongruent to a third segment,then the first segment is alsocongruent to the third segment.

a. Given: W�X� � U�V�; U�V� � R�T�Conclusion: W�X� � R�T� true

b. Given: U�V�; W�X� � R�T�Conclusion: W�X� � U�V� and U�V� � R�T� false

LAW OF SYLLOGISM

PROM Use the Law ofSyllogism to determinewhether a valid conclusioncan be reached from each setof statements.

a. (1) If Salline attends the prom,she will go with Mark.(2) Mark is a 17-year-oldstudent. not valid

b. (1) If Mel and his date eat atthe Peddler Steakhouse beforegoing to the prom, they willmiss the senior march. (2) The Peddler Steakhousestays open until 10 P.M.not valid

In-Class Example 3 is on p. 84.Lesson 2-4 Deductive Reasoning 83

LAW OF SYLLOGISM Another law of logic is the . It issimilar to the Transitive Property of Equality.

Law of Syllogism

Law of Syllogism• Words If p → q and q → r are true, then p → r is also true.

• Symbols [(p → q) � (q → r)] → (p → r)

Determine Valid Conclusions From Two ConditionalsCHEMISTRY Use the Law of Syllogism to determine whether a valid conclusioncan be reached from each set of statements.a. (1) If the symbol of a substance is Pb, then it is lead.

(2) The atomic number of lead is 82.Let p, q, and r represent the parts of the statement.p: the symbol of a substance is Pbq: it is leadr: the atomic number is 82Statement (1): p → qStatement (2): q → rSince the given statements are true, use the Law of Syllogism to conclude p → r.That is, If the symbol of a substance is Pb, then its atomic number is 82.

b. (1) Water can be represented by H2O.(2) Hydrogen (H) and oxygen (O) are in the atmosphere.There is no valid conclusion. While both statements are true, the conclusion ofeach statement is not used as the hypothesis of the other.

Example 2Example 2

Analyze ConclusionsDetermine whether statement (3) follows from statements (1) and (2) by the Lawof Detachment or the Law of Syllogism. If it does, state which law was used. Ifit does not, write invalid.a. (1) Vertical angles are congruent.

(2) If two angles are congruent, then their measures are equal.(3) If two angles are vertical, then their measures are equal.p: two angles are verticalq: they are congruentr: their measures are equalStatement (3) is a valid conclusion by the Law of Syllogism.

b. (1) If a figure is a square, then it is a polygon.(2) Figure A is a polygon.(3) Figure A is a square.Statement (1) is true, but statement (3) does not follow from statement (2). Not all polygons are squares.Statement (3) is invalid.

Example 3Example 3

ConditionalStatementsLabel the hypotheses and conclusions of aseries of statementsbefore applying the Law of Syllogism.

Study Tip


Lesson 2-4 Deductive Reasoning 83

2 Teach2 Teach

Verbal/Linguistic Have students write a paragraph to explain andprovide an example for the Law of Detachment. Repeat for the Law ofSyllogism. Then students can write another paragraph to point outsimilarities and differences between the two laws. They can place theirwritten explanations in their study notebooks.

Differentiated Instruction ELL


33




Determine whether statement(3) follows from statements(1) and (2) by the Law ofDetachment or the Law ofSyllogism. If it does, statewhich law was used. If itdoes not, write invalid.

a. (1) If the sum of the squares oftwo sides of a triangle is equalto the square of the third side,then the triangle is a righttriangle. (2) For �XYZ, (XY)2 � (YZ)2 � (ZX)2.(3) �XYZ is a right triangle.Law of Detachment

b. (1) If Ling wants to participatein the wrestling competition,he will have to meet an extrathree times a week to practice.(2) If Ling adds anythingextra to his weekly schedule,he cannot take karate lessons.(3) If Ling wants to participatein the wrestling competition,he cannot take karate lessons.Law of Syllogism

Have students—• add the definitions/examples of



FIND THE ERRORExplain that if

you have p → q and p → r, you cannot assume q → r,as Suzanne tries to do. Tellstudents that the same hypothesiscan imply two differentconclusions, but the conclusionsare not related to one another.


1. OPEN ENDED Write an example to illustrate the correct use of the Law of Detachment. 1–3. See margin.

2. Explain how the Transitive Property of Equality is similar to the Law of Syllogism.

3. FIND THE ERROR An article in a magazine states that if you get seasick, thenyou will get dizzy. It also says that if you get seasick, you will get an upsetstomach. Suzanne says that this means that if you get dizzy, then you will get an upset stomach. Lakeisha says that she is wrong. Who is correct? Explain.

Determine whether the stated conclusion is valid based on the given information.If not, write invalid. Explain your reasoning.If two angles are vertical angles, then they are congruent.4. Given: �A and �B are vertical angles.

Conclusion: �A � �B valid

5. Given: �C � �DConclusion: �C and �D are vertical angles.

Use the Law of Syllogism to determine whether a valid conclusion can be reachedfrom each set of statements. If a valid conclusion is possible, write it. If not, writeno conclusion.6. If you are 18 years old, you are in college.

You are in college. no conclusion

7. The midpoint divides a segment into two congruent segments.If two segments are congruent, then their measures are equal.

Determine whether statement (3) follows from statements (1) and (2) by the Lawof Detachment or the Law of Syllogism. If it does, state which law was used. If itdoes not, write invalid. 8. valid; Law of Syllogism8. (1) If Molly arrives at school at 7:30 A.M., she will get help in math.

(2) If Molly gets help in math, then she will pass her math test.(3) If Molly arrives at school at 7:30 A.M., then she will pass her math test.

9. (1) Right angles are congruent.(2) �X � �Y(3) �X and �Y are right angles. invalid

INSURANCE For Exercises 10 and 11, use the following information.An insurance company advertised the following monthly rates for life insurance.

10. If Ann is 35 years old and she wants to purchase $30,000 of insurance from thiscompany, then what is her premium? $14.35

11. If Terry paid $21.63 for life insurance, can you conclude that Terry is 35?Explain. No; Terry could be a man or a woman. She could be 45 and have purchased $30,000 of life insurance.

If you are a:Female, age 35

Male, age 35

Female, age 45

Male, age 45

Premium for $30,000Coverage

$14.35

$16.50

$21.63

$23.75

Premium for $50,000Coverage

$19.00

$21.63

$25.85

$28.90

Invalid; congruent anglesdo not have to be vertical.

Concept Check


4, 5 16, 7 28, 9 3

7. The midpoint of asegment divides it intotwo segments withequal measures.

Application

Guided Practice


Answers

1. Sample answer: a: If it is rainy, the game will be cancelled. b: Itis rainy. c: The game will be cancelled.

2. Transitive Property of Equality: a � b and b � c implies a � c. Law of Syllogism: a implies band b implies c implies a implies c. Each statement establishesa relationship between a and c through their relationships to b.

3. Lakeisha; if you are dizzy, that does not necessarily mean thatyou are seasick and thus have an upset stomach.


13. Valid; since 5 and7 are odd, the Law ofDetachment indicatesthat their sum is even.14. Valid; since 11and 23 are odd, theLaw of Detachmentindicates that theirsum is even.16. Valid; A, B, and Care noncollinear, andby definition threenoncollinear pointsdetermine a plane.17. Invalid; E, F, andG are not necessarilynoncollinear.18. Invalid; thehypothesis is false asthere are only twopoints.19. Valid; the verticesof a triangle are non-collinear, andtherefore determine a plane.21. If the measure ofan angle is less than90, then it is notobtuse.22. If X is the mid-point of Y�Z�, then Y�X� � X�Z�.


ForExercises

12–1920–2324–29

SeeExamples

123



For Exercises 12–19, determine whether the stated conclusion is valid based on thegiven information. If not, write invalid. Explain your reasoning.If two numbers are odd, then their sum is even.12. Given: The sum of two numbers is 22.

Conclusion: The two numbers are odd. invalid; 10 � 12 � 2213. Given: The numbers are 5 and 7.

Conclusion: The sum is even.

14. Given: 11 and 23 are added together.Conclusion: The sum of 11 and 23 is even.

15. Given: The numbers are 2 and 6.Conclusion: The sum is odd. Invalid; the sum is even.

If three points are noncollinear, then they determine a plane.16. Given: A, B, and C are noncollinear.

Conclusion: A, B, and C determine a plane.

17. Given: E, F, and G lie in plane M.Conclusion: E, F, and G are noncollinear.

18. Given: P and Q lie on a line.Conclusion: P and Q determine a plane.

19. Given: �XYZConclusion: X, Y, and Z determine a plane.

Use the Law of Syllogism to determine whether a valid conclusion can be reachedfrom each set of statements. If a valid conclusion is possible, write it. If not, writeno conclusion.20. If you spend money on it, then it is a business.

If you spend money on it, then it is fun. no conclusion21. If the measure of an angle is less than 90, then it is acute.

If an angle is acute, then it is not obtuse.

22. If X is the midpoint of segment YZ, then YX � XZ.If the measures of two segments are equal, then they are congruent.

23. If two lines intersect to form a right angle, then they are perpendicular.Lines � and m are perpendicular. no conclusion

Determine whether statement (3) follows from statements (1) and (2) by the Lawof Detachment or the Law of Syllogism. If it does, state which law was used. If itdoes not, write invalid.24. (1) In-line skaters live dangerously.

(2) If you live dangerously, then you like to dance.(3) If you are an in-line skater, then you like to dance. yes; Law of Syllogism

25. (1) If the measure of an angle is greater than 90, then it is obtuse.(2) m�ABC > 90(3) �ABC is obtuse. yes; Law of Detachment

26. (1) Vertical angles are congruent.(2) �3 � �4(3) �3 and �4 are vertical angles. invalid

27. (1) If an angle is obtuse, then it cannot be acute.(2) �A is obtuse.(3) �A cannot be acute. yes; Law of Detachment


About the Exercises…Organization by Objective• Law of Detachment: 12–19• Law of Syllogism: 20–29

Odd/Even AssignmentsExercises 12–29 are structuredso that students practice thesame concepts whether theyare assigned odd or evenproblems.Alert! Exercise 31 requires theInternet or other researchmaterials.

Assignment GuideBasic: 13–31 odd, 32–58Average: 13–31 odd, 32–58Advanced: 12–30 even, 32,34–55 (optional: 56–58)

Study Guide and InterventionDeductive Reasoning

NAME ______________________________________________ DATE ____________ PERIOD _____

2-42-4


Less

on

2-4

Law of Detachment Deductive reasoning is the process of using facts, rules,definitions, or properties to reach conclusions. One form of deductive reasoning that drawsconclusions from a true conditional p → q and a true statement p is called the Law ofDetachment.

Law of Detachment If p → q is true and p is true, then q is true.

Symbols [(p → q)] � p] → q

The statement If two angles are supplementary to the same angle,then they are congruent is a true conditional. Determine whether each conclusionis valid based on the given information. Explain your reasoning.

a. Given: �A and �C are supplementary to �B.Conclusion: �A is congruent to �C.

The statement �A and �C are supplementary to �B is the hypothesis of the conditional. Therefore, by the Lawof Detachment, the conclusion is true.

b. Given: �A is congruent to �C.Conclusion: �A and �C are supplementary to �B.The statement �A is congruent to �C is not the hypothesis of the conditional, so the Law of Detachment cannot be used.The conclusion is not valid.

Determine whether each conclusion is valid based on the true conditional given.If not, write invalid. Explain your reasoning.If two angles are complementary to the same angle, then the angles are congruent.

1. Given: �A and �C are complementary to �B.Conclusion: �A is congruent to �C.

The given statement is the hypothesis of the conditional statement.Since the conditional is true, the conclusion �A � �C is true.

2. Given: �A � �CConclusion: �A and �C are complements of �B.

The given statement is not the hypothesis of the conditional.Therefore, the conclusion is invalid.

3. Given: �E and �F are complementary to �G.Conclusion: �E and �F are vertical angles.

While the given statement is the hypothesis of the conditional statement,the statement that �E and �F are vertical angles is not the conclusion ofthe conditional. The conclusion is invalid.

A D

EF

H

J

C

GB

ExampleExample

ExercisesExercises



Determine whether the stated conclusion is valid based on the given information.If not, write invalid. Explain your reasoning.If a point is the midpoint of a segment, then it divides the segment into twocongruent segments.

1. Given: R is the midpoint of Q�S�.Conclusion: Q�R� � R�S�Valid; since R is the midpoint of Q�S�, the Law of Detachment indicatesthat it divides Q�S� into two congruent segments.

2. Given: A�B� � B�C�Conclusion: B divides A�C� into two congruent segments.Invalid; the points A, B, and C may not be collinear, and if they are not,then B will not be the midpoint of A�C�.

Use the Law of Syllogism to determine whether a valid conclusion can be reachedfrom each set of statements. If a valid conclusion is possible, write it.

3. If two angles form a linear pair, then the two angles are supplementary.If two angles are supplementary, then the sum of their measures is 180.If two angles form a linear pair, then the sum of their measures is 180.

4. If a hurricane is Category 5, then winds are greater than 155 miles per hour.If winds are greater than 155 miles per hour, then trees, shrubs, and signs are blown down.If a hurricane is Category 5, then trees, shrubs, and signs are blown down.

Determine whether statement (3) follows from statements (1) and (2) by the Lawof Detachment or the Law of Syllogism. If it does, state which law was used. If itdoes not, write invalid.

5. (1) If a whole number is even, then its square is divisible by 4.(2) The number I am thinking of is an even whole number.(3) The square of the number I am thinking of is divisible by 4.yes; Law of Detachment

6. (1) If the football team wins its homecoming game, then Conrad will attend the schooldance the following Friday.(2) Conrad attends the school dance on Friday.(3) The football team won the homecoming game.invalid

7. BIOLOGY If an organism is a parasite, then it survives by living on or in a hostorganism. If a parasite lives in or on a host organism, then it harms its host. Whatconclusion can you draw if a virus is a parasite?If a virus is a parasite, then it harms its host.

Practice (Average)

Deductive Reasoning

NAME ______________________________________________ DATE ____________ PERIOD _____


Reading to Learn MathematicsDeductive Reasoning

NAME ______________________________________________ DATE ____________ PERIOD _____

2-42-4

Pre-Activity How does deductive reasoning apply to health?


Suppose a doctor wants to use the dose chart in your textbook to prescribean antibiotic, but the only scale in her office gives weights in pounds. Howcan she use the fact that 1 kilogram is about 2.2 pounds to determine thecorrect dose for a patient? Sample answer: The doctor can dividethe patient’s weight in pounds by 2.2 to find the equivalentmass in kilograms. She can then use the dose chart.

Reading the LessonIf s, t, and u are three statements, match each description from the list on the leftwith a symbolic statement from the list on the right.1. negation of t e a. s � u

2. conjunction of s and u g b. [(s → t) � s] → t

3. converse of s → t h c. �s → �u

4. disjunction of s and u a d. �u → �s

5. Law of Detachment b e. �t

6. contrapositive of s → t j f. [(u → t) � (t → s)] → (u → s)

7. inverse of s → u c g. s � u

8. contrapositive of s → u d h. t → s

9. Law of Syllogism f i. t

10. negation of �t i j. �t → �s

11. Determine whether statement (3) follows from statements (1) and (2) by the Law ofDetachment or the Law of Syllogism. If it does, state which law was used. If it does not,write invalid.a. (1) Every square is a parallelogram.

(2) Every parallelogram is a polygon.(3) Every square is a polygon. yes; Law of Syllogism

b. (1)If two lines that lie in the same plane do not intersect, they are parallel.(2) Lines � and m lie in plane U and do not intersect.(3) Lines � and m are parallel. yes; Law of Detachment

c. (1) Perpendicular lines intersect to form four right angles.(2) �A, �B, �C, and �D are four right angles.(3) �A, �B, �C, and �D are formed by intersecting perpendicular lines. invalid

Helping You Remember12. A good way to remember something is to explain it to someone else. Suppose that a

classmate is having trouble remembering what the Law of Detachment means?Sample answer: The word detach means to take something off of anotherthing. The Law of Detachment says that when a conditional and itshypothesis are both true, you can detach the conclusion and feelconfident that it too is a true statement.


Valid and Faulty ArgumentsConsider the statements at the right.What conclusions can you make?

From statements 1 and 3, it is correct to conclude that Bootspurrs if it is happy. However, it is faulty to conclude from only statements 2 and 3 that Boots is happy. The if-then form of statement 3 is If a cat is happy, then it purrs.

Advertisers often use faulty logic in subtle ways to help selltheir products. By studying the arguments, you can decide whether the argument is valid or faulty.

Decide if each argument is valid or faulty.

1. (1) If you buy Tuff Cote luggage, it 2. (1) If you buy Tuff Cote luggage, itwill survive airline travel. will survive airline travel.

(2) Justin buys Tuff Cote luggage. (2) Justin’s luggage survived airline travel.C l ’ l ll C l h ff C

(1) Boots is a cat.(2) Boots is purring.(3) A cat purrs if it is happy.

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____


Determine whether statement (3) follows from statements (1) and (2) by the Lawof Detachment or the Law of Syllogism. If it does, state which law was used. If itdoes not, write invalid.28. (1) If you drive safely, then you can avoid accidents.

(2) Tika drives safely.(3) Tika can avoid accidents. yes; Law of Detachment

29. (1) If you are a customer, then you are always right.(2) If you are a teenager, then you are always right.(3) If you are a teenager, then you are a customer. invalid

30. LITERATURE John Steinbeck, a Pulitzer Prize winning author, lived inMonterey, California, for part of his life. In 1945, he published the book, CanneryRow, about many of his local working-class heroes from Monterey. If you visitedCannery Row in Monterey during the 1940s, then you could hear the gratingnoise of the fish canneries. Write a valid conclusion to the following hypothesis.If John Steinbeck lived in Monterey in 1941, . . .

31. SPORTS In the 2002 Winter Olympics, Canadian speed skater Catriona Le MayDoan won her second Olympic title in 500-meter speed skating. Ms. Doan wasin the last heat for the second round of that race. Use the two true conditionalstatements to reach a valid conclusion about Ms. Doan’s 2002 competition.(1) If Catriona Le May Doan skated her second 500 meters in 37.45 seconds,

then she would beat the time of Germany’s Monique Garbrecht-Enfeldt.(2) If Ms. Doan beat the time of Monique Garbrecht-Enfeldt, then she would

win the race. If Catriona Le May Doan skated her second 500 meters in37.45 seconds, then she would win the race.Online Research Data Update Use the Internet or another resource to find the winning times for other Olympic events. Write statements using these times that can lead to a valid conclusion. Visitwww.geometryonline.com/data_update to learn more.

32. CRITICAL THINKING An advertisement states that “If you like to ski, thenyou’ll love Snow Mountain Resort.” Stacey likes to ski, but when she went toSnow Mountain Resort, she did not like it very much. If you know that Staceysaw the ad, explain how her reasoning was flawed.

33. Answer the question that was posed at the beginning of the lesson. See p. 123C.

How does deductive reasoning apply to health?

Include the following in your answer:• an explanation of how doctors may use deductive reasoning to prescribe

medicine, and• an example of a doctor’s uses of deductive reasoning to diagnose an illness,

such as strep throat or chickenpox.

34. Based on the following statements, which statement must be true? CI If Yasahiro is an athlete and he gets paid, then he is a professional athlete.

II Yasahiro is not a professional athlete.III Yasahiro is an athlete.

Yasahiro is an athlete and he gets paid.Yasahiro is a professional athlete or he gets paid.Yasahiro does not get paid.Yasahiro is not an athlete.D

C

B

A

WRITING IN MATH


30. then he couldhear the grating noiseof the fish canneries

32. Sample answer:Stacey assumed thatthe conditional state-ment was true.

AP/Wide World Photos

LiteratureThe Pulitzer Prize isawarded annually foroutstanding contributionsin the fields of journalism,literature, drama, andmusic.Source: www.pulitzer.org



ELL

http://www.geometryonline.com/data_update

Open-Ended AssessmentModeling Students can useshapes of paper to model the Lawof Detachment and the Law ofSyllogism. For example, youcould provide students with twoyellow squares of laminatedconstruction paper marked with p,two blue triangles marked withq, and two red circles markedwith r. Students can physicallyarrange the shapes to representthe symbolic relationships of thetwo laws. They could also securethese shapes in their studynotebooks for a colorfulreminder of these laws.

Getting Ready forLesson 2-5Prerequisite Skill Students willlearn about postulates andparagraph proofs in Lesson 2-5.They will extract information fromfigures to test postulates andwrite proofs. Use Exercises 56–58to determine your students’familiarity with informationfrom figures.


Mid-Chapter Test (Lessons 2-1through 2-4) is available on p. 121 of the Chapter 2 ResourceMasters.



35. ALGEBRA At a restaurant, a diner uses a coupon for 15% off the cost of onemeal. If the diner orders a meal regularly priced at $16 and leaves a tip of 20%of the discounted meal, how much does she pay in total? B

$15.64 $16.32 $16.80 $18.72

ADVERTISING For Exercises 36–38, use the following information. (Lesson 2-3)

Advertising writers frequently use if-then statements to relay a message andpromote their product. An ad for a type of Mexican food reads, If you’re looking for a fast, easy way to add some fun to your family’s menu, try Casa Fiesta.36. Write the converse of the conditional.37. What do you think the advertiser wants people to conclude about

Casa Fiesta products? 38. Does the advertisement say that Casa Fiesta adds fun to your family’s menu?

Construct a truth table for each compound statement. (Lesson 2-2) 39–42. See p. 123C.39. q � r 40. ~p � r 41. p � (q � r) 42. p � (~q � r)

For Exercises 43–47, refer to the figure at the right. (Lesson 1-5)

43. Which angle is complementary to �FDG? �HDG44. Name a pair of vertical angles.45. Name a pair of angles that are noncongruent

and supplementary.46. Identify �FDH and �CDH as congruent, adjacent,

vertical, complementary, supplementary, and/or a linear pair.

47. Can you assume that D�C� � C�K�? Explain. Yes, slashes on the segments indicate that they are congruent.

Use the Pythagorean Theorem to find the distance between each pair of points.(Lesson 1-3)

48. A(1, 5), B(�2, 9) 5 49. C(�4, �2), D(2, 6) 1050. F(7, 4), G(1, 0) �52� � 7.2 51. M(�5, 0), N(4, 7) �130� � 11.4

For Exercises 52–55, draw and label a figure for each relationship. (Lesson 1-1)

52. FG�� lies in plane M and contains point H. 52–55. See margin.53. Lines r and s intersect at point W.54. Line � contains P and Q, but does not contain R.55. Planes A and B intersect in line n.

PREREQUISITE SKILL Write what you can assume about the segments or angleslisted for each figure. (To review information from figures, see Lesson 1-5.)

56. A�M�, C�M�, C�N�, B�N� 57. �1, �2 58. �4, �5, �6

4 56

12

A

C N B

M

D

F

G

H

C K J

DCBA

Mixed Review36. If you try CasaFiesta, then you’relooking for a fast,easy way to add somefun to your family’smenu.37. They are a fast,easy way to add fun toyour family’s menu.38. No; the conclusionis implied.

44. Sample answer:�KHJ and �DHG45. Sample answer:�JHK and �DHK46. congruent, adja-cent, supplementary,linear pair


56–58. See margin.



4 Assess4 Assess

Answers52.

FH G

M53.

54. Q

R

�P

W r

s

55.A

B

n

56. Sample answer: C�M� � A�M�, C�N� �B�N�, AM � CM, CN � BN, M ismidpoint of A�C�, N is midpoint of B�C�.

57. Sample answer: �1 and �2 arecomplementary, m�1 � m�2 �90.

58. Sample answer: �4 and �5 aresupplementary, m�4 � m�5 �180, �5 and �6 are supple-mentary, m�5 � m�6 � 180,�4 � �6, m�4 � m�6.


GeometryActivity

Getting StartedGetting Started

TeachTeach

AssessAssess

A Follow-Up of Lesson 2-4


A Follow-Up of Lesson 2-4

Exercises1. Nate, John, and Nick just began after-school jobs. One works at a veterinarian’s office,

one at a computer store, and one at a restaurant. Nate buys computer games on the way to work. Nick is allergic to cat hair. John receives free meals at his job. Who works at which job? Nate, veterinarian’s office; John, restaurant; Nick, computer store

2. Six friends live in consecutive apartments on the same side of their apartment building. Anita lives in apartment C. Kelli’s apartment is just past Scott’s. Anita’s closest neighbors are Eric and Ava. Scott’s apartment is not A through D. Eric’s apartment is before Ava’s. If Roberto lives in one of the apartments, who lives in which apartment?A, Roberto; B, Eric; C, Anita; D, Ava; E, Scott; F, Kelli

Matrix LogicDeductive reasoning can be used in problem-solving situations. One method ofsolving problems uses a table. This method is called .

ExampleGEOLOGY On a recent test, Rashaun was given five different mineral samples to identify, along with the chart at right. Rashaun observed the following.

• Sample C is brown.• Samples B and E are harder

than glass.• Samples D and E are red.

Identify each of the samples.

Make a table to organize the information.Mark each false condition with an � and eachtrue condition with a �. The first observationis that Sample C is brown. Only one of theminerals, biotite, is brown, so place a check inthe box that corresponds to biotite andSample C. Then place an � in each of theother boxes in the same column and row.

The second observation is that Samples B andE are harder than glass. Place an � in eachbox for minerals that are softer than glass.The third observation is that Samples D andE are red. Mark the boxes accordingly. Noticethat Sample E has an � in all but one box.Place a check mark in the remaining box, andan � in all other boxes in that row.

Then complete the table. Sample A is Halite, Sample B is Feldspar, Sample C is Biotite,Sample D is Hematite, and Sample E is Jaspar.

matrix logic

�

Sample

Biotite

Halite

Hematite

Feldspar

Jaspar

A B C D E

� � � �

� � � �

� � �

� � �

� � � �

�

Sample

Biotite

Halite

Hematite

Feldspar

Jaspar

A B C D E

� � � �

� � � �

� � � �

� � � �

� � � �

�

�

�

�

�

Mineral Color Hardness

Biotite brown or black softer than glass

Halite white softer than glass

Hematite red softer than glass

Feldspar white, pink, or harder than glassgreen

Jaspar red harder than glass


You could provide students witha sheet of three blank logic tablesin which they would fill in thetitles of the rows and columns,or ask students to use a ruler todraw the tables as they go along.

Objective Apply deductivereasoning by using matrix logictables to solve problems.

• When students are consideringthe second observation, tellthem they can think in terms ofboxes to mark out or boxes toleave open. The only twominerals that are harder thanglass are feldspar and jasper,so these boxes are left open inthe columns B and E. Thenthey can place an X in the restof the boxes in these columns.

• For Exercise 2, advise studentsto draw a model with sixadjacent boxes representing theapartments. After marking allobvious information, studentscan use the model to placeeach person in the correctapartment.

In Exercises 1 and 2 studentspractice their thinking anddeductive reasoning skills byworking more logic tables.

Study NotebookStudy NotebookAsk students to summarize whatthey have learned about usingdeductive reasoning in problem-solving situations.

Teaching Geometry withManipulatives• p. 46 (student recording sheet)


11





were postulates usedby the founding fathers

of the United States?Ask students:• How would you interpret the

words of William Douglas?Sample answer: The constitutionassumes that people have thematurity to handle theresponsibilities of democracy.

• Do you think the foundingfathers set up a test to find outwhether or not people do havecommon sense and maturity?Sample answer: No; they assumedthat people did.

POINTS, LINES, ANDPLANES

SNOW CRYSTALS Some snowcrystals are shaped likeregular hexagons. How manylines must be drawn tointerconnect all vertices of ahexagonal snow crystal? 15

Postulates and Paragraph Proofs

Lesson 2-5 Postulates and Paragraph Proofs 89

Vocabulary• postulate• axiom• theorem• proof• paragraph proof• informal proof

were postulates used by the founding fathers of the United States?were postulates used by the founding fathers of the United States?

• Identify and use basic postulates about points, lines, and planes.

• Write paragraph proofs.

U.S. Supreme Court Justice William Douglas stated“The First Amendment makes confidence in thecommon sense of our people and in the maturity oftheir judgment the great postulate of our democracy.”The writers of the constitution assumed that citizenswould act and speak with common sense and maturity.Some statements in geometry also must be assumed oraccepted as true.

PostulatesPostulates

2.1 Through any two points, there is exactly one line.

2.2 Through any three points not on the same line, there is exactly one plane.

Points and LinesCOMPUTERS Jessica is setting up a network for her father’s business. There arefive computers in his office. Each computer needs to be connected to every othercomputer. How many connections does Jessica need to make?

Explore There are five computers, and each is connected to four others.

Plan Draw a diagram to illustrate the solution.

Solve Let noncollinear points A, B, C, D, and E representthe five computers. Connect each point with everyother point. Then, count the number of segments.

Between every two points there is exactly onesegment. So, the connection between computer Aand computer B is the same as the connectionbetween computer B and computer A. For the fivepoints, ten segments can be drawn.

Examine A�B�, A�C�, A�D�, A�E�, B�C�, B�D�, B�E�, C�D�, C�E�, and D�E� each represent aconnection between two computers. So there will be ten connectionsamong the five computers.

A

D

E B

C

Example 1Example 1

Drawing DiagramsWhen listing segments,start with one vertex anddraw all of the segmentsfrom that vertex. Thenmove on to the othervertices until all possiblesegments have beendrawn.

Study Tip

POINTS, LINES, AND PLANES In geometry, a , or , is astatement that describes a fundamental relationship between the basic terms ofgeometry. Postulates are accepted as true. The basic ideas about points, lines, andplanes can be stated as postulates.

axiompostulate

Jeff Hunter/Getty Images








Transparencies

LessonNotes

1 Focus1 Focus

2 Teach2 Teach

22


Teaching Tip Tell students thatmost postulates are very obviousand make very good sense, butthey do not have a formal proofbehind them. Nonetheless,students are to accept them astrue and use them to proveother statements and theorems.

Determine whether eachstatement is always, sometimes,or never true. Explain.

a. If plane T contains EF�� and EF��

contains point G, then planeT contains point G. Always;Postulate 2.5 states that if twopoints lie in a plane, then theentire line containing thosepoints lies in the plane.

b. For XY��, if X lies in plane Q andY lies in plane R , then plane Q intersects plane R .Sometimes; planes Q and R canbe parallel, and XY�� can intersectboth planes.

c. GH�� contains threenoncollinear points. Never;noncollinear points do not lie onthe same line by definition.

Building on PriorKnowledge

Students learned basic principlesabout points, lines, and planes inChapter 1. In this lesson, they willrevisit those concepts in the formof postulates that they can use towrite informal proofs andparagraph proofs.

Answers

Use PostulatesDetermine whether each statement is always, sometimes, or never true. Explain.a. If points A, B, and C lie in plane M, then they are collinear.

Sometimes; A, B, and C do not necessarily have to be collinear to lie in plane M.b. There is exactly one plane that contains noncollinear points P, Q, and R.

Always; Postulate 2.2 states that through any three noncollinear points, there isexactly one plane.

c. There are at least two lines through points M and N.Never; Postulate 2.1 states that through any two points, there is exactly one line.

Example 2Example 2


PostulatesPostulates

2.3 A line contains at least two points.

2.4 A plane contains at least three points not on the same line.

2.5 If two points lie in a plane, then the entire line containing those points liesin that plane.

2.6 If two lines intersect, then their intersection is exactly one point.

2.7 If two planes intersect, then their intersection is a line.

PARAGRAPH PROOFS Undefined terms, definitions, postulates, and algebraicproperties of equality are used to prove that other statements or conjectures are true.Once a statement or conjecture has been shown to be true, it is called a , andit can be used like a definition or postulate to justify that other statements are true.

You will study and use various methods to verify or prove statements andconjectures in geometry. A is a logical argument in which each statement youmake is supported by a statement that is accepted as true. One type of proof iscalled a or . In this type of proof, you write aparagraph to explain why a conjecture for a given situation is true.

informal proofparagraph proof

proof

theorem

ProofsFive essential parts of a good proof:

• State the theorem or conjecture to be proven.

• List the given information.

• If possible, draw a diagram to illustrate the given information.

• State what is to be proved.

• Develop a system of deductive reasoning.

In Lesson 1-2, you learned the relationship between segments formed by themidpoint of a segment. This statement can be proven, and the result stated as atheorem.

There are other postulates that are based on relationships among points, lines, and planes.

ProofsBefore writing a proof,you should have a plan.One strategy is to workbackward. Start with whatyou want to prove, andwork backward step bystep until you reach thegiven information.

Study Tip


Intrapersonal Tell students to read quietly over the postulates andexamples in this lesson and note the differences in the postulatestatements and the statements they are to write proofs for. Advisestudents to go through the text and their study notebooks to compile alist of useful information they could use to write the proofs in this lesson.


1. Deductive reasoning is used tosupport claims that are made in aproof.

2.

3. postulates, theorems, algebraicproperties, definitions



33


PARAGRAPH PROOFS

Given AC�� intersecting CD��,write a paragraph proof toshow that A, C, and Ddetermine a plane.AC�� and CD�� must intersect at Cbecause if two lines intersect,then their intersection is exactly one point. Point A is on AC�� and point D is on CD��. Thereforepoints A and D are not collinear.Therefore ACD is a plane as itcontains three points not on thesame line.

Have students—• add the definitions/examples of



Theorem 2.8Theorem 2.8

Once a conjecture has been proven true, it can be stated as a theorem and used inother proofs. The conjecture in Example 3 is known as the Midpoint Theorem.

Write a Paragraph ProofGiven that M is the midpoint of P�Q�, write a paragraph proof to show that PM—– � MQ–—.Given: M is the midpoint of P�Q�.

Prove: P�M� � M�Q�.

From the definition of midpoint of a segment, PM � MQ. This means that P�M� and M�Q� have the same measure. By the definition of congruence, if two segments have the same measure, then they are congruent. Thus, P�M� � M�Q�.

P

M

Q

Example 3Example 3



4–5, 11 16 2

7–10 3

1. Explain how deductive reasoning is used in a proof. 1–3. See margin.2. OPEN ENDED Draw figures to illustrate Postulates 2.6 and 2.7.3. List the types of reasons that can be used for justification in a proof.

Determine the number of segments that can be drawn connecting each pair of points.4. 6 5. 15

6. Determine whether the following statement is always, sometimes, or nevertrue. Explain. See p. 123C.The intersection of three planes is two lines.

In the figure, BD�� and BR�� are in plane P, and W is on BD��. State thepostulate or definition that can be used toshow each statement is true.7. B, D, and W are collinear.8. E, B, and R are coplanar.9. R and W are collinear.

10. In the figure at the right, P is the midpoint of Q�R� and S�T�, and Q�R� � S�T�. Write a paragraph proof to show that PQ � PT. See p. 123C.

11. DANCING Six students are participating in a dance to celebrate the opening ofa new community center. The students, each connected to each of the otherstudents with wide colored ribbons, will move in a circular motion. How manyribbons are needed? 15 ribbons

Q T

S R

P

PROOF

PD

BW

R

E

Midpoint Theorem If M is the midpoint of AB��, then A�M� � M�B�.

Concept Check

Guided Practice

Application

7. definition ofcollinear.8. Through any threepoints not on the sameline, there is exactlyone plane.9. Through any twopoints, there is exactlyone line.



About the Exercises…Organization by Objective• Points, Lines, and Planes:

12–21• Paragraph Proofs: 22–28


Assignment GuideBasic: 13–19 odd, 23–31 odd,33–48Average: 13–31 odd, 33–48Advanced: 12–30 even, 31–42(optional: 43–48)

Unlocking Misconceptions

Writing Proofs Explain to students that a common mistake in writingproofs is skipping a step or assuming a step that should be included inthe proof. Sometimes, the missed step can be quite obvious, but it stillhas to be included. Tell students to make a habit of listing each piece ofinformation with a separate explanation for each and to avoid using tworeasons or postulates for the same statement when they are writingproofs.


Study Guide and InterventionPostulates and Paragraph Proofs

NAME ______________________________________________ DATE ____________ PERIOD _____

2-52-5


Less

on

2-5

Points, Lines, and Planes In geometry, a postulate is a statement that is accepted astrue. Postulates describe fundamental relationships in geometry.

Postulate: Through any two points, there is exactly one line.Postulate: Through any three points not on the same line, there is exactly one plane.Postulate: A line contains at least two points.Postulate: A plane contains at least three points not on the same line.Postulate: If two points lie in a plane, then the line containing those points lies in the plane.Postulate: If two lines intersect, then their intersection is exactly one point.Postulate: If two planes intersect, then their intersection is a line.

Determine whether each statement is always,sometimes, or never true.a. There is exactly one plane that contains points A, B, and C.

Sometimes; if A, B, and C are collinear, they are contained in many planes. If they arenoncollinear, then they are contained in exactly one plane.

b. Points E and F are contained in exactly one line.Always; the first postulate states that there is exactly one line through any two points.

c. Two lines intersect in two distinct points M and N.Never; the intersection of two lines is one point.

Use postulates to determine whether each statement is always, sometimes, ornever true.

1. A line contains exactly one point. never

2. Noncollinear points R, S, and T are contained in exactly one plane. always

3. Any two lines � and m intersect. sometimes

4. If points G and H are contained in plane M, then G�H� is perpendicular to plane M. never

5. Planes R and S intersect in point T. never

6. If points A, B, and C are noncollinear, then segments A�B�, B�C�, and C�A� are contained inexactly one plane. always

In the figure, A�C� and D�E� are in plane Q and A�C� || D�E�.State the postulate that can be used to show each statement is true.

7. Exactly one plane contains points F, B, and E. Through any three points not on the same line, there isexactly one plane.

8. BE�� lies in plane Q. If two points lie in a plane, then the line containing those points lies in the plane.

QB

C

A

DE

F

G

ExampleExample

ExercisesExercises



Determine the number of line segments that can be drawn connecting each pairof points.

1. 21 2. 28

Determine whether the following statements are always, sometimes, or never true.Explain.

3. The intersection of two planes contains at least two points.Always; the intersection of two planes is a line, and a line contains atleast two points.

4. If three planes have a point in common, then they have a whole line in common.Sometimes; they might have only that single point in common.

In the figure, line m and TQ�� lie in plane A . State the postulate that can be used to show that each statement is true.

5. L, T, and line m lie in the same plane.Postulate 2.5: If two points lie in a plane, then theentire line containing those points lies in that plane.

6. Line m and S�T� intersect at T.Postulate 2.6: If two lines intersect, then their intersection is exactly onepoint.

7. In the figure, E is the midpoint of A�B� and C�D�, and AB � CD. Write a paragraph proof to prove that A�E� � E�D�.

Given: E is the midpoint of A�B� and C�D�AB � CD

Prove: A�E� � E�D�Proof: Since E is the midpoint of A�B� and C�D�, we know by the MidpointTheorem, that A�E� � E�B� and C�E� � E�D�. By the definition of congruent

segments, AE � EB � �12

�AB and CE � ED � �12

�CD. Since AB � CD,

�12

�AB � �12

�CD by the Multiplication Property. So AE � ED, and by the

definition of congruent segments, A�E� � E�D�.

8. LOGIC Points A, B, and C are not collinear. Points B, C, and D are not collinear. PointsA, B, C, and D are not coplanar. Describe two planes that intersect in line BC.the plane that contains A, B, and C and the plane that contains B, C,and D

BE

C

D

A

AmT

QL

S

Practice (Average)

Postulates and Paragraph Proofs

NAME ______________________________________________ DATE ____________ PERIOD _____


Reading to Learn MathematicsPostulates and Paragraph Proofs

NAME ______________________________________________ DATE ____________ PERIOD _____

2-52-5

Pre-Activity How are postulates used by the founding fathers of the United States?


Postulates are often described as statements that are so basic and so clearlycorrect that people will be willing to accept them as true without asking forevidence or proof. Give a statement about numbers that you think mostpeople would accept as true without evidence. Sample answer: Everynumber is equal to itself.

Reading the Lesson1. Determine whether each of the following is a correct or incorrect statement of a

geometric postulate. If the statement is incorrect, replace the underlined words to makethe statement correct. incorrect;a. A plane contains at least that do not lie on the same line. three pointsb. If intersect, then the intersection is a line. correct incorrect;c. Through any not on the same line, there is exactly one plane. three pointsd. A line contains at least . incorrect; two points incorrect;e. If two lines , then their intersection is exactly one point. intersectf. Through any two points, there is one line. incorrect; exactly

2. Determine whether each statement is always, sometimes, or never true. If the statementis not always true, explain why.

a. If two planes intersect, their intersection is a line. alwaysb. The midpoint of a segment divides the segment into two congruent segments. alwaysc. There is exactly one plane that contains three collinear points. never; Sample

answer: There are infinitely many planes if the three points arecollinear, but only one plane if the points are noncollinear.

d. If two lines intersect, their intersection is one point. always

3. Use the walls, floor, and ceiling of your classroom to describe a model for each of thefollowing geometric situations.

a. two planes that intersect in a line Sample answer: two adjacent walls thatintersect at an edge of both walls in the corner of the room

b. two planes that do not intersect Sample answer: the ceiling and the floor (ortwo opposite walls)

c. three planes that intersect in a point Sample answer: the floor (or ceiling)and two adjacent walls that intersect at a corner of the floor (or ceiling)

Helping You Remember4. A good way to remember a new mathematical term is to relate it to a word you already

know. Explain how the idea of a mathematical theorem is related to the idea of a scientifictheory. Sample answer: Scientists do experiments to prove theories;mathematicians use deductive reasoning to prove theorems. Bothprocesses involve using evidence to show that certain statements are true.

at most

are parallel

one point

four points

two planes

two points


Logic ProblemsThe following problems can be solved by eliminating possibilities.It may be helpful to use charts such as the one shown in the firstproblem. Mark an X in the chart to eliminate a possible answer.

Solve each problem.

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

2-52-5

Nancy Olivia Mario Kenji

Peach X X XO X X X

1. Nancy, Olivia, Mario, and Kenji each haveone piece of fruit in their school lunch.They have a peach, an orange, a banana,and an apple. Mario does not have apeach or a banana. Olivia and Mario justcame from class with the student who hasan apple. Kenji and Nancy are sittingnext to the student who has a banana.Nancy does not have a peach. Whichstudent has each piece of fruit?

2. Victor, Leon, Kasha, and Sheri each playone instrument. They play the viola,clarinet, trumpet, and flute. Sheri doesnot play the flute. Kasha lives near thestudent who plays flute and the onewho plays trumpet. Leon does not playa brass or wind instrument. Whichstudent plays each instrument?

Victor—flute,Leon—viola,Kasha—clarinet,Sheri—trumpet

Enrichment, p. 86

Spencer Grant/PhotoEdit



Determine the number of segments that can be drawn connecting each pair of points.

12. 6 13. 10

14. 15 15. 21

Determine whether the following statements are always, sometimes, or never true.Explain. 16–21. See p. 123C.16. Three points determine a plane.

17. Points G and H are in plane X. Any point collinear with G and H is in plane X.

18. The intersection of two planes can be a point.

19. Points S, T, and U determine three lines.

20. Points A and B lie in at least one plane.

21. If line � lies in plane P and line m lies in plane Q , then lines � and m lie in plane R.

In the figure at the right, AC�� and BD�� lie in plane J, and BY�� and CX�� lie in plane K.State the postulate that can be used to show each statement is true. 22–27. See p. 123C.22. C and D are collinear.

23. XB�� lies in plane K.

24. Points A, C, and X are coplanar.

25. AD�� lies in plane J.

26. X and Y are collinear.

27. Points Y, D, and C are coplanar.

28. Point C is the midpoint of A�B�and B is the midpoint of C�D�. Prove that A�C� � B�D�. See margin.

29. MODELS Faith’s teacher asked her to make a figure showing the number oflines and planes formed from four points that are noncollinear and noncoplanar.Faith decided to make a mobile of straws, pipe cleaners, and colored sheets oftissue paper. She plans to glue the paper to the straws and connect the strawstogether to form a group of connected planes. How many planes and lines willshe have? She will have 4 different planes and 6 lines.

30. CAREERS Many professions use deductive reasoning and paragraph proofs.For example, a police officer uses deductive reasoning investigating a trafficaccident and then writes the findings in a report. List a profession, and describehow it can use paragraph proofs. Sample answer: Lawyers make final arguments, which is a speech that uses deductive reasoning, in court cases.

PROOF

J

K

B CA

Y

X

D

ForExercises

12–1516–2122–28

SeeExamples

123



Online ResearchFor information about a career as a detective, visit:www.geometryonline.com/careers

★

★

DetectiveA police detective gathersfacts and collects evidencefor use in criminal cases.The facts and evidence areused together to prove asuspect’s guilt in court.


ELL

http://www.geometryonline.com/careers

Open-Ended AssessmentSpeaking Have students choosea paragraph proof they wroteand explain each step aloud.

Getting Ready forLesson 2-6Prerequisite Skill Students willlearn about algebraic proof inLesson 2-6. Use Exercises 43–48 to determine your students’familiarity with solving equations.

Answers

32. Sample answer: The forms andstructures of different types ofwriting are accepted as valid,such as the structure of a poem.Answers should include thefollowing.• The Declaration of

Independence, “We hold thesetruths to be self-evident, …”

• Through any two points, there isexactly one line.

36. Converse: If you have a computer,then you have access to theInternet at your house. False; youcan have a computer and not haveaccess to the Internet. Inverse: Ifyou do not have access to theInternet at your house, then youdo not have a computer. False; itis possible to not have access tothe Internet and still have acomputer. Contrapositive: If youdo not have a computer, then youdo not have access to the Internetat your house. False; you couldhave Internet access through yourtelevision or wireless phone.

37. Converse: If �ABC has an anglewith measure greater than 90, then�ABC is a right triangle. False; thetriangle would be obtuse. Inverse:If �ABC is not a right triangle,none of its angle measures aregreater than 90. False; it could bean obtuse triangle. Contrapositive:If �ABC does not have an anglemeasure greater than 90, �ABC isnot a right triangle. False; m�ABCcould still be 90 and �ABC be aright triangle.


31. CRITICAL THINKING You know that three noncollinear points lie in a singleplane. In Exercise 29, you found the number of planes defined by fournoncollinear points. What are the least and greatest number of planes definedby five noncollinear points? one, ten

32. Answer the question that was posed at the beginning of thelesson. See margin.

How are postulates used in literature?

Include the following in your answer:• an example of a postulate in historic United States’ documents, and• an example of a postulate in mathematics.

33. Which statement cannot be true? CA plane can be determined using three noncollinear points.Two lines intersect at exactly one point.At least two lines can contain the same two points.A midpoint divides a segment into two congruent segments.

34. ALGEBRA For all values of x, (8x4 � 2x2 � 3x � 5) � (2x4 � x3 � 3x � 5) � A6x4 � x3 � 2x2 � 10. 6x4 � x3 � 2x2 � 6x.

6x4 � 3x2 � 6x � 10. 6x4 � 3x2.DB

CA

D

C

B

A

WRITING IN MATH


35. Determine whether statement (3) follows from statements (1) and (2) by the Lawof Detachment or the Law of Syllogism. If it does, state which law was used. If itdoes not, write invalid. (Lesson 2-4) yes; Law of Detachment(1) Part-time jobs require 20 hours of work per week.(2) Jamie has a part-time job.(3) Jamie works 20 hours per week.

Write the converse, inverse, and contrapositive of each conditional statement.Determine whether each related conditional is true or false. If a statement is false,find a counterexample. (Lesson 2-3) 36–37. See margin.36. If you have access to the Internet at your house, then you have a computer.

37. If �ABC is a right triangle, one of its angle measures is greater than 90.

38. BIOLOGY Use a Venn diagram to illustrate the following statement.If an animal is a butterfly, then it is an arthropod. (Lesson 2-2) See p. 123C.

Use the Distance Formula to find the distance between each pair of points. (Lesson 1-3)

39. D(3, 3), F(4, �1) �17� � 4.1 40. M(0, 2), N(�5, 5) �34� � 5.8

41. P(�8, 2), Q(1, �3) �106� � 10.3 42. R(�5, 12), S(2, 1) �170� � 13.0

PREREQUISITE SKILL Solve each equation.(To review solving equations, see pages 737 and 738.)

43. m � 17 � 8 25 44. 3y � 57 19 45. �y6

� � 12 � 14 12

46. �t � 3 � 27 �24 47. 8n � 39 � 41 10 48. �6x � 33 � 0 �121�

Mixed Review





4 Assess4 Assess

28. Given: C is the midpoint of A�B�.B is the midpoint of C�D�.Prove: A�C� � B�D�Proof: We are given that C is the midpoint of A�B�, and B is themidpoint of C�D�. By the definition of midpoint A�C� � C�B� and C�B� � B�D�.Using the definition of congruent segments, AC � CB, and CB � BD. AC � BD by the Transitive Property of Equality. Thus, A�C� � B�D� by the definition of congruent segments.





is mathematicalevidence similar to

evidence in law?Ask students:• In math, what one thing do you

need to prove a statement isfalse? What evidence can alawyer use to prove thatsomeone is innocent (or hasbeen falsely accused)?a counterexample; sample answer:an alibi

• How does the use of evidence inlaw differ from its use in math?Sample answer: Lawyers presentevidence to sway opinions,sometimes without knowing thetruth. A mathematician presentsevidence to prove factualstatements.

Vocabulary• deductive argument• two-column proof• formal proof

Algebraic Proof


• Use algebra to write two-column proofs.

• Use properties of equality in geometry proofs.

Lawyers develop their cases using logicalarguments based on evidence to lead a jury to a conclusion favorable to their case. At the endof a trial, a lawyer will make closing remarkssummarizing the evidence and testimony that they feel proves their case. These closingarguments are similar to a proof in mathematics.

Verify Algebraic RelationshipsSolve 3(x � 2) � 42.

Algebraic Steps Properties3(x � 2) � 42 Original equation

3x � 6 � 42 Distributive Property

3x � 6 � 6 � 42 � 6 Addition Property

3x � 48 Substitution Property

�33x� � �

438� Division Property

x � 16 Substitution Property

Example 1Example 1

Bob Daemmrich/The Image Works

Commutative and AssociativePropertiesThroughout this text, we shall assume theCommutative andAssociative Properties for addition andmultiplication.

Study Tip

Properties of Equality for Real NumbersReflexive Property For every number a, a � a.

Symmetric Property For all numbers a and b, if a � b, then b � a.

Transitive Property For all numbers a, b, and c, if a � b and b � c, then a � c.

Addition and For all numbers a, b, and c, if a � b, then a � c � b � cSubtraction Properties and a � c � b � c.

Multiplication and For all numbers a, b, and c, if a � b, then a � c � b � cDivision Properties and if c � 0, �a

c� � �b

c�.

Substitution Property For all numbers a and b, if a � b, then a may be replaced by b in any equation or expression

Distributive Property For all numbers a, b, and c, a(b � c) � ab � ac.

ALGEBRAIC PROOF Algebra is a system with sets of numbers, operations,and properties that allow you to perform algebraic operations.

The properties of equality can be used to justify each step when solving an equation.A group of algebraic steps used to solve problems form a .deductive argument

is mathematical evidencesimilar to evidence in law?is mathematical evidencesimilar to evidence in law?

LessonNotes

1 Focus1 Focus


Prerequisite Skills Workbook, pp. 41–44,83–86, 93–94



TechnologyInteractive ChalkboardMultimedia Applications: Virtual Activities



Transparencies

11

22


ALGEBRAIC PROOFS

Solve 2(5 � 3a) � 4(a � 7) � 92.2(5 � 3a) � 4(a � 7) � 92(Original eqn.)10 � 6a � 4a � 28 � 92 (Distr. Prop.)�18 � 10a � 92 (Subst. Prop.)�18 � 10a � 18 � 92 � 18(Add. Prop.)�10a � 110 (Subst. Prop.)

��

1100a

� � ��11

100� (Div. Prop.)

a � �11 (Subst. Prop.)

Teaching Tip Explain thatsince students may havedifferent preferences whensolving algebraic equations,their proofs might vary slightlyfrom the examples. Forexample, one student maydistribute a variable first, whileanother uses addition orsubtraction. Assure students thatas long as they use properties ofequality appropriately, theirproofs will be correct.

Write a two-column proof foreach of the following.

a. If �7d4� 3� � 6, then d � 3.

Statements (Reasons)

1. �7d4� 3� � 6 (Given)

2. 4��7d4� 3� � 4(6) (Mult. Prop.)

3. 7d � 3 � 24 (Substitution)4. 7d � 3 � 3 � 24 � 3

(Subtr. Prop.)5. 7d � 21 (Substitution)

6. �77d� � �

271� (Div. Prop.)

7. d � 3 (Substitution)(continued on the next page)

Lesson 2-6 Algebraic Proof 95

Example 1 is a proof of the conditional statement If 5x � 3(x � 2) � 42, then x � 6.Notice that the column on the left is a step-by-step process that leads to a solution.The column on the right contains the reason for each statement.

In geometry, a similar format is used to prove conjectures and theorems. A, or , contains statements and reasons organized in

two columns. In a two-column proof, each step is called a statement, and theproperties that justify each step are called reasons.

formal prooftwo-column proof

GEOMETRIC PROOF Since geometry also uses variables, numbers, andoperations, many of the properties of equality used in algebra are also true ingeometry. For example, segment measures and angle measures are real numbers, so properties from algebra can be used to discuss their relationships. Some examples of these applications are shown below.

Write a Two-Column ProofWrite a two-column proof.

a. If 3�x � �53

� � 1, then x � 2

Statements Reasons

1. 3�x � �53

�� 1 1. Given

2. 3x � 3��53

�� 1 2. Distributive Property

3. 3x � 5 � 1 3. Substitution4. 3x � 5 � 5 � 1 � 5 4. Addition Property5. 3x � 6 5. Substitution

6. �33x� � �

63

� 6. Division Property

7. x � 2 7. Substitution

b. Given: �72

� � n � 4 � �12

�n

Prove: n � �1Proof:Statements Reasons

1. �72

� � n � 4 � �12

�n 1. Given

2. 2��72

� � n� � 2�4 � �12

�n� 2. Multiplication Property

3. 7 � 2n � 8 � n 3. Distributive Property4. 7 � 2n � n � 8 � n � n 4. Addition Property5. 7 � n � 8 5. Substitution6. 7 � n � 7 � 8 � 7 6. Subtraction Property7. �n � 1 7. Substitution

8. ��

n1� � �

�11� 8. Division Property

9. n � �1 9. Substitution

Mental MathIf your teacher permitsyou to do so, some stepsmay be eliminated byperforming mentalcalculations. For example,in part a of Example 2,statements 4 and 6 couldbe omitted. Then thereason for statements 5would be AdditionProperty and DivisionProperty for statement 7.

Study Tip

Example 2Example 2

Property Segments Angles

Reflexive AB � AB m�1 � m�1

Symmetric If AB � CD, then CD � AB. If m�1 � m�2, then m�2 � m�1.

Transitive If AB � CD and CD � EF, If m�1 � m�2 and m�2 � m�3, then AB � EF. then m�1 � m�3.


Lesson 2-6 Algebraic Proofs 95

2 Teach2 Teach


33

44


b. If 3p � �95� � �

1110� � �1

p0�, then

p � 1.Statements (Reasons)

1. 3p � �95� � �

1110� � �1

p0� (Given)

2. 10�3p � �95� � 10��

1110� � �1

p0�

(Mult. Prop.)3. 30p � 18 � 11 � p

(Distr. Prop.)4. 30p � p � 18 � 11 � p � p

(Subtr. Prop.)5. 29p � 18 � 11 (Substitution)6. 29p � 18 � 18 � 11 � 18

(Add. Prop.)7. 29p � 29 (Substitution)

8. �2299p

� � �2299� (Div. Prop.)

9. p � 1 (Substitution)

GEOMETRIC PROOFS

If GH � JK � ST and S�T� � R�P�, then which of thefollowing is a validconclusion? B

I. GH � JK � RPII. PR � TS

III. GH � JK � ST � RPA I only B I and IIC I and III D I, II, and III

SEA LIFE A starfish has fivearms. If the length of arm 1 is22 cm, and arm 1 is congruentto arm 2, and arm 2 iscongruent to arm 3, prove thatarm 3 has length 22 cm. Weare given arm 1 � arm 2 andarm 2 � arm 3, so by thedefinition of congruence, themeasure of arm 1 � the measureof arm 2 and the measure of arm 2 � the measure of arm 3.By the Transitive Property ofEquality, we know that themeasure of arm 1 � the measureof arm 3. We can then substitute22 cm for the measure of arm 1to prove that the measure of arm3 is 22 cm.

In Example 3, each conclusion was justified using a definition or property. Thisprocess is used in geometry to verify and prove statements.


Justify Geometric RelationshipsMultiple-Choice Test Item

Read the Test ItemDetermine whether the statements are true based on the given information.

Solve the Test ItemStatement I:Examine the given information, A�B� � C�D� and C�D� � E�F�. From the definition ofcongruent segments, if A�B� � C�D� and C�D� � E�F�, then AB � CD and CD � EF. Thus,Statement I is true.Statement II:By the definition of congruent segments, if AB � EF, then A�B� � E�F�. Statement II is true also.Statement III:If AB � CD and CD � EF, then AB � EF by the Transitive Property. Thus, Statement III is true.

Because Statements I, II, and III are true, choice D is correct.

Aaron Haupt

Example 3Example 3

Test-Taking TipMore than one statementmay be correct. Workthrough each problemcompletely beforeindicating your answer.

If A�B� � C�D�, and C�D� � E�F�, then which of the following is a valid conclusion?

I AB � CD and CD � EFII A�B� � E�F�

III AB � EF

I only I and II

I and III I, II, and IIIDC

BA

A

B

C

D

E F

Geometric ProofTIME On a clock, the angle formed by the hands at 2:00 is a 60� angle. If theangle formed at 2:00 is congruent to the angle formed at 10:00, prove that theangle at 10:00 is a 60� angle.

Given: m�2 � 60�2 � �10

Prove: m�10 � 60

Proof:Statements Reasons1. m�2 � 60 1. Given

�2 � �102. m�2 � m�10 2. Definition of congruent angles3. 60 � m�10 3. Substitution4. m�10 � 60 4. Symmetric Property

Example 4Example 4



Interpersonal Let groups of students work one or two selectedproblems from Exercises 24–29 on p. 98. Stipulate that each groupmember should contribute at least one step of the proof. Encouragegroups to brainstorm beforehand to determine the properties they willuse and the order they will use them in. Allow the groups to check andcompare their proofs when they are done to see if any two groupsfound different ways to prove the same statement.





• include a sample algebraic proofand a sample geometric proof.



ForExercises15, 16, 20

14, 17–19, 2122–2728, 29

SeeExamples

1234



State the property that justifies each statement.14. If m�A � m�B and m�B � m�C, m�A � m�C. Trans. Prop.

15. If HJ � 5 � 20, then HJ � 15. Subt. Prop.16. If XY � 20 � YW and XY � 20 � DT, then YW � DT. Substitution17. If m�1 � m�2 � 90 and m�2 � m�3, then m�1 � m�3 � 90. Substitution

18. If �12

�AB � �12

�EF, then AB � EF. Div. or Mult. Prop.

19. AB � AB Reflexive Property


Guided Practice

1. OPEN ENDED Write a statement that illustrates the Substitution Property of Equality. 1–2. See margin.

2. Describe the parts of a two-column proof.3. State the part of a conditional that is related to the Given statement of a proof.

What part is related to the Prove statement? hypothesis; conclusion

State the property that justifies each statement.

4. If 2x � 5, then x � �52

� Division Property

5. If �x2

� � 7, then x � 14. Multiplication Property

6. If x � 5 and b � 5, then x � b. Substitution Property

7. If XY � AB � WZ � AB, then XY � WZ. Addition Property

8. Solve �x2

� � 4x � 7 � 11. List the property that justifies each step. See margin.

9. Complete the following proof.

Given: 5 � �23

�x � 1

Prove: x � 6

Proof:

Statements Reasonsa. 5 � �2

3�x � 1 a. Given

b. 3�5 � �23

�x� � 3(1) b. Mult. Prop.c. 15 � 2x � 3 c. Dist. Prop.d. �2x � �12 d. Subtraction Prop.e. x � 6 e. Div. Prop.

Write a two-column proof. 10–12. See pp. 123C–123D.10. Prove that if 25 � �7(y � 3) � 5y, then �2 � y.11. If rectangle ABCD has side lengths AD � 3 and AB � 10, then AC � BD.12. The Pythagorean Theorem states that in a right triangle ABC, c2 � a2 � b2.

Prove that a � �c2 � b�2�.

13. ALGEBRA If 8 � x = 12, then 4 � x = . C28 24 0 4DCBA

?

PROOF

??

?

?

?


4–7 18 3

9, 10, 12 211 4

Concept Check



About the Exercises…Organization by Objective• Algebraic Proofs: 14–21• Geometric Proofs: 22–29


Assignment GuideBasic: 15–31 odd, 32–33, 35–51Average: 15–31 odd, 32–33,35–51Advanced: 14–30 even, 32–48(optional: 49–51)All: Quiz 2 (1–5)

Answers

1. Sample answer: If x � 2 and x � y � 6, then 2 � y � 6.

2. given and prove statements andtwo columns, one of statementsand one of reasons

8. Given: �2x

� � 4x � 7 � 11Prove: x � 4Proof:Statements (Reasons)

1. �2x

� � 4x � 7 � 11 (Given)

2. 2��2x

� � 4x � 7 � 2(11)

(Mult. Prop.) 3. x � 8x � 14 � 22 (Dist.Prop.)4. 9x � 14 � 22 (Substitution)5. 9x � 36 (Add. Prop.)6. x � 4 (Div. Prop.)

Answers

32. Given: Ek � hf � W

Prove: f � �Ek �

hW

�

Proof:Statements (Reasons)1. Ek � hf � W (Given)2. Ek � W � hf (Subt. Prop.)

3. �Ek �

hW

� � f (Div. Prop.)

4. f � �Ek �

hW

� (Sym. Prop.)

36. Sample answer: Lawyers useevidence and testimony asreasons for justifying statementsand actions. All of the evidenceand testimony are linked togetherto prove a lawyer’s case, much asin a proof in mathematics.Answers should include thefollowing.• Evidence is used to verify facts

from witnesses or materials.• Postulates, theorems,

definitions, and properties canbe used to justify statementsmade in mathematics.

20. If 2�x � �32

�� 5, which property can be used to support the statement 2x � 3 � 5?

21. Which property allows you to state m�4 � m�5, if m�4 � 35 and m�5 � 35?Substitution

22. If �12

�AB � �12

�CD, which property can be used to justify the statement AB � CD?

23. Which property could be used to support the statement EF � JK, given that EF � GH and GH � JK? Transitive Prop.

Complete each proof.

24. Given: �3x

2� 5� � 7

Prove: x � 3

Proof:Statements Reasons

a. �3x

2� 5� � 7 a. Given

b. 2��3x2� 5� � 2(7) b. Mult. Prop.

c. 3x � 5 � 14 c. Substitutiond. 3x � 9 d. Subt. Prop.e. x � 3 e. Div. Prop.

25. Given: 2x � 7 � �13

�x � 2Prove: x � 3


a. 2x � 7 � �13

�x � 2 a. Given

b. 3(2x � 7) � 3��13

�x � 2 b. Mult. Prop.c. 6x � 21 � x � 6 c. Dist. Prop.d. 5x � 21 � �6 d. Subt. Prop.e. 5x � 15 e. Add. Prop.f. x � 3 f. Div. Prop.

Write a two-column proof. 26–31. See p. 123D.

26. If 4 � �12

�a � �72

� � a, then a � �1. 27. If �2y � �32

� � 8, then y � ��143�.

28. If � �12

� m � 9, then m � �18. 29. If 5 � �23

�z � 1, then z � 6.

30. If XZ � ZY, XZ � 4x � 1, 31. If m�ACB � m�ABC,and ZY � 6x � 13, then x � 7. then m�XCA � m�YBA.

32. PHYSICS Kinetic energy is the energy of motion. The formula for kinetic energy is Ek � h � f � W, where h represents Planck’s Constant, f represents thefrequency of its photon, and W represents the work function of the materialbeing used. Solve this formula for f and justify each step. See margin.

X C B Y

AX Z

Y

4x � 1

6x � 13

PROOF

??

??

?

?

???

?

?


Dist. Prop.

Duomo/CORBIS

PhysicsA gymnast exhibits kineticenergy when performingon the balance beam. Themovements and flips showthe energy that is beingdisplayed while thegymnast is moving.Source: www.infoplease.com

Div. or Mult. Prop.


Study Guide and InterventionAlgebraic Proof

NAME ______________________________________________ DATE ____________ PERIOD _____

2-62-6


Less

on

2-6

Algebraic Proof The following properties of algebra can be used to justify the stepswhen solving an algebraic equation.

Property Statement

Reflexive For every number a, a � a.

Symmetric For all numbers a and b, if a � b then b � a.

Transitive For all numbers a, b, and c, if a � b and b � c then a � c.

Addition and Subtraction For all numbers a, b, and c, if a � b then a � c � b � c and a � c � b � c.

Multiplication and Division For all numbers a, b, and c, if a � b then a � c � b � c, and if c � 0 then �ac

� � �bc

�.

Substitution For all numbers a and b, if a � b then a may be replaced by b in any equation or expression.

Distributive For all numbers a, b, and c, a(b � c) � ab � ac.

Solve 6x � 2(x � 1) � 30.

Algebraic Steps Properties6x � 2(x � 1) � 30 Given

6x � 2x � 2 � 30 Distributive Property

8x � 2 � 30 Substitution

8x � 2 � 2 � 30 � 2 Addition Property

8x � 32 Substitution

�88x� � �

382� Division Property

x � 4 Substitution


ExampleExample

ExercisesExercises

1. Given: �4x

2� 6� � 9

Prove: x � 3

Statements Reasons

a. �4x

2� 6� � 9 a. Given

b. 2��4x2� 6�� 2(9) b. Mult. Prop.

c. 4x � 6 � 18 c. Subst.d. 4x � 6 � 6 � 18 � 6 d. Subtr. Prop.e. 4x � 12 e. Substitution

f. �44x� � �

142� f. Div. Prop.

g. x � 3 g. Substitution

2. Given: 4x � 8 � x � 2Prove: x � �2

Statements Reasons

a. 4x � 8 � x � 2 a. Givenb. 4x � 8 � x �

x � 2 � x b. Subtr. Prop.c. 3x � 8 � 2 c. Substitution

d. 3x � 8 � 8 �2 � 8 d. Subtr. Prop.

e. 3x � �6 e. Substitution

f. �33x� � �

�36� f. Div. Prop.

g. x � �2 g. Substitution



PROOF Write a two-column proof.

1. If m�ABC � m�CBD � 90, m�ABC � 3x � 5,

and m�CBD � �x �

21

�, then x � 27.

Given: m�ABC � m�CBD � 90m�ABC � 3x � 5m� CBD � �

x �2

1�

Prove: x � 27Proof:Statements Reasons

1. m�ABC � m�CBD � 90 1. Givenm�ABC � 3x � 5m�CBD � �

x �2

1�

2. 3x � 5 � �x �

21

� � 90 2. Substitution Property

3. (2)(3x � 5) � (2)��x �2

1� � (2)90 3. Multiplication Property

4. 6x � 10 � x � 1 � 180 4. Substitution Property5. 7x � 9 � 180 5. Substitution Property6. 7x � 9 � 9 � 180 � 9 6. Addition Property7. 7x � 189 7. Substitution Property

8. �77x� � �

1879


9. x � 27 9. Substitution Property

2. FINANCE The formula for simple interest is I � prt, where I is interest, p is principal,r is rate, and t is time. Solve the formula for r and justify each step.Given: I � prtProve: r � �

pIt�

Proof:Statements Reasons1. I � prt 1. Given

2. �pIt� � �

pprtt


3. �pIt� � r 3. Substitution Property

4. r � �pIt� 4. Symmetric Property

A

D C

B

Practice (Average)

Algebraic Proof

NAME ______________________________________________ DATE ____________ PERIOD _____


Reading to Learn MathematicsAlgebraic Proof

NAME ______________________________________________ DATE ____________ PERIOD _____

2-62-6

Less

on

2-6

Pre-Activity How is mathematical evidence similar to evidence in law?


What are some of the things that lawyers might use in presenting theirclosing arguments to a trial jury in addition to evidence gathered prior tothe trial and testimony heard during the trial? Sample answer: Theymight tell the jury about laws related to the case, courtrulings, and precedents set by earlier trials.

Reading the Lesson1. Name the property illustrated by each statement.

a. If a � 4.75 and 4.75 � b, then a � b. Transitive Property of Equalityb. If x � y, then x � 8 � y � 8. Addition Property of Equalityc. 5(12 � 19) � 5 � 12 � 5 � 19 Distributive Property Substitution Propertyd. If x � 5, then x may be replaced with 5 in any equation or expression. of Equalitye. If x � y, then 8x � 8y. Multiplication Property of Equalityf. If x � 23.45, then 23.45 � x. Symmetric Property of Equalityg. If 5x � 7, then x � �

75�. Division Property of Equality

h. If x � 12, then x � 3 � 9. Subtraction Property of Equality

2. Give the reason for each statement in the following two-column proof.Given: 5(n � 3) � 4(2n � 7) � 14Prove: n � 9Statements Reasons

1. 5(n � 3) � 4(2n � 7) � 14 1. Given2. 5n � 15 � 8n � 28 � 14 2. Distributive Property3. 5n � 15 � 8n � 42 3. Substitution Property4. 5n � 15 � 15 � 8n � 42 � 15 4. Addition Property5. 5n � 8n � 27 5. Substitution Property6. 5n � 8n � 8n � 27 � 8n 6. Subtraction Property7. �3n � �27 7. Substitution Property

8. ��33n

� � ��237


9. n � 9 9. Substitution Property

Helping You Remember3. A good way to remember mathematical terms is to relate them to words you already know.

Give an everyday word that is related in meaning to the mathematical term reflexive andexplain how this word can help you to remember the Reflexive Property and to distinguishit from the Symmetric and Transitive Properties. Sample answer: Reflection: If youlook at your reflection, you see yourself. The Reflexive Property says thatevery number is equal to itself. The Reflexive Property involves only onenumber, while the Symmetric and Transitive Properties each involve twoor three numbers.


Symmetric, Reflexive, and Transitive PropertiesEquality has three important properties.

ReflexiveSymmetricTransitive

Other relations have some of the same properties. Consider therelation “is next to” for objects labeled X, Y, and Z. Which of theproperties listed above are true for this relation?

X is next to X. FalseIf X is next to Y, then Y is next to X. TrueIf X is next to Y and Y is next to Z, then X is next to Z. False

Only the symmetric property is true for the relation “is next to.”

For each relation, state which properties (symmetric, reflexive,transitive) are true.

a � aIf a � b, then b � a.If a � b and b � c, then a � c.

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____



33. GARDENING Areas in the southwestand southeast have cool but mild winters.In these areas, many people plant pansiesin October so that they have flowersoutside year-round. In the arrangement ofpansies shown, the walkway divides thetwo sections of pansies into four beds thatare the same size. If m�ACB � m�DCE,what could you conclude about therelationship among �ACB, �DCE, �ECF,and �ACG? All of the angle measureswould be equal.

CRITICAL THINKING For Exercises 34 and 35, use the following information.Below is a family tree of the Gibbs family. Clara, Carol, Cynthia, and Cheryl are alldaughters of Lucy. Because they are sisters, they have a transitive and symmetricrelationship. That is, Clara is a sister of Carol, Carol is a sister of Cynthia, so Clara is a sister of Cynthia.

34. What other relationships in a family have reflexive, symmetric, or transitiverelationships? Explain why. Remember that the child or children of each personare listed beneath that person’s name. Consider relationships such as firstcousin, ancestor or descendent, aunt or uncle, sibling, or any other relationship.

35. Construct your family tree on one or both sides of your family and identify thereflexive, symmetric, or transitive relationships. See students’ work.

36. Answer the question that was posed at the beginning ofthe lesson. See margin.

How is mathematical evidence similar to evidence in law?Include the following in your answer:• a description of how evidence is used to influence jurors’ conclusions

in court, and• a description of the evidence used to make conclusions in mathematics.

37. In �PQR, m�P � m �Q and m�R � 2(m�Q). Find m�P if m�P � m�Q + m�R � 180. B

30 45

60 90

38. ALGEBRA If 4 more than x is 5 less than y, what is x in terms of y? By � 1 y � 9 y � 9 y � 5DCBA

DC

BA

P

QR

WRITING IN MATH

Lucy

Cheryl

Michael Chris Kevin

CynthiaCarolClara

Cyle Ryan Allycia Maria

Diane Dierdre Steven

34. Sample answersare: Michael has asymmetric relationshipof first cousin withChris, Kevin, Diane,Dierdre, and Steven.Diane, Dierdre, and Steve have asymmetric andtransitive relationshipof sibling. Any directline from bottom to top has a transitivedescendentrelationship.

A

EF

G

D

B

C




ELL


Open-Ended AssessmentWriting Select some statementsto prove and write them on theboard. Have different volunteerscome up to the board and writeone statement and reason toadvance the proof until thestudents have proven theoriginal statement.

Getting Ready forLesson 2-7Prerequisite Skill Students willlearn about proving segmentrelationships in Lesson 2-7. Theywill learn about segment additionand will use segment measuresto prove segment congruence.Use Exercises 51–53 to determineyour students’ familiarity withsegment measures.

Assessment OptionsPractice Quiz 2 The quizprovides students with a briefreview of the concepts and skillsin Lessons 2-4 through 2-6.Lesson numbers are given to theright of the exercises orinstruction lines so students canreview concepts not yetmastered.Quiz (Lessons 2-5 and 2-6) isavailable on p. 120 of the Chapter 2Resource Masters.

Answers

43. If people are happy, then theyrarely correct their faults.

44. If you don’t know where you aregoing, then you will probably endup somewhere else.

45. If a person is a champion, thenthe person is afraid of losing.

46. If we would have new knowledge,then we must get a whole newworld of questions.


40. Valid; since 24 isdivisible by 6, the Lawof Detachment says itis divisible by 3.41. Invalid; 27 � 6 �4.5, which is not aninteger.42. Valid; since 85 isnot divisible by 3, thecontrapositive of thestatement and the Lawof Detachment saythat 85 is not divisibleby 6.


39. CONSTRUCTION There are four buildings on the Medfield High SchoolCampus, no three of which stand in a straight line. How many sidewalks needto be built so that each building is directly connected to every other building?(Lesson 2-5) 6

Determine whether the stated conclusion is valid based on the given information.If not, write invalid. Explain your reasoning. A number is divisible by 3 if it isdivisible by 6. (Lesson 2-4)

40. Given: 24 is divisible by 6. Conclusion: 24 is divisible by 3.41. Given: 27 is divisible by 3. Conclusion: 27 is divisible by 6.42. Given: 85 is not divisible by 3. Conclusion: 85 is not divisible by 6.

Write each statement in if-then form. (Lesson 2-3) 43–46. See margin.43. “Happy people rarely correct their faults.” (La Rochefoucauld)44. “If you don’t know where you are going, you will probably end up

somewhere else.” (Laurence Peters)45. “A champion is afraid of losing.” (Billie Jean King)46. “If we would have new knowledge, we must get a whole new world of

questions.” (Susanne K. Langer)

Find the precision for each measurement. (Lesson 1-2)

47. 13 feet 48. 5.9 meters 49. 74 inches 50. 3.1 kilometers�12

� ft 0.05 m 0.5 in. 0.05 kmPREREQUISITE SKILL Find the measure of each segment.(To review segment measures, see Lesson 1-2.)

51. K�L� 11 52. Q�S� 28 53. W�Z� 47

938

W X ZY

5123

P Q SR

2514

J K L


Practice Quiz 2Practice Quiz 2

1. Determine whether statement (3) follows from statements (1) and (2) by the Law of Detachment or theLaw of Syllogism. If it does, state which law was used. If it does not, write invalid. (Lesson 2-4)

(1) If n is an integer, then n is a real number.(2) n is a real number.(3) n is an integer. invalid

In the figure at the right, A, B, and C are collinear. Points A, B, C,and D lie in plane N. State the postulate or theorem that can beused to show each statement is true. (Lesson 2-5) 2–4. See margin.2. A, B, and D determine plane N.

3. BE�� intersects AC�� at B.

4. � lies in plane N.

5. If 2(n � 3) � 5 � 3(n � 1), prove that n � 2. (Lesson 2-6)

NBA

CD

E

�

PROOF

Lessons 2-4 through 2-6

Mixed Review

See p. 123E.


4 Assess4 Assess

Answers

Practice Quiz 22. Through any three points not on the same line,

there is exactly one plane.3. If two lines intersect, then their intersection is

exactly one point.4. If two points lie in a plane, then the entire line

containing those points lies in that plane.




can segmentrelationships be used

for travel?Ask students:• Use a ruler to measure the

length in millimeters from SanDiego to Phoenix and fromPhoenix to Dallas. about 11 mm;about 32 mm

• Given the pilot’s information,how many miles will Janelle beflying from San Diego toDallas? 1430

• How are segment lengthshelpful for air travel? Sampleanswer: Pilots can use segmentlengths to calculate distance, flighttime and necessary fuelrequirements.

Segment Measures

CA B

AB = 1.79 cmBC = 3.21 cmAC = 5.00 cm

Proving Segment Relationships

Lesson 2-7 Proving Segment Relationships 101

• Write proofs involving segment addition.

• Write proofs involving segment congruence.

When leaving San Diego, the pilot saidthat the flight would be about 360 milesto Phoenix before continuing on toDallas. When the plane left Phoenix,the pilot said that the flight would beflying about 1070 miles to Dallas.

Ruler Postulate The points on any line or line segment can be paired with real numbers so that, given any two points A and B on a line, A corresponds to zero, and B corresponds to a positive real number.

0

A B

Postulate 2.8Postulate 2.8

The Ruler Postulate can be used to further investigate line segments.

Adding Segment Measures

Construct a Figure• Use The Geometer’s Sketchpad to

construct A�C�.• Place point B on A�C�.• Find AB, BC, and AC.

Analyze the Model1. What is the sum AB � BC?2. Move B. Find AB, BC and AC.

What is the sum of AB � BC? 3. Repeat moving B, measuring the segments, and finding the sum AB � BC

three times. Record your results.

Make a Conjecture4. What is true about the relationship of AB, BC, and AC? AB � BC � AC5. Is it possible to place B on A�C� so that this relationship is not true? no

1–3. See students’ work. The sum AB � BCshould always equal AC.

Phoenix

Dallas

SanDiego

1/2 inch = 400 mi.

can segment relationships be used for travel?can segment relationships be used for travel?

SEGMENT ADDITION In Lesson 1-2, you measured segments with a ruler byplacing the mark for zero on one endpoint, then finding the distance to the otherendpoint. This illustrates the Ruler Postulate.



School-to-Career Masters, p. 4Prerequisite Skills Workbook, pp. 89–90,

101–104Teaching Geometry With Manipulatives

Masters, p. 8





Transparencies

LessonNotes

1 Focus1 Focus

11


SEGMENT ADDITION

Prove the following. Use thefigure from Example 1 in theStudent Edition.Given: PR � QSProve: PQ � RSStatements (Reasons)1. PR � QS (Given)2. PR � QR � QS � QR

(Subtr. Prop.)3. PR � QR � PQ ;

QS � QR � RS(Seg. Add. Post.)

4. PQ � RS (Substitution)

Teaching Tip Tell students thatwith each new lesson, they areaccumulating more postulates andtheorems that they can use forwriting proofs. Encourage studentsto practice using these concepts asmuch as possible before moving onto the next lesson to strengthentheir ability to recall important factsfor proof-writing skills.

Examine the measures AB, BC, and AC in the Geometry Activity. Notice thatwherever B is placed between A and C, AB � BC � AC. This suggests the followingpostulate.


Proof With Segment AdditionProve the following.

Given: PQ � RS

Prove: PR � QS

Proof:

Statements Reasons

1. PQ � RS 1. Given2. PQ � QR � QR � RS 2. Addition Property3. PQ � QR � PR 3. Segment Addition Postulate

QR � RS � QS4. PR � QS 4. Substitution

P Q R S

Example 1Example 1

SEGMENT CONGRUENCE In Lesson 2-5, you learned that once a theorem isproved, it can be used in proofs of other theorems. One theorem we can prove issimilar to properties of equality from algebra.

Transitive Property of CongruenceGiven: M�N� � P�Q�

P�Q� � R�S�

Prove: M�N� � R�S�

Proof:

Method 1 Paragraph Proof

Since M�N� � P�Q� and P�Q� � R�S�, MN � PQ and PQ � RS by the definition of congruent segments. By the Transitive Property of Equality, MN � RS. Thus, M�N� � R�S� by the definition of congruent segments.

R

M Q

N P

S

Segment Addition Postulate If B is between A and C, then AB � BC � AC.

If AB � BC � AC, then B is between A and C. AC

A B C

BCAB


BetweennessIn general, the definitionof between is that B isbetween A and C if A, B,and C are collinear andAB � BC � AC.

Study Tip

Segment CongruenceCongruence of segments is reflexive, symmetric, and transitive.

Reflexive Property A�B� � A�B�

Symmetric Property If A�B� � C�D�, then C�D� � A�B�.

Transitive Property If A�B� � C�D�, and C�D� � E�F�, then A�B� � E�F�.


ProofProof

You will prove the first two properties in Exercises 10 and 24.


2 Teach2 Teach

Geometry Software Investigation

Adding Segment Measures Have students repeat the activity for different lengths of A�C�. Students can also construct A�C� vertically and at variousdiagonals. Tell students that this activity provides several examples tosubstantiate the Segment Addition Postulate.

22


SEGMENT CONGRUENCE

Prove the following.

Given: WY � YZY�Z� � X�Z�X�Z� � W�X�

Prove: W�X� � W�Y�Proof:Statements (Reasons)1. WY � YZ (Given)2. W�Y� �Y�Z� (Def. of � Segs.)3. Y�Z� � X�Z�; X�Z� � W�X� (Given)4. W�Y� � W�X� (Trans. Prop.)5. W�X� � W�Y� (Symmetric)

Answers

1. Sample answer: The distancefrom Cleveland to Chicago is thesame as the distance fromCleveland to Chicago.

2. Sample answer: If A�B� � X�Y� andX�Y� � P�Q�, then A�B� � P�Q�.

P

Q

A

B

X

Y

3 cm

3 cmY Z

XW

C Squared Studios/PhotoDisc (t) file photo (b)


Method 2 Two-Column Proof

Statements Reasons

1. M�N� � P�Q�, P�Q� � R�S� 1. Given2. MN � PQ, PQ � RS 2. Definition of congruent segments3. MN � RS 3. Transitive Property4. M�N� � R�S� 4. Definition of congruent segments

The theorems about segment congruence can be used to prove segmentrelationships.


6 14, 5, 7–10 2

Concept Check

Guided Practice

1. Choose two cities from a United States road map. Describe the distance betweenthe cities using the Reflexive Property. See margin.

2. OPEN ENDED Draw three congruent segments, and illustrate the TransitiveProperty using these segments. See margin.

3. Describe how to determine whether a point B is between points A and C.If A, B, and C are collinear and AB � BC � AC, then B is between A and C.

Justify each statement with a property of equality or a property of congruence.4. X�Y� � X�Y� Reflexive5. If G�H� � M�N�, then M�N� � G�H�. Symmetric6. If AB � AC � CB, then AB � AC � CB. Subtraction

7. Copy and complete the proof.Given: P�Q� � R�S�,� Q�S� � S�T�Prove: P�S� � R�T�Proof:

Statements Reasons

a. , a. Givenb. PQ � RS, QS � ST b. Def. of � segs.c. PS � PQ � QS, RT � RS � ST c. Segment Addition Post.d. PQ � QS � RS � ST d. Addition Propertye. PS � RT e. Substitutionf. P�S� � R�T� f. Def. of � segs.?

??

??

??

R S

Q

P

T

Proof With Segment CongruenceProve the following.Given: J�K� � K�L�, H�J� � G�H�, K�L� � H�J�Prove: G�H� � J�K�Proof:Statements Reasons

1. J�K� � K�L�, K�L� � H�J� 1. Given2. J�K� � H�J� 2. Transitive Property3. H�J� � G�H� 3. Given4. J�K� � G�H� 4. Transitive Property5. G�H� � J�K� 5. Symmetric Property

Example 2Example 2

7a. P�Q� � R�S�, Q�S� � SS�T�

J K

H

G

L



Visual/Spatial When students are first examining figures to determinethe steps necessary for writing proofs, encourage them to use their spatialskills to locate obvious and hidden congruent segments and parts ofsegments that qualify for the Segment Addition Postulate. Advise studentsto use the given information to mark the figures so they can easily refer tothe relationships in the figures while they are writing their proofs.






• include a sample proof usingsegment addition and one usingsegment congruence.



Application


Justify each statement with a property of equality or a property of congruence.12. If J�K� � L�M�, then L�M� � J�K�. Symmetric13. If AB � 14 and CD � 14, then AB � CD. Substitution14. If W, X, and Y are collinear, in that order, then WY � WX � XY.15. If M�N� � P�Q� and P�Q� � R�S�, then M�N� � R�S�. Transitive16. If EF � TU and GH � VW, then EF � GH � TU � VW. Addition17. If JK � MN � JK � QR, then MN � QR. Subtraction

18. Copy and complete the proof.Given: A�D� � C�E�, D�B� � E�B�

Prove: A�B� � C�B�


a. A�D� � C�E�, D�B� � E�B� a. Givenb. AD � CE, DB � EB b. Def. of � segs.c. AD � DB � CE � EB c. Add. Prop.d. d. Segment Addition Postulatee. AB � CB e. Substitutionf. A�B� � C�B� f. Def. of � segs.

Write a two-column proof. 19–20. See p. 123E.19. If X�Y� � W�Z� and W�Z� � A�B�, 20. If A�B� � A�C� and P�C� � Q�B�,

then X�Y� � A�B�. then A�P� � A�Q�.

C

A

P Q

B

A

BW

Z

X

Y

PROOF

??

???

?

B

A C

D E

SegmentAddition

For Exercises 8–10, write a two-column proof. 8–9. See p. 123E.8. Given: A�P� � C�P� 9. Given: H�I� � T�U�

B�P� � D�P� H�J� � T�V�Prove: A�B� � C�D� Prove: I�J� � U�V�

10. Symmetric Property of Congruence (Theorem 2.2) See margin.

11. GEOGRAPHY Aberdeen in South Dakota and Helena, Miles City, andMissoula, all in Montana, are connected in a straight line by interstate highways.Missoula is 499 miles from Miles City and 972 miles from Aberdeen. Aberdeen is473 miles from Miles City and 860 miles from Helena. Between which cities doesHelena lie? Helena is between Missoula and Miles City.

H

I

J T U V

A

C

PD

B

PROOF

18d. AB � AD � DB, CB � CE � EB

ForExercises14, 16, 1712, 13, 15,

18–24

SeeExamples

12




About the Exercises…Organization by Objective• Segment Addition: 14, 16, 17• Segment Congruence: 12, 13,

15, 18–24


Assignment GuideBasic: 13–27 odd, 29–45Average: 13–27 odd, 29–45Advanced: 12–26 even, 27–39(optional: 40–45)

Answer

10. Given: A�B� � C�D�Prove: C�D� � A�B�

Proof:Statements (Reasons)

1. A�B� � C�D� (Given)2. AB � CD (Def. of � segs.)3. CD � AB (Symmetric Prop.)4. C�D� � A�B� (Def. of � segs.)

A

B

CD

Study Guide and InterventionProving Segment Relationships

NAME ______________________________________________ DATE ____________ PERIOD _____

2-72-7


Less

on

2-7

Segment Addition Two basic postulates for working with segments and lengths arethe Ruler Postulate, which establishes number lines, and the Segment Addition Postulate,which describes what it means for one point to be between two other points.

Ruler PostulateThe points on any line or line segment can be paired with real numbers so that, given any twopoints A and B on a line, A corresponds to zero and B corresponds to a positive real number.

Segment Addition Postulate

B is between A and C if and only if AB � BC � AC.

Write a two-column proof.Given: Q is the midpoint of P�R�.

R is the midpoint of Q�S�.Prove: PR � QS

Statements Reasons

1. Q is the midpoint of P�R�. 1. Given2. PQ � QR 2. Definition of midpoint3. R is the midpoint of Q�S�. 3. Given4. QR � RS 4. Definition of midpoint5. PQ � QR � QR � RS 5. Addition Property6. PQ � QR � PR, QR � RS � QS 6. Segment Addition Postulate7. PR � QS 7. Substitution


PQ

RS

ExampleExample

ExercisesExercises

1. Given: BC � DEProve: AB � DE � AC

Statements Reasonsa. BC � DE a. Givenb. AB � BC � AC b. Seg. Add. Post.

c. AB � DE � AC c. Substitution

A BC

DE

2. Given: Q is betweenP and R, R is between Q and S, PR � QS.Prove: PQ � RS

Statements Reasons

a.Q is between a. GivenP and R.

b.PQ � QR � PR b. Seg. Add. Post.c. R is between c. Given

Q and S.d.QR � RS � QS d. Seg. Add. Post.e. PR � QS e. Givenf. PQ � QR � f. Substitution

QR � RSg.PQ � QR � QR � g. Subtraction

QR � RS � QR Prop.h.PQ � RS h. Substitution

P QR S



Complete the following proof.

1. Given: A�B� � D�E�B is the midpoint of A�C�.E is the midpoint of D�F�.

Prove: B�C� � E�F�Proof:

Statements Reasons

a. A�B� � D�E� a. Given

B is the midpoint of A�C�.E is the midpoint of D�F�.

b. AB � DE b. Definition of � segmentsc. AB � BC c. Definition of Midpoint

DE � EFd. AC � AB � BC d. Segment Addition Postulate

DF � DE � EF

e. AB � BC � DE � EF e. Substitution Propertyf. AB � BC � AB � EF f. Substitution Propertyg. AB � BC � AB � AB � EF � AB g. Subtraction Property

h. BC � EF h. Substitution Propertyi. B�C� � E�F� i. Definition of � segments

2. TRAVEL Refer to the figure. DeAnne knows that the distance from Grayson to Apex is the same as the distancefrom Redding to Pine Bluff. Prove that the distance fromGrayson to Redding is equal to the distance from Apex to Pine Bluff.

Given: G�A� � R�P�Prove: G�R� � A�P�Proof:Statements Reasons

1. G�A� � R�P� 1. Given2. GA � RP 2. Definition of � segments3. GA � AR � AR � RP 3. Addition Property4. GR � GA � AR, AP � AR � RP 4. Segment Addition Postulate5. GR � AP 5. Substitution Property6. G�R� � A�P� 6. Definition of � segments

Grayson Apex Redding Pine Bluff

G A R P

CA B

FD

E

Practice (Average)

Proving Segment Relationships

NAME ______________________________________________ DATE ____________ PERIOD _____


Reading to Learn MathematicsProving Segment Relationships

NAME ______________________________________________ DATE ____________ PERIOD _____

2-72-7

Less

on

2-7

Pre-Activity How can segment relationships be used for travel?


• What is the total distance that the plane will fly to get from San Diego toDallas? 1430 mi

• Before leaving home, a passenger used a road atlas to determine that thedistance between San Diego and Dallas is about 1350 miles. Why is theflying distance greater than that? Sample answer: Phoenix is noton a straight line between San Diego and Dallas, so the stopadded to the distance traveled. A nonstop flight would havebeen shorter.

Reading the Lesson1. If E is between Y and S, which of the following statements are always true? B, E

A. YS � ES � YE B. YS � ES � YEC. YE ES D. YE � ES � YSE. SE � EY � SY F. E is the midpoint of Y�S�.

2. Give the reason for each statement in the following two-column proof.Given: C is the midpoint of B�D�.

D is the midpoint of C�E�.Prove: B�D� � C�E�Statements Reasons

1. C is the midpoint of B�D�. 1. Given2. BC � CD 2. Definition of midpoint3. D is the midpoint of C�E�. 3. Given4. CD � DE 4. Definition of midpoint5. BC � DE 5. Transitive Property of Equality6. BC � CD � CD � DE 6. Addition Property of Equality7. BC � CD � BD 7. Segment Addition Postulate

CD � DE � CE

8. BD � CE 8. Substitution Property9. B�D� � C�E� 9. Def. of � segments

Helping You Remember3. One way to keep the names of related postulates straight in your mind is to associate

something in the name of the postulate with the content of the postulate. How can you usethis idea to distinguish between the Ruler Postulate and the Segment Addition Postulate?Sample answer: There are two words in “Ruler Postulate” and three wordsin “Segment Addition Postulate.”The statement of the Ruler Postulatementions two points, and the statement of the Segment AdditionPostulate mentions three points.

A

B EDC


Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

2-72-7

Geometry Crossword Puzzle

C O L L I

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1110

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Enrichment, p. 98

(t)David Madison/Getty Images, (b)Dan Sears


21. Copy and complete the proof.Given: W�Y� � Z�X�

A is the midpoint of W�Y�.A is the midpoint of Z�X�.

Prove: W�A� � Z�A�Proof:Statements Reasons

a. W�Y� � Z�X� a. GivenA is the midpoint of W�Y�.A is the midpoint of Z�X�.

b. WY � ZX b. Def. of � segs.c. WA � AY, ZA � AX c. Definition of midpointd. WY � WA � AY, ZX � ZA � AX d. Segment Addition Post.e. WA � AY � ZA � AX e. Substitutionf. WA � WA � ZA � ZA f. Substitutiong. 2WA = 2ZA g. Substitutionh. WA � ZA h. Division Propertyi. W�A� � Z�A� i. Def. of � segs.

For Exercises 22–24, write a two-column proof. 22–24. See p. 123E.22. If L�M� � P�N� and X�M� � X�N�, 23. If AB � BC,

then L�X� � P�X�. then AC � 2BC.

24. Reflexive Property of Congruence (Theorem 2.2)

25. DESIGN The front of a building has a triangular window. If A�B� � D�E� and C is the midpoint of B�D�, prove that A�C� � C�E�.See p. 123E.

26. LIGHTING The light fixture in Gerrard Hall of the University of North Carolina is shown at the right. If A�B� � E�F� and B�C� � D�E�, prove that A�C� � D�F�. See p. 123F.

27. CRITICAL THINKING Given that L�N� � R�T�, R�T� � Q�O�, L�Q� � N�O�, M�P� � N�O�, S is themidpoint of R�T�, M is the midpoint of L�N�, and Pis the midpoint of Q�O�, list three statements thatyou could prove using the postulates, theorems,and definitions that you have learned. OPQ

NML

R TS

A FEB D

C

F

A B C D E

A B CML

P

N

X

PROOF

??

????

??

?

A

W

Y

Z X

DesignWindows come in manydifferent shapes and sizes.Some commonly usedshapes for windows arecircles, rectangles, squares,triangles, pentagons, andoctagons.Source: www.pella.com

See p. 123F.



ELL


Open-Ended AssessmentModeling Make a“Reasons/Statements” boardwith Velcro in positions where youcould place given information,statements and reasons. Createthree or four proofs using segmentaddition and segment congruence,and write the given informationand each statement and reason ona separate rectangular piece ofposter board (large enough to readfrom the back of the classroom).Affix Velcro to the back of theboards so they can be easilyplaced on the R/S board. Placethe given information at the topof the R/S board. Have studentsselect each statement, match itwith its corresponding reasonand place it in the correct orderon the R/S board.

Getting Ready forLesson 2-8Prerequisite Skill In Lesson 2-8,students will apply properties ofsupplementary andcomplementary angles to proveangle relationships. UseExercises 40�45 to determineyour students’ familiarity withcomplementary andsupplementary angles.

Answers

28. Sample answer: You can usesegment addition to find the totaldistance between two destinationsby adding the distances of variouspoints in between. Answersshould include the following.• A passenger can add the

distance from San Diego toPhoenix and the distance fromPhoenix to Dallas to find thedistance from San Diego toDallas.

• The Segment Addition Postulatecan be useful if you aretraveling in a straight line.


How can segment relationships be used for travel?

Include the following in your answer:• an explanation of how a passenger can use the distances the pilot announced

to find the total distance from San Diego to Dallas, and• an explanation of why the Segment Addition Postulate may or may not be

useful when traveling.

29. If P is the midpoint of B�C� and Q is the midpoint of A�D�, what is PQ? B

�12

� 1

2 2�12

�

30. GRID IN A refreshment stand sells a large tub of popcorn for twice the price ofa box of popcorn. If 60 tubs were sold for a total of $150 and the total popcornsales were $275, how many boxes of popcorn were sold? 100

DC

BA A B C D

12 3414 1

4 12 14

WRITING IN MATH




State the property that justifies each statement. (Lesson 2-6)

31. If m�P � m�Q � 110 and m�R � 110, then m�P � m�Q � m�R. Substitution32. If x(y � z) � a, then xy � xz = a. Dist. Prop.33. If n � 17 � 39, then n � 56. Add. Prop.34. If cv � md and md � 15, then cv � 15. Trans. Prop.

Determine whether the following statements are always, sometimes, or never true.Explain. (Lesson 2-5) 35–38. See margin for explanations.35. A midpoint divides a segment into two noncongruent segments. never36. Three lines intersect at a single point. sometimes37. The intersection of two planes forms a line. always38. Three single points determine three lines. sometimes

39. If the perimeter of rectangle ABCDis 44 centimeters, find x and thedimensions of the rectangle.(Lesson 1-6) 3; 9 cm by 13 cm

PREREQUISITE SKILL Find x.(To review complementary and supplementary angles, see Lesson 1-5.)

40. 41. 42.

43. 44. 45.

(4x � 10)̊ (3x � 5)̊26x° 10x°x° 3x°

(3x � 2)̊

x°2x°

4x°2x°

x°

Mixed Review

C B

D A

(x � 6) cm

(2x � 7) cm

30

45

15

5

22

25



4 Assess4 Assess

35. The midpoint of a segment divides it into two congruent segments.36. If the lines have a common intersection point, then it is a single point.37. If two planes intersect, they intersect in a line.38. If the points are noncollinear, then they lie on three distinct lines.




do scissors illustratesupplementary angles?

Ask students:• In the figure, label �4 vertical

to �2 and name all pairs ofsupplementary angles.�1 and �2, �2 and �3, �3 and�4, �4 and �1

• Use a protractor to measureangles 1 and 2. What is thesum of these two measures?about 40°; about 140°; 180°.

• Will the same angles still formlinear pairs if the scissors wereopened wider? narrower?yes; yes

1/16/2003 1:06 PM T_Maria_Manko 107-114 GEO C2L8-

Proving Angle Relationships

Lesson 2-8 Proving Angle Relationships 107

• Write proofs involving supplementary and complementary angles.

• Write proofs involving congruent and right angles.

Protractor Postulate Given AB�� and a number rbetween 0 and 180, there is exactly one ray withendpoint A, extending on either side of AB��, suchthat the measure of the angle formed is r.


B r°10 170

2016

030

150

4014

0

50

130

60

120

70

110

80

100

10080

90 11070 120

60 13050

14040

15030

16020

170

10

A

Angle Addition Postulate If R is in the interiorof �PQS, then m�PQR � m�RQS � m�PQS.

If m�PQR � m�RQS � m�PQS, then R is in theinterior of �PQS.

Postulate 2.11Postulate 2.11P

R

S

Q

A

C

DB

D

SUPPLEMENTARY AND COMPLEMENTARY ANGLES Recall that whenyou measure angles with a protractor, you position the protractor so that one of therays aligns with zero degrees and then determine the position of the second ray.This illustrates the Protractor Postulate.

HistoryThe Grand Union flag wasthe first flag used by thecolonial United States thatresembles the current flag.It was made up of thirteenstripes with the flag ofGreat Britain in the corner.Source: www.usflag.org

In Lesson 2-7, you learned about the Segment Addition Postulate. A similarrelationship exists between the measures of angles.

Notice that when a pair of scissors isopened, the angle formed by the twoblades, �1, and the angle formed by ablade and a handle, �2, are a linear pair.Likewise, the angle formed by a blade anda handle, �2, and the angle formed by thetwo handles, �3, also forms a linear pair.

do scissors illustratesupplementary angles?do scissors illustratesupplementary angles?

12

3

Angle AdditionHISTORY The Grand Union Flag at the left contains several angles. If m�ABD � 44 and m�ABC � 88, find m�DBC.m�ABD � m�DBC � m�ABC Angle Addition Postulate

44 � m�DBC � 88 m�ABD � 44, m�ABC � 88m�DBC � 44 Subtraction Property

Example 1Example 1

(t)C Squared Studios, (b)file photo



Prerequisite Skills Workbook, pp. 81–82,85–86

Teaching Geometry With ManipulativesMasters, pp. 8, 16, 48





Transparencies

LessonNotes

1 Focus1 Focus

11

22


SUPPLEMENTARY ANDCOMPLEMENTARY ANGLES

TIME At 4 o’ clock, the anglebetween the hour and minutehands of a clock is 120°. If thesecond hand stops where itbisects the angle between thehour and minute hands, whatare the measures of the anglesbetween the minute andsecond hands and betweenthe second and hour hands?They are both 60° by thedefinition of angle bisector andthe Angle Addition Postulate.

If �1 and �2 form a linearpair and m�2 � 166, findm�1. 14

The Angle Addition Postulate can be used with other angle relationships toprovide additional theorems relating to angles.


Supplementary AnglesIf �1 and �2 form a linear pair and m�2 � 67, find m�1.m�1 � m�2 � 180 Supplement Theorem

m�1 � 67 � 180 m�2 � 67

m�1 � 113 Subtraction Property

Example 2Example 2

TheoremsTheorems

2.3 Supplement Theorem If two angles forma linear pair, then they are supplementaryangles.

2.4 Complement Theorem If the noncommonsides of two adjacent angles form a rightangle, then the angles arecomplementary angles.

Look BackTo review supplementaryand complementary angles,see Lesson 1-5.

Study Tip

m�1 � m�2 � 180

1 2

m�1 � m�2 � 90

12

You will prove Theorems 2.3 and 2.4 in Exercises 10 and 11.

1 2

Congruence of angles is reflexive, symmetric, and transitive.

Reflexive Property �1 � �1

Symmetric Property If �1 � �2, then �2 � �1.

Transitive Property If �1 � �2, and �2 � �3, then �1 � �3.


You will prove the Reflexive and Transitive Properties of Angle Congruence in Exercises 26 and 27.

Symmetric Property of CongruenceGiven: �A � �B

Prove: �B � �A

Paragraph Proof:

We are given �A � �B. By the definition of congruent angles, m�A � m�B.Using the Symmetric Property, m�B � m�A. Thus, �B � �A by the definition ofcongruent angles.

ProofProof

A B

TEACHING TIPThe Symmetric Propertyis often assumed in proofsto condense the numberof steps in a proof. Therigor of proof is left up to the teacher, but wewill assume symmetric property statements infuture chapters.

CONGRUENT AND RIGHT ANGLES The properties of algebra that appliedto the congruence of segments and the equality of their measures also hold true forthe congruence of angles and the equality of their measures.

Algebraic properties can be applied to prove theorems for congruencerelationships involving supplementary and complementary angles.


2 Teach2 Teach

Auditory/Musical Ask students to close their books. Read Theorems2.3–2.13 aloud for students one by one. After each one, ask studentsto discuss how they know the theorem is true and how they might usethe theorem in a proof.


33


CONGRUENT AND RIGHTANGLES

In the figure, �1 and �4form a linear pair, and m�3� m�1 � 180. Prove that �3and �4 are congruent.

Statements (Reasons)1. m�3 � m�1 � 180; �1 and

�4 form a linear pair. (Given)2. �1 and �4 are supplementary.

(Linear pairs are suppl.)3. �3 and �1 are supplementary.

(Def. of suppl. �)4. �3 � �4 (� supplementary

to same � are �.)

12 3

4


TheoremsTheorems

2.6 Angles supplementary to the same angle or to congruent angles are congruent.Abbreviation: � suppl. to same �

or � � are �.

Example: If m�1 � m�2 � 180 and m�2 � m�3 � 180, then �1 � �3.

2.7 Angles complementary to the same angle or to congruent angles are congruent.Abbreviation: � compl. to same � or

� � are �.

Example: If m�1 � m�2 � 90 andm�2 � m�3 � 90, then�1 � �3.

2

13

You will prove Theorem 2.6 in Exercise 6.

21

3

Theorem 2.7Given: �1 and �3 are complementary.

�2 and �3 are complementary.Prove: �1 � �2Proof:Statements Reasons1. �1 and �3 are complementary. 1. Given

�2 and �3 are complementary.2. m�1 � m�3 � 90 2. Definition of complementary angles

m�2 � m�3 � 903. m�1 � m�3 � m�2 � m�3 3. Substitution4. m�3 � m�3 4. Reflective Property5. m�1 � m�2 5. Subtraction Property6. �1 � �2 6. Definition of congruent angles

ProofProof

2

1 3

Use Supplementary AnglesIn the figure, �1 and �2 form a linear pair and �2 and �3 form a linear pair. Prove that �1 and �3 are congruent.Given: �1 and �2 form a linear pair.

�2 and �3 form a linear pair.Prove: �1 � �3Proof:Statements Reasons1. �1 and �2 form a linear pair. 1. Given

�2 and �3 form a linear pair.2. �1 and �2 are supplementary. 2. Supplement Theorem

�2 and �3 are supplementary.3. �1 � �3 3. � suppl. to same � or � � are �.

Example 3Example 3

1

34 2




44


Teaching Tip Tell students toread problems carefully so theycan be sure to provide theinformation requested. For thisexample, point out that studentsare to find angle measures, notjust the value of the variable;however, they have to use thevalue of the variable to find theanswer.

If �1 and �2 are verticalangles and m�1 � d � 32and m�2 � 175 � 2d, findm�1 and m�2. 37; 37


Look BackTo review vertical angles,see Lesson 1-5.

Study TipVertical Angles Theorem If two angles arevertical angles, then they are congruent.Abbreviation: Vert. � are �.


1 34

2

�1 � �3 and �2 � �4

Right Angles

Make a Model• Fold the paper so that one corner is folded downward.• Fold along the crease so that the top edge meets the

side edge.• Unfold the paper and measure each of the angles

formed.• Repeat the activity three more times.

Analyze the Model 1. The lines are perpendicular.1. What do you notice about the lines formed? 2. What do you notice about each pair of adjacent

angles?3. What are the measures of the angles formed? 90

Make a Conjecture 4. They form right angles.4. What is true about perpendicular lines? 5. What is true about all right angles? They all measure 90 and are congruent.

Vertical AnglesIf �1 and �2 are vertical angles and m�1 � x and m�2 � 228 � 3x, find m�1 and m�2.

�1 � �2 Vertical Angles Theorem

m�1 � m�2 Definition of congruent angles

x � 228 � 3x Substitution

4x � 228 Add 3x to each side.

x � 57 Divide each side by 4.

m�1 � x m�2 � m�1

� 57 � 57

Example 4Example 4

Note that in Example 3, �1 and �3 are vertical angles. The conclusion in theexample is a proof for the following theorem.

The theorems you have learned can be applied to right angles. You can createright angles and investigate congruent angles by paper folding.

The following theorems support the conjectures you made in the Geometry Activity.

They are congruent and they form linear pairs.


Geometry Activity

Materials: paper, protractor• When students are repeating the activity, tell them to use different folds

from the right and the left sides of the paper each time.• Ask students what they notice about each pair of vertical angles (they are

congruent and form right angles).




• include an example each of aproof involving supplementary,complementary, congruent, andright angles.


FIND THE ERRORExplain that

when two anglemeasures are added using the

Angle Addition Postulate, theymust share a common ray.Students can note that thecommon ray in Tomas’s answer is BE��, and a combination of theseletters appears in both anglesthat are being added (ABE andEBC).


TheoremsTheorems Right Angles2.9 Perpendicular lines intersect to form four right angles.

2.10 All right angles are congruent.

2.11 Perpendicular lines form congruent adjacent angles.

2.12 If two angles are congruent and supplementary, then each angle is a right angle.

2.13 If two congruent angles form a linear pair, then they are right angles.

1. FIND THE ERROR Tomas and Jacob wrote equations involving the anglemeasures shown.

Who is correct? Explain your reasoning. 1–2. See margin.2. OPEN ENDED Draw three congruent angles. Use these angles to illustrate the

Transitive Property for angle congruence.

Find the measure of each numbered angle.3. m�1 � 65 4. �6 and �8 are 5. m�11 � x � 4,

complementary. m�12 � 2x � 5m�8 � 47

6. Copy and complete the proof of Theorem 2.6.Given: �1 and �2 are supplementary.

�3 and �4 are supplementary.�1 � �4

Prove: �2 � �3Proof:Statements Reasonsa. �1 and �2 are supplementary. a. Given

�3 and �4 are supplementary.�1 � �4

b. m�1 � m�2 � 180 b. Def. of suppl. �m�3 � m�4 � 180

c. m�1 � m�2 � m�3 � m�4 c. Substitutiond. m�1 � m�4 d. Def. of � �

e. m�2 � m�3 e. Subtr. Prop.f. �2 � �3 f. Def. of � �?

???

?

?

1 2 3 4

PROOF

6 87

21

Jacob

m∆ABE + m∆FBC = m∆ABC

Tomas

m∆ABE + m∆EBC = m∆ABC

Concept Check


4 15 26 3

7–11 4

A B

E F

C

Guided Practice

m�2 � 65

1112

m�11 � 59,m�12 � 121

m�6 � 43, m�7 � 90


Answers

About the Exercises…Organization by Objective• Supplementary and

Complementary Angles:16–18

• Congruent and RightAngles: 19–39

Odd/Even AssignmentsExercises 20–36 and 42–43 arestructured so that studentspractice the same conceptswhether they are assignedodd or even problems.

Assignment GuideBasic: 17–41 odd, 42–55Average: 17–41 odd, 42–55Advanced: 16–42 even, 44–55

1. Tomas; Jacob’s answerleft out the part of �ABCrepresented by �EBF.

2. Sample answer: If �1 � �2 and�2 � �3, then �1 � �3.

1 2 3

Answers

7. Given: VX��bisects �WVY, VY�� bisects �XVZ.

Prove: �WVX � �YVZ


1.VX��bisects �WVY; VY�� bisects�XVZ. (Given)

2.�WVX � �XVY (Def. of �bisector)

3.�XVY � �YVZ (Def. of �bisector)

4.�WVX � �YVZ (Tran. Prop.)10. Given: Two angles form a

linear pair.Prove: The angles are

supplementary

Paragraph Proof: When two anglesform a linear pair, the resultingangle is a straight angle whosemeasure is 180. By definition, twoangles are supplementary if thesum of their measures is 180. Bythe Angle Addition Postulate,m�1 � m�2 � 180. Thus, if twoangles form a linear pair, then theangles are supplementary.

11. Given: �ABC is a right angle.Prove: �1 and �2 are

complementary angles.

Proof:Statements (Reasons)1.�ABC is a right angle. (Given)2.m�ABC � 90 (Def. of rt. �)3.m�ABC � m�1 � m�2

(� Add. Post.)4.90 � m�1 � m�2 (Subst.)5.�1 and �2 are complementary

angles. (Def. of comp. �)

CB

A

1 2

1 2

W

Z

Y

XV

7. Write a two-column proof. See margin.Given: VX�� bisects �WVY.

VY�� bisects �XVZ.

Prove: �WVX � �YVZ

Determine whether the following statements are always, sometimes, or never true.8. Two angles that are nonadjacent are vertical. sometimes9. Two angles that are congruent are complementary to the same angle.

Write a proof for each theorem. 10–11. See margin.10. Supplement Theorem 11. Complement Theorem

ALGEBRA For Exercises 12–15, use the following information.�1 and �X are complementary, �2 and �X are complementary, m�1 � 2n � 2, and m�2 � n � 32.12. Find n. 30 13. Find m�1. 6214. What is m�2? 62 15. Find m�X. 28

1X

2

PROOF

??

V

W

X

Y

Z

PROOF


Application


18. m�5 � 61, m�7 � 29, m�8 � 61

m�11 � 124,m�12 � 56

ForExercises

16–1819–2425–39

SeeExamples

1, 243



sometimes

Find the measure of each numbered angle.16. m�2 = 67 m�1 � 113 17. m�3 = 38 m�4 � 52 18. �7 and �8 are

complementary. �5 ��8and m�6 = 29.

19. m�9 � 2x � 4, 20. m�11 � 4x, 21. m�13 � 2x � 94, m�10 � 2x � 4 m�12 � 2x � 6 m�14 � 7x � 49

22. m�15 � x, 23. m�17 � 2x � 7, 24. m�19 � 100 � 20x, m�16 � 6x � 290 m�18 � x � 30 m�20 � 20x

19

2017 18

15 16

13

1411 12

109

85

6 7

43

1 2

m�13 � 112,m�14 � 112

m�15 � 58,m�16 � 58 m�17 � 53,

m�18 � 53

m�19 � 140,m�20 � 40

m�9 � 86, m�10 � 94


Study Guide and InterventionProving Angle Relationships

NAME ______________________________________________ DATE ____________ PERIOD _____

2-82-8


Less

on

2-8

Supplementary and Complementary Angles There are two basic postulates forworking with angles. The Protractor Postulate assigns numbers to angle measures, and theAngle Addition Postulate relates parts of an angle to the whole angle.

Protractor Given AB�� and a number r between 0 and 180, there is exactly one ray Postulate with endpoint A, extending on either side of AB��, such that the measure

of the angle formed is r.

Angle Addition R is in the interior of �PQS if and only if Postulate m�PQR � m�RQS � m�PQS.

The two postulates can be used to prove the following two theorems.

Supplement If two angles form a linear pair, then they are supplementary angles.Theorem If �1 and �2 form a linear pair, then m�1 � m�2 � 180.

Complement If the noncommon sides of two adjacent angles form a right angle, Theorem then the angles are complementary angles.

If GF�� ⊥ GH��, then m�3 � m�4 � 90. HG 43

FJ

CB1 2

A

D

S

R

P

Q

If �1 and �2 form alinear pair and m�2 � 115, find m�1.

m�1 � m�2 � 180 Suppl. Theorem

m�1 � 115 � 180 Substitution

m�1 � 65 Subtraction Prop.

P

Q

NM12

If �1 and �2 form aright angle and m�2 � 20, find m�1.

m�1 � m�2 � 90 Compl. Theorem

m�1 � 20 � 90 Substitution

m�1 � 70 Subtraction Prop.

T

WR

S 12


ExercisesExercises

Find the measure of each numbered angle.

1. 2. 3.

m�7 � 5x � 5, m�5 � 5x, m�6 � 4x � 6, m�11 � 11x,m�8 � x � 5 m�7 � 10x, m�12 � 10x � 10m�7 � 155, m�8 � 12x � 12 m�11 � 110,m�8 � 25 m�5 � 30, m�6 � 30, m�12 � 110,

m�7 � 60, m�8 � 60 m�13 � 70

F CJ

HA 11

1213

WU

X YZ

V58

67R

TP

Q

S

87



Find the measure of each numbered angle.

1. m�1 � x � 10 2. m�4 � 2x � 5 3. m�6 � 7x � 24m�2 � 3x � 18 m�5 � 4x � 13 m�7 � 5x � 14

m�1 � 48, m�3 � 90, m�4 � 31, m�6 � 109,m�2 � 132 m�5 � 59 m�7 � 109

Determine whether the following statements are always, sometimes, or never true.

4. Two angles that are supplementary are complementary.

never

5. Complementary angles are congruent.

sometimes

6. Write a two-column proof.Given: �1 and �2 form a linear pair.

�2 and �3 are supplementary.Prove: �1 � �3

Proof:Statements Reasons1. �1 and �2 form a linear pair. 1. Given

�2 and �3 are supplementary.2. �1 and �2 are supplementary. 2. Supplement Theorem3. �1 � �3 3. � suppl. to the same � or � �

are �.

7. STREETS Refer to the figure. Barton Road and Olive Tree Lane form a right angle at their intersection. Tryon Street forms a 57°angle with Olive Tree Lane. What is the measure of the acute angleTryon Street forms with Barton Road? 33 Olive Tree Lane

BartonRd

TryonSt

1 23

6

7

453

1 2

Practice (Average)

Proving Angle Relationships

NAME ______________________________________________ DATE ____________ PERIOD _____


Reading to Learn MathematicsProving Angle Relationships

NAME ______________________________________________ DATE ____________ PERIOD _____

2-82-8

Less

on

2-8

Pre-Activity How do scissors illustrate supplementary angles?


Is it possible to open a pair of scissors so that the angles formed by the twoblades, a blade and a handle, and the two handles, are all congruent? If so,explain how this could happen. Sample answer: Yes; open thescissors so that the two blades are perpendicular. Then all theangles will be right angles and will be congruent.

Reading the Lesson1. Complete each sentence to form a statement that is always true.

a. If two angles form a linear pair, then they are adjacent and .

b. If two angles are complementary to the same angle, then they are .

c. If D is a point in the interior of �ABC, then m�ABC � m�ABD � .

d. Given RS�� and a number x between and , there is exactly one raywith endpoint R, extended on either side of RS, such that the measure of the angleformed is x.

e. If two angles are congruent and supplementary, then each angle is a(n)

angle.

f. lines form congruent adjacent angles.

g. “Every angle is congruent to itself” is a statement of the Propertyof angle congruence.

h. If two congruent angles form a linear pair, then the measure of each angle is .

i. If the noncommon sides of two adjacent angles form a right angle, then the angles are

.

2. Determine whether each statement is always, sometimes, or never true.a. Supplementary angles are congruent. sometimesb. If two angles form a linear pair, they are complementary. neverc. Two vertical angles are supplementary. sometimesd. Two adjacent angles form a linear pair. sometimese. Two vertical angles form a linear pair. neverf. Complementary angles are congruent. sometimesg. Two angles that are congruent to the same angle are congruent to each other. alwaysh. Complementary angles are adjacent angles. sometimes

Helping You Remember3. A good way to remember something is to explain it to someone else. Suppose that a

classmate thinks that two angles can only be vertical angles if one angle lies above theother. How can you explain to him the meaning of vertical angles, using the word vertexin your explanation? Sample answer: Two angles are vertical angles if theyshare the same vertex and their sides are opposite rays. It doesn’t matterhow the angles are positioned.

complementary

90

ReflexivePerpendicular

right

1800m�DBC

congruentsupplementary


Bisecting a Hidden AngleThe vertex of �BAD at the right is hidden in a region.Within the region, you are not allowed to use a compass.Can you bisect the angle?

Follow these instructions to bisect �BAD.

1. Use a straightedge to draw lines CE and BD.

AD

E

C

PQ

B

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____



25. Prove that congruence of angles is reflexive. See p. 123F.26. Write a proof of the Transitive Property of Angle Congruence. See p. 123F.

Determine whether the following statements are always, sometimes, or never true.27. Two angles that are complementary form a right angle. sometimes28. Two angles that are vertical are nonadjacent. always29. Two angles that form a right angle are complementary. always30. Two angles that form a linear pair are congruent. sometimes31. Two angles that are supplementary are congruent. sometimes32. Two angles that form a linear pair are supplementary. always

Use the figure to write a proof of each theorem.33. Theorem 2.934. Theorem 2.1035. Theorem 2.1136. Theorem 2.1237. Theorem 2.13

Write a two-column proof. 38–39. See p. 123G.38. Given: �ABD � �YXZ 39. Given: m�RSW � m�TSU

Prove: �CBD � �WXZ Prove: m�RST � m�WSU

40. RIVERS Tributaries of rivers sometimes form a linear pair of angles when they meetthe main river. The Yellowstone River formsthe linear pair �1 and �2 with the MissouriRiver. If m�1 is 28, find m�2. 152

41. HIGHWAYS Near the city of Hopewell, Virginia, Route 10 runs perpendicular toInterstate 95 and Interstate 295. Show that theangles at the intersections of Route 10 withInterstate 95 and Interstate 295 are congruent.Because the lines are perpendicular, theangles formed are right angles. All rightangles are congruent. Therefore, �1 iscongruent to �2.

42. CRITICAL THINKING What conclusion can you make about the sum of m�1 and m�4 if m�1 � m�2and m�3 � m�4? Explain. See margin.

341

2

29595

1 2Route 10

21

R T

S

W

UA B C

D Z

W X Y

PROOF

1 23 4

�

m

33–37. See p. 123F.PROOF

??

??

??

HighwaysInterstate highways thatrun from north to southare odd-numbered withthe lowest numbers in thewest. East-west interstatesare even-numbered, andbegin in the south.Source: www.infoplease.com

(l)Richard Pasley/Stock Boston, (r)Sam Abell/National Geographic Image Collection



ELL

Answer

42. m�1 � m�4 � 90;m�1 � m�2 � m�3 � m�4 � 180m�1 � m�1 � m�4 � m�4 � 180

2(m�1) � 2(m�4) � 1802(m�1 � m�4) � 180

m�1 � m�4 � 90


Open-Ended AssessmentWriting Give students a list oftheorems from this chapter. Havestudents choose a theorem andwrite a proof of it with theirbooks closed.


Answers

43. Two angles that aresupplementary to the same angleare congruent. Answers shouldinclude the following.• �1 and �2 are supplementary;

�2 and �3 are supplementary.• �1 and �3 are vertical angles,

and are therefore congruent.• If two angles are

complementary to the sameangle, then the angles arecongruent.

46. Given: G is between F and H.H is between F and J.Prove: FG � GJ � FH � HJ

Proof:Statements (Reasons)1.G is between F and H ; H is

between F and J. (Given)2.FG � GJ � FJ, FH � HJ � FJ

(Seg. Add. Post.)3.FJ � FH � HJ (Sym. Prop.)4.FG � GJ � FH � HJ (Transitive

Prop.)47. Given: X is the midpoint of W�Y�.

Prove: WX � YZ � XZ

Proof:Statements (Reasons)1.X is the midpoint of W�Y�. (Given) 2.WX � XY (Def. of midpoint)3.XY � YZ � XZ (Segment Add.

Post.)4.WX � YZ � XZ (Substitution)

W X Y Z

F G HJ


How do scissors illustrate supplementary angles?

Include the following in your answer:• a description of the relationship among �1, �2, and �3,• an example of another way that you can tell the relationship between �1

and �3, and• an explanation of whether this relationship is the same for two angles

complementary to the same angle.

44. The measures of two complementary angles are in the ratio 4:1. What is themeasure of the smaller angle? B

15 18 24 36

45. ALGEBRA T is the set of all positive numbers n such that n � 50 and �n� is aninteger. What is the median of the members of set T? B

4 16 20 25DCBA

DCBA

WRITING IN MATH



Mixed Review Write a two-column proof. (Lesson 2-7) 46–47. See margin.46. Given: G is between F and H.

H is between G and J.Prove: FG � GJ � FH � HJ

47. Given: X is the midpoint of W�Y�.

Prove: WX � YZ � XZ

48. PHOTOGRAPHY Film is fed through a camera by gears that catch the perforation in the film. Thedistance from the left edge of the film, A, to the rightedge of the image, C, is the same as the distancefrom the left edge of the image, B, to the right edgeof the film, D. Show that the two perforated stripsare the same width. (Lesson 2-6) See p. 123G.

For Exercises 49–55, refer to the figure at the right.(Lesson 1-4)

49. Name two angles that have N as a vertex. �ONM, �MNR50. If M��Q� bisects �PMN, name two congruent angles.51. Name a point in the interior of �LMQ. N or R52. List all the angles that have O as the vertex. 53. Does �QML appear to be acute, obtuse, right, or

straight? obtuse54. Name a pair of opposite rays. Sample answer: NR�� and NP��

55. List all the angles that have M�N� as a side. �NML, �NMP, �NMO, �RNM, �ONM

OPI

M NR

L

Q

A DB C

W X Y Z

F G HJ

PROOF

50. �PMQ � �QMN

52. �POQ, �QON,�NOM, �MOP



4 Assess4 Assess

Answers (page 115)

9.

10.X Y Z

A45 135

B

11. M

L

N

O

Study Guide and Review

Chapter 2 Study Guide and Review 115

A complete list of postulates and theorems can be found on pages R1–R8.

Exercises Choose the correct term to complete each sentence.1. A (counterexample, ) is an educated guess based on known information.

2. The truth or falsity of a statement is called its (conclusion, ).

3. Two or more statements can be joined to form a (conditional, ) statement.

4. A conjunction is a compound statement formed by joining two or more statements using (or, ).

5. The phrase immediately following the word if in a conditional statement is called the ( , conclusion).

6. The (inverse, ) is formed by exchanging the hypothesis and the conclusion.

7. (Theorems, ) are accepted as true without proof.

8. A paragraph proof is a (an) ( , formal proof ).informal proof

Postulates

converse

hypothesis

and

compound

truth value

conjecture

www.geometryonline.com/vocabulary_review

Vocabulary and Concept CheckVocabulary and Concept Check

axiom (p. 89)biconditional (p. 81)compound statement (p. 67)conclusion (p. 75)conditional statement (p. 75)conjecture (p. 62)conjunction (p. 68)contrapositive (p. 77)

converse (p. 77)counterexample (p. 63)deductive argument (p. 94)deductive reasoning (p. 82)disjunction (p. 68)formal proof (p. 95)hypothesis (p. 75)if-then statement (p. 75)

inductive reasoning (p. 62)informal proof (p. 90)inverse (p. 77)Law of Detachment (p. 82)Law of Syllogism (p. 83)logically equivalent (p. 77)negation (p. 67)paragraph proof (p. 90)

postulate (p. 89)proof (p. 90)related conditionals (p. 77)statement (p. 67)theorem (p. 90)truth table (p. 70)truth value (p. 67)two-column proof (p. 95)

See pages62–66.

2-12-1

ExampleExample

Inductive Reasoning and ConjectureConcept Summary

• Conjectures are based on observations and patterns.

• Counterexamples can be used to show that a conjecture is false.

Given that points P, Q, and R are collinear, determine whether the conjecture that Q is between P and R is true or false. If the conjecture is false, give a counterexample.In the figure, R is between P and Q. Since we can find a counterexample, the conjecture is false.

Exercises Make a conjecture based on the given information. Draw a figure toillustrate your conjecture. See Example 2 on page 63. 9–11. See margin for figures.

9. �A and �B are supplementary. m�A � m�B � 18010. X, Y, and Z are collinear and XY � YZ. Y is the midpoint of XZ.11. In quadrilateral LMNO, LM � LO � MN � NO, and m �L � 90. LMNO is a square.

P R Q


Have students look through the chapter to make sure they haveincluded notes and examples in their Foldables for each lesson ofChapter 2.Encourage students to refer to their Foldables while completingthe Study Guide and Review and to use them in preparing for theChapter Test.

TM

For more informationabout Foldables, seeTeaching Mathematicswith Foldables.

Lesson-by-LessonReviewLesson-by-LessonReview

Vocabulary and Concept CheckVocabulary and Concept Check

• This alphabetical list ofvocabulary terms in Chapter 2includes a page referencewhere each term wasintroduced.

• Assessment A vocabularytest/review for Chapter 2 isavailable on p. 118 of theChapter 2 Resource Masters.

For each lesson,• the main ideas are

summarized,• additional examples review

concepts, and• practice exercises are provided.

The Vocabulary PuzzleMakersoftware improves students’ mathematicsvocabulary using four puzzle formats—crossword, scramble, word search using aword list, and word search using clues.Students can work on a computer screenor from a printed handout.

Vocabulary PuzzleMaker

ELL

MindJogger Videoquizzesprovide an alternative review of conceptspresented in this chapter. Students workin teams in a game show format to gainpoints for correct answers. The questionsare presented in three rounds.

Round 1 Concepts (5 questions)Round 2 Skills (4 questions)Round 3 Problem Solving (4 questions)

MindJogger Videoquizzes

ELL

http://www.geometryonline.com/vocabulary_review


Answers

12. �1 � 0 and in a right trianglewith right angle C, a2 � b2 � c2.

13. In a right triangle with right angleC, a2 � b2 � c2 or the sum of themeasures of two supplementaryangles is 180.

14. The sum of the measures of twosupplementary angles is 180 and�1 � 0.

15. �1 � 0, and in a right trianglewith right angle C, a2 � b2 � c2,or the sum of the measures of twosupplementary angles is 180.

16. In a right triangle with right angleC, a2 � b2 � c2, or �1 � 0 or thesum of the measures of twosupplementary angles is 180.

17. In a right triangle with right angleC, a2 � b2 � c2 and the sum of themeasures of two supplementaryangles is 180, and �1 � 0.

18. Converse: If an angle is obtuse,then it measures 120. False; themeasure could be any valuebetween 90 and 180. Inverse: Ifan angle measure does not equal120, then it is not obtuse. False;the measure could be any valueother than 120 between 90 and180. Contrapositive: If an angle isnot obtuse, then its measure doesnot equal 120; true.

19. Converse: If a month has 31 days,then it is March. False; July has31 days. Inverse: If a month is notMarch, then it does not have 31 days. False; July has 31 days.Contrapositive: If a month doesnot have 31 days, then it is notMarch; true.

20. Converse: If a point lies on the y-axis, then its ordered pair has 0for its x-coordinate; true.Inverse: If an ordered pair doesnot have 0 for its x-coordinate,then the point does not lie on they-axis; true. Contrapositive: If apoint does not lie on the y-axis,then its ordered pair does nothave 0 for its x-coordinate; true.

• Extra Practice, see pages xxx-xxx.• Mixed Problem Solving, see page xxx.

1/16/2003 1:48 PM T_Maria_Manko 115-121 GEO C2SGT-

Chapter X Study Guide and ReviewChapter X Study Guide and Review

LogicConcept Summary

• The negation of a statement has the opposite truth value of the originalstatement.

• Venn diagrams and truth tables can be used to determine the truth values ofstatements.

Use the following statements to write a compound statement for each conjunction.Then find its truth value.p: �15� � 5 q: The measure of a right angle equals 90.a. p and q

�15� � 5, and the measure of a right angle equals 90.p and q is false because p is false and q is true.

b. p � q�15� � 5, or the measure of a right angle equals 90.p � q is true because q is true. It does not matter that p is false.

Exercises Use the following statements to write a compound statement for eachconjunction. Then find its truth value. See Examples 1 and 2 on pages 68 and 69.

p: �1 � 0 q: In a right triangle with right angle C, a2 � b2 � c2.r: The sum of the measures of two supplementary angles is 180.12. p and q false 13. q or r true 14. r � p false15. p � (q � r) false 16. q � (p � r) true 17. (q � r) � p false

12–17. See margin for statements.

Conditional StatementsConcept Summary

• Conditional statements are written in if-then form.

• Form the converse, inverse, and contrapositive of an if-then statement by usingnegations and by exchanging the hypothesis and conclusion.

Identify the hypothesis and conclusion of the statement The intersection of two planes is a line. Then write the statement in if-then form.Hypothesis: two planes intersectConclusion: their intersection is a lineIf two planes intersect, then their intersection is a line.

Exercises Write the converse, inverse, and contrapositive of each conditionalstatement. Determine whether each related conditional is true or false. If a statement is false, find a counterexample. See Example 4 on page 77. 18–20. See margin.18. If an angle measure equals 120, then the angle is obtuse.19. If the month is March, then it has 31 days.20. If an ordered pair for a point has 0 for its x-coordinate, then the point lies on

the y-axis.

See pages67–74.

2-22-2


Chapter 2 Study Guide and ReviewChapter 2 Study Guide and Review

ExampleExample

ExampleExample

See pages75–80.

2-32-3




Determine the truth value of the following statement for each set of conditions.If the temperature is at most 0°C, then water freezes. See Example 3 on page 76.

21. The temperature is �10°C, and water freezes. true22. The temperature is 15°C, and water freezes. true23. The temperature is �2°C, and water does not freeze. false24. The temperature is 30°C, and water does not freeze. true

Deductive ReasoningConcept Summary

• The Law of Detachment and the Law of Syllogism can be used todetermine the truth value of a compound statement.

Use the Law of Syllogism to determine whether a valid conclusion can bereached from the following statements.(1) If a body in our solar system is the Sun, then it is a star.(2) Stars are in constant motion.p: a body in our solar system is the sunq: it is a starr: stars are in constant motionStatement (1): p → q Statement (2): q → r

Since the given statements are true, use the Law of Syllogism to conclude p → r. Thatis, If a body in our solar system is the Sun, then it is in constant motion.

Exercises Determine whether the stated conclusion is valid based on the giveninformation. If not, write invalid. Explain your reasoning. See Example 1 on page 82.

If two angles are adjacent, then they have a common vertex.25. Given: �1 and �2 are adjacent angles. Valid; by definition, adjacent angles

Conclusion: �1 and �2 have a common vertex. have a common vertex.26. Given: �3 and �4 have a common vertex. Invalid; vertical angles also have

Conclusion: �3 and �4 are adjacent angles. a common vertex.

Determine whether statement (3) follows from statements (1) and (2) by the Lawof Detachment or the Law of Syllogism. If it does, state which law was used. If itdoes not follow, write invalid. See Example 3 on page 83.

27. (1) If a student attends North High School, then the student has an ID number.(2) Josh Michael attends North High School.(3) Josh Michael has an ID number. yes; Law of Detachment

28. (1) If a rectangle has four congruent sides, then it is a square.(2) A square has diagonals that are perpendicular.(3) A rectangle has diagonals that are perpendicular. invalid

29. (1) If you like pizza with everything, then you’ll like Cardo’s Pizza. yes; Law(2) If you like Cardo’s Pizza, then you are a pizza connoisseur. of Syllogism(3) If you like pizza with everything, then you are a pizza connoisseur.


ExampleExample

See pages82–87.

2-42-4



Answers

30. Never; the intersection of twolines is a point.

31. Always; if P is the midpoint of X�Y�,then X�P� � P�Y�. By definition ofcongruent segments, XP � PY.

32. Sometimes; if M, X, and Y arecollinear.

33. Sometimes; if the points arecollinear.

34. Always; there is exactly one linethrough Q and R. The line lies inat least one plane.

35. Sometimes; if the right anglesform a linear pair.

36. Always; the Reflexive Propertystates that �1 � �1.

37. Never; adjacent angles must sharea common side, and verticalangles do not.

38. If M is the midpoint of A�B�, then AM � �

12�(AB). Since Q is the

midpoint of A�M�, AQ � �12�AM or

�12��

12�(AB) � �

14�AB.


Postulates and Paragraph ProofsConcept Summary

• Use undefined terms, definitions, postulates, and theorems to provethat statements and conjectures are true.

Determine whether the following statement is always, sometimes, or nevertrue. Explain. Two points determine a line.According to a postulate relating to points and lines, two points determine a line.Thus, the statement is always true.

Exercises Determine whether the following statements are always, sometimes, or never true. Explain. See Example 2 on page 90. 30–37. See margin.30. The intersection of two lines can be a line.31. If P is the midpoint of X�Y�, then XP � PY.32. If MX � MY, then M is the midpoint of XY.33. Three points determine a line.34. Points Q and R lie in at least one plane.35. If two angles are right angles, they are adjacent.36. An angle is congruent to itself.37. Vertical angles are adjacent.

38. Write a paragraph proof to prove that

if M is the midpoint of A�B� and Q is the midpoint

of A�M�, then AQ � �14

�AB. See margin.

Algebraic ProofConcept Summary

• The properties of equality used in algebra can be applied to themeasures of segments and angles to verify and prove statements.

Given: 2x � 6 � 3 � �53

�xProve: x � �9Proof:Statements Reasons

1. 2x � 6 � 3 � �53

�x 1. Given

2. 3(2x � 6) � 3�3 � �53

�x� 2. Multiplication Property

3. 6x � 18 � 9 � 5x 3. Distributive Property4. 6x � 18 � 5x � 9 � 5x � 5x 4. Subtraction Property5. x � 18 � 9 5. Substitution6. x � 18 � 18 � 9 � 18 6. Subtraction Property7. x � � 9 7. Substitution

PROOF

See pages89–93.

2-52-5

ExampleExample


See pages94–100.

2-62-6

A MQ B

ExampleExample



Answers

43. Given: 5 � 2 � �12�x

Prove: x � �6Proof:Statements (Reasons)

1.5 � 2 � �12�x (Given)

2.5 � 2 � 2 � �12�x � 2

(Subt. Prop.)3.3 � ��

12�x (Substitution)

4.�2(3) � �2��12�x (Mult. Prop)

5.�6 � x (Substitution)6.x � �6 (Sym. Prop.)

44. Given: x � 1 � �x �

�210

�

Prove: x � 4Proof:Statements (Reasons)

1.x � 1 � �x �

�210

� (Given)

2.�2(x � 1) � �2��x ��2

10�

(Mult. Prop.)3.�2x � 2 � x � 10 (Dist. Prop.)4.�2x � 2 � 2 � x � 10 � 2

(Subt. Prop.)5.�2x � x � 12 (Substitution)6.�2x � x � x � 12 � x

(Subt. Prop.)7.�3x � �12 (Substitution)

8.��

33x

� � ��

132

� (Div. Prop.)

9.x � 4 (Substitution)45. Given: AC � AB, AC � 4x � 1,

AB � 6x �13Prove: x � 7

Proof:Statements (Reasons)1.AC � AB,AC � 4x � 1,

AB � 6x �13 (Given)2.4x � 1 � 6x �13 (Subst.)3.4x � 1 � 1 � 6x � 13 � 1

(Subt. Prop.)4.4x � 6x � 14 (Subst.)5.4x � 6x � 6x � 14 � 6x

(Subt. Prop.)6.�2x � �14 (Subst.)

7.��

22x

� � ��

124

� (Div. Prop.)

8.x � 7 (Subst.)

A B6x � 13

4x � 1 C


Exercises State the property that justifies each statement. See Example 1 on page 94.

39. If 3(x � 2) � 6, then 3x � 6 � 6. Dist. Prop.40. If 10x � 20, then x � 2. Div. Prop.41. If AB � 20 � 45, then AB � 25. Subt. Prop.42. If 3 � CD and CD � XY, then 3 � XY. Transitive Prop.

Write a two-column proof. See Examples 2 and 4 on pages 95 and 96.

43. If 5 � 2 � �12

�x, then x � �6.

44. If x � 1 � �x �

�210

�, then x � 4.

45. If AC = AB, AC = 4x � 1, and AB � 6x � 13, then x � 7.

46. If MN � PQ and PQ � RS, then MN � RS.


PROOF


See pages101–106.

2-72-7

ExampleExample

Proving Segment RelationshipsConcept Summary

• Use properties of equality and congruence to write proofs involving segments.

Write a two-column proof.Given: QT � RT, TS � TPProve: QS � RP


1. QT � RT, TS � TP 1. Given2. QT � TS � RT � TS 2. Addition Property3. QT � TS � RT � TP 3. Substitution4. QT � TS � QS, RT � TP � RP 4. Segment Addition Postulate5. QS � RP 5. Substitution

Exercises Justify each statement with a property of equality or a property of congruence. See Example 1 on page 102.

47. PS � PS Reflexive Prop.48. If XY � OP, then OP � XY. Symmetric Prop.49. If AB � 8 � CD � 8, then AB � CD. Add. Prop.50. If EF � GH and GH � LM, then EF � LM. Transitive Prop.

51. If 2(XY) � AB, then XY � �12

�(AB). Div. or Mult. Prop.

52. If AB � CD, then AB � BC � CD � BC. Add. Prop.

P Q

T

S R


46. Given: MN � PQ, PQ � RSProve: MN � RS

Proof:Statements (Reasons)1.MN � PQ, PQ � RS (Given)2.MN � RS (Transitive Prop.)

M N R

SP Q


Answers (page 121)

1. Sample answer: Formal is thetwo-column proof, informal canbe paragraph proofs.

2. Sample answer: You can use acounterexample.

3. Sample answer: statements andreasons to justify statements

7. �3 � 2 and 3x � 12 when x � 4.8. �3 � 2 or 3x � 12 when x � 4.9. �3 � 2, or 3x � 12 when x � 4

and an equilateral triangle is alsoequiangular.

10. H: you eat an apple a day; C: thedoctor will stay away; If you eatan apple a day, then the doctorwill stay away. Converse: If thedoctor stays away, then you eat anapple a day. Inverse: If you do noteat an apple a day, then thedoctor will not stay away.Contrapositive: If the doctor doesnot stay away, then you do not eatan apple a day.

11. H: a stone is rolling; C: it gathersno moss; If a stone is rolling, thenit gathers no moss. Converse: If astone gathers no moss, then it isrolling. Inverse: If a stone is notrolling, then it gathers moss.Contrapositive: If a stone gathersmoss, then it is not rolling.

16. Given: y � 4x � 9; x � 2Prove: y � 17Proof:Statements (Reasons)1.y � 4x � 9; x � 2 (Given)2.y � 4(2) � 9 (Substitution)3.y � 8 � 9 (Substitution)4.y � 17 (Substitution)

17. Given: AM � CN, MB � ND


Write a two-column proof. See Examples 1 and 2 on pages 102 and 103.

53. Given: BC � EC, CA � CD 54. Given: AB � CDProve: BA � DE Prove: AC � BD

53–54. See p. 123G.

PROOF

See pages107–114.

2-82-8

ExampleExample 255°

1

678

157° 35°

A B C D

C

B E

DA

• Extra Practice, see pages 756–758.• Mixed Problem Solving, see page 783.

Proving Angle RelationshipsConcept Summary

• The properties of equality and congruence can be applied to anglerelationships.

Find the measure of each numbered angle. m�1 � 55, since �1 is a vertical angle to the 55° angle.�2 and the 55° angle form a linear pair.

55 + m�2 � 180 Def. of supplementary �m�2 � 125 Subtract 55 from each side.

Exercises Find the measure of each numbered angle. See Example 2 on page 108.

55. m�6 14556. m�7 2357. m�8 90

58. Copy and complete the proof.See Example 3 on page 109.

Given: �1 and �2 form a linear pair. m�2 � 2(m�1)

Prove: m�1 � 60Proof:Statements Reasons

a. �1 and �2 form a linear pair. a. Givenb. �1 and �2 are supplementary. b. Supplement Theoremc. m�1 � m�2 � 180 c. Definition of supplementary angles

d. m�2 � 2(m�1) d. Givene. m�1 � 2(m�1) � 180 e. Substitution

f. 3(m�1) � 180 f. Substitution

g. �3(m

3�1)� � �

1830

� g. Division Property

h. m�1 � 60 h. Substitution?

?

?

?

?

?

?

?

PROOF


Prove: AB � CDParagraph Proof:

We are given that AM � CN, MB � ND. By theAddition Property, AM � MB � CN � MB. By

Substitution, AM � MB � CN � ND. Using the Segment Addition Postulate, AB � AM � MB, and CD � CN � ND. Then, by Substitution AB � CD.

A M B

D N C

18. H: you are a hard-working person;C: you deserve a great vacation; If you are a hard-working person,then you deserve a great vacation.

Practice Test

Chapter 2 Practice Test 121

Vocabulary and ConceptsVocabulary and Concepts

Skills and ApplicationsSkills and Applications

1. Explain the difference between formal and informal proofs. 1–3. See margin.2. Explain how you can prove that a conjecture is false.3. Describe the parts of a two-column proof.

Determine whether each conjecture is true or false. Explain your answer andgive a counterexample for any false conjecture.

4. Given: �A �B 5. Given: y is a real number 6. Given: 3a2 � 48Conjecture: �B �A Conjecture: �y 0 Conjecture: a � 4true; Symmetric Prop. false; y � 2 false; a � �4

Use the following statements to write a compound statement for each conjunction ordisjunction. Then find its truth value. 7–9. See margin for statements.p: �3 � 2 q: 3x � 12 when x � 4. r: An equilateral triangle is also equiangular.

7. p and q false 8. p or q true 9. p � (q � r) true

Identify the hypothesis and conclusion of each statement and write each statement in if-then form. Then write the converse, inverse, and contrapositive of each conditional. 10–11. See margin.10. An apple a day keeps the doctor away. 11. A rolling stone gathers no moss.

12. Determine whether statement (3) follows from statements (1) and (2) by the Law of Detachment or the Law of Syllogism. If it does, state which law was used. If it does not, write invalid.(1) Perpendicular lines intersect.(2) Lines m and n are perpendicular.(3) Lines m and n intersect. valid; Law of Detachment

Find the measure of each numbered angle.13. �1 2214. �2 8515. �3 85

16.Write a two-column proof. 17. Write a paragraph proof.If y � 4x � 9 and x � 2, then y � 17. Given: AM � CN, MB � ND

Prove: AB � CD

18. ADVERTISING Identify the hypothesis and conclusion of the following statement, then write it in if-then form. Hard working people deserve a great vacation. See margin.

19. STANDARDIZED TEST PRACTICE If two planes intersect, their intersection can be AI a line. II three noncollinear points. III two intersecting lines.

I only II only III only I and II onlyDCBA

A M B

D N C

1

2 3

73˚

95˚


www.geometryonline.com/chapter_test

Chapter 2 Practice Test 121

Introduction In a chapter of diverse material, highlight concepts that areimportant but may not be used often or regularly, so that they will remainfresh in students’ minds.Ask Students Search the chapter for items you found the most difficult.Record these items in your portfolio and write about how you were able tomaster the concepts. If you are still having difficulty with the concepts, writeabout the steps you could take to better your understanding of them.

Portfolio Suggestion

Assessment OptionsVocabulary Test A vocabularytest/review for Chapter 2 can befound on p. 118 of the Chapter 2Resource Masters.

Chapter Tests There are sixChapter 2 Tests and an Open-Ended Assessment task availablein the Chapter 2 Resource Masters.

Open-Ended AssessmentPerformance tasks for Chapter 2can be found on p. 117 of theChapter 2 Resource Masters. Asample scoring rubric for thesetasks appears on p. A31.

ExamView® Pro Use the networkable ExamView® Pro to:• Create multiple versions of

tests.• Create modified tests for

Inclusion students.• Edit existing questions and

add your own questions.• Use built-in state curriculum

correlations to create testsaligned with state standards.

• Apply art to your tests from aprogram bank of artwork.

Chapter 2 TestsForm Type Level Pages

1 MC basic 105–106

2A MC average 107–108

2B MC average 109–110

2C FR average 111–112

2D FR average 113–114

3 FR advanced 115–116

MC = multiple-choice questionsFR = free-response questions

http://www.geometryonline.com/chapter_test


Standardized Test PracticeStudent Record Sheet (Use with pages 122–123 of the Student Edition.)

22

An

swer

s

Select the best answer from the choices given and fill in the corresponding oval.

1 4 7

2 5 8

3 6 DCBADCBA

DCBADCBADCBA

DCBADCBADCBA

Part 1 Multiple ChoicePart 1 Multiple Choice

Part 2 Short Response/Grid InPart 2 Short Response/Grid In

Part 3 Extended ResponsePart 3 Extended Response

Solve the problem and write your answer in the blank.

For Questions 9 and 11, also enter your answer by writing each number or symbolin a box. Then fill in the corresponding oval for that number or symbol.

9 (grid in) 9 11

10

11 (grid in)

12

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

Record your answers for Questions 13–15 on the back of this paper.

Standardized Test PracticeStudent Recording Sheet, p. A1

Additional PracticeSee pp. 123–124 in the Chapter 2Resource Masters for additionalstandardized test practice.


Record your answers on the answer sheetprovided by your teacher or on a sheet of paper.

1. Arrange the numbers �7 , �17

�, �7�, �72 in orderfrom least to greatest. (Prerequisite Skill) D

�7 , �7�, �17

�, �72

�72, �7 , �17

�, �7�

�7 , �17

�, �7�, �72

�72, �17

�, �7�, �7

2. Points A and B lie on the line y � 2x � 3.Which of the following are coordinates of apoint noncollinear with A and B? (Lesson 1-1)C

(7, 11) (4, 5)

(–2, –10) (–5, –13)

3. Dana is measuring distance on a map. Whichof the following tools should Dana use tomake the most accurate measurement? (Lesson 1-2) A

centimeter ruler protractor

yardstick calculator

4. Point E is the midpoint of D�F�. If DE � 8x � 3and EF � 3x � 7, what is x? (Lesson 1-3) B

1 2 4 13

5. What is the relationship between �ACF and �DCF? (Lesson 1-6) A

complementary angles

congruent angles

supplementary angles

vertical angles

6. Which of the following is an example ofinductive reasoning? (Lesson 2-1) C

Carlos learns that the measures of all acute angles are less than 90. Heconjectures that if he sees an acute angle, its measure will be less than 90.

Carlos reads in his textbook that themeasure of all right angles is 90. Heconjectures that the measure of each right angle in a square equals 90.

Carlos measures the angles of severaltriangles and finds that their measures all add up to 180. He conjectures that the sum of the measures of the angles in any triangle is always 180.

Carlos knows that the sum of themeasures of the angles in a square isalways 360. He conjectures that if hedraws a square, the sum of the measuresof the angles will be 360.

7. Which of the following is the contrapositive ofthe statement If Rick buys hamburgers for lunch,then Denzel buys French fries and a large soda?(Lesson 2-2) A

If Denzel does not buy French fries and a large soda, then Rick does not buyhamburgers for lunch.

If Rick does not buy hamburgers forlunch, then Denzel does not buy Frenchfries and a large soda.

If Denzel buys French fries and a largesoda, then Rick buys hamburgers for lunch.

If Rick buys hamburgers for lunch, thenDenzel does not buy French fries and alarge soda.

8. Which property could justify the first step in

solving 3 � �14x

8� 6� � 18? (Lesson 2-5) A

Division Property of Equality

Substitution Property of Equality

Addition Property of Equality

Transitive Property of EqualityD

C

B

A

D

C

B

A

D

C

B

A

D

C

B

A

F D

E

CBA

DCBA

DC

BA

DC

BA

D

C

B

A

Part 1 Multiple Choice


These two pages contain practicequestions in the various formatsthat can be found on the mostfrequently given standardizedtests.

A practice answer sheet for thesetwo pages can be found on p. A1of the Chapter 2 Resource Masters.

ExamView® Pro Special banks of standardized test questions similar to those on the SAT,ACT, TIMSS 8, NAEP 8, and state proficiency tests can be found on this CD-ROM.

Evaluating ExtendedResponse QuestionsExtended Response questions aregraded by using a multilevelrubric that guides you inassessing a student’s knowledgeof a particular concept.Goal: Find measures and provean angle measure.Sample Scoring Rubric: Thefollowing rubric is a samplescoring device. You may wish toadd more detail to this sample tomeet your individual scoringneeds.

Record your answers on the answer sheet provided by your teacher or on a sheet of paper.

9. Two cheerleaders stand at opposite cornersof a football field. What is the shortestdistance between them, to the nearest yard? (Lesson 1-3) 131 yd

10. Consider the conditional If I call in sick, thenI will not get paid for the day. Based on theoriginal conditional, what is the name of theconditional If I do not call in sick, then I willget paid for the day? (Lesson 2-2) inverse

11. Examine the following statements.

p: Martina drank a cup of soy milk. q: A cup is 8 ounces.r: Eight ounces of soy milk contains

300 milligrams of calcium.

Using the Law of Syllogism, how manymilligrams of calcium did Martina get from drinking a cup of soy milk?(Lesson 2-4) 300

12. In the following proof, what propertyjustifies statement c? (Lesson 2-7)

Given: A�C� � M�N�Prove: AB � BC � MN

Proof: Statements Reasons

a. A�C� � M�N� a. Givenb. AC � MN b. Definition of

congruent segments

c. AC � AB � BC c.d. AC � BC � MN d. Substitution

Record your answers on a sheet of paper.Show your work.

13. In any right triangle, the sum of the squaresof the lengths of the legs equals the squareof the length of the hypotenuse. From asingle point in her yard, Marti measures andmarks distances of 18 feet and 24 feet fortwo sides of her garden. Explain how Martican ensure that the two sides of her gardenform a right angle. (Lesson 1-3) See margin.

14. A farmer needs to make a 100-square-footrectangular enclosure for her chickens. Shewants to save money by purchasing theleast amount of fencing possible to enclosethe area. (Lesson 1-4) a–c. See margin.

a. What whole-number dimensions, to thenearest yard, will require the leastamount of fencing?

b. Explain your procedure for finding thedimensions that will require the leastamount of fencing.

c. Explain how the amount of fencingrequired to enclose the area changes as the dimensions change.

15. Given: �1 and �3 are vertical angles.m�1 � 3x � 5, m�3 � 2x � 8

Prove: m�1 � 14 (Lesson 2-8)

132

4

See p. 123G.

?1 02 03 04 05 04 03 02 01 0

1 0 2 0 3 0 4 0 5 0 4 0 3 0 2 0 1 0

120 yd

53 yd13

Part 2 Short Response/Grid In

Chapter 2 Standardized Test Practice 123

Part 3 Extended Response

A

BC

N

M

Segment AdditionPostulate

Preparing for Standardized TestsFor test-taking strategies and more practice, see pages 795–810.

Test-Taking TipQuestion 6When answering a multiple-choice question, always readevery answer choice and eliminate those you decide aredefinitely wrong. This way, you may deduce the correctanswer.

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Chapter 2 Standardized Test Practice 123

Score Criteria4 A correct solution that is

supported by well-developed,accurate explanations

3 A generally correct solution, but may contain minor flaws in reasoning or computation

2 A partially correct interpretationand/or solution to the problem

1 A correct solution with no supporting evidence or explanation

0 An incorrect solution indicating no mathematical understandingof the concept or task, or no solution is given

Answers

13. Sample answer: Marti can measurea third distance c, the distancebetween the ends of the twosides, and make sure it satisfiesthe equation a2 � b2 � c2.

14a. 10 yd by 10 yd14b. Sample answer: Make a list of all

possible whole-number lengthsand widths that will form a 100-square-foot area. Then findthe perimeter of each rectangle.Choose the length and widthcombination that has thesmallest perimeter.

14c. As the length and width get closerto having the same measure asone another, the amount offencing required decreases.

http://www.geometryonline.com/standardized_test

Pages 63–66, Lesson 2-1

1. Sample answer: After the news is over, it’s time fordinner.

2. Sometimes; the conjecture is true when E is betweenD and F; otherwise it is false.

3. Sample answer: When it is cloudy, it rains.Counterexample: It is often cloudy and it does not rain.

19. 30 20. 20

21. 22.

23.

24. 25.

26.

27. 28.


11. 12.

13.

14.

18. ��64� � 8 and an equilateral triangle has threecongruent sides.

19. ��64� � 8 or an equilateral triangle has threecongruent sides.

20. ��64� � 8 and 0 � 0.

21. 0 � 0 and an obtuse angle measures greater than 90°and less than 180°.

22. An equilateral triangle has three congruent sides or 0 � 0.

23. An equilateral triangle has three congruent sides andan obtuse angle measures greater than 90° and lessthan 180°.

24. ��64� � 8 and an obtuse angle measures greaterthan 90° and less than 180°.

25. An equilateral triangle has three congruent sides and 0 � 0.

26. 0 � 0 or ��64� � 8

27. An obtuse angle measures greater than 90° and lessthan 180° or an equilateral triangle has threecongruent sides.

28. ��64� � 8 and an equilateral triangle has threecongruent sides, or an obtuse angle measures greaterthan 90° and less than 180°.

29. An obtuse angle measures greater than 90° and lessthan 180°, or an equilateral triangle has threecongruent sides and 0 � 0.

33. 34.

35. 36. p q p and q

T T T

T F F

F T F

F F F

p r p or r

T T T

T F T

F T T

F F F

p q p or q

T T T

T F T

F T T

F F F

q r q and r

T T T

T F F

F T F

F F F

p q r p � q ( p � q ) � r

T T T T T

T T F T T

T F T T T

T F F T T

F T T T T

F T F T T

F F T F T

F F F F F

p r �p �p � r

T T F F

T F F F

F T T T

F F T F

q r q � r

T T T

T F T

F T T

F F F

p q p � q

T T T

T F F

F T F

F F F

B

A

C

P

S

Q

R

H

K

I

J

x

y

O

Q(6, –2)

R(6, 5)(–1, 7)P

BC

D

A

3 4

x

y

O

C(5, 10)

4 8–4–8B(2, 1)

A(–2, –11)

8

4

–4

–8

�

m

123A Chapter 2 Additional Answers

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37. 38.

39.

40.


15. If you are in Colorado, then aspen trees cover highareas of the mountains. If you are in Florida, thencypress trees rise from the swamps. If you are inVermont, then maple trees are prevalent.

16. H: 2x � 6 � 10, C: x � 217. H: you are a teenager; C: you are at least 13 years old18. H: you have a driver’s license; C: you are at least

16 years old19. H: three points lie on a line; C: the points are collinear20. H: a man hasn’t discovered something he will die for;

C: he isn’t fit to live21. H: an angle measures between 0 and 90; C: the angle

is acute22. If you buy a 1-year fitness plan, then you get a free visit.23. If you are a math teacher, then you love to solve

problems.24. If I think, then I am.25. If two angles are adjacent, then they have a common

side.26. If two angles are vertical, then they are congruent.

27. If two triangles are equiangular, then they areequilateral.

40. Converse: If you live in Texas, then you live in Dallas.False; you could live in Austin. Inverse: If you don’t livein Dallas, then you don’t live in Texas. False; youcould live in Austin. Contrapositive: If you don’t live inTexas, then you don’t live in Dallas; true.

41. Converse: If you are in good shape, then you exerciseregularly; true. Inverse: If you do not exerciseregularly, then you are not in good shape; true.Contrapositive: If you are not in good shape, then youdo not exercise regularly. False; an ill person mayexercise a lot, but still not be in good shape.

42. Converse: If the sum of two angles is 90, then they arecomplementary; true. Inverse: If two angles are notcomplementary, then their sum is not 90; true.Contrapositive: If the sum of two angles is not 90, thenthey are not complementary; true.

43. Converse: If a figure is a quadrilateral, then it is arectangle; false, rhombus. Inverse: If a figure is not arectangle, then it is not a quadrilateral; false, rhombus.Contrapositive: If a figure is not a quadrilateral, then itis not a rectangle; true.

44. Converse: If an angle has a measure of 90, then it is aright angle; true. Inverse: If an angle is not a rightangle, then its measure is not 90; true. Contrapositive:If an angle does not have a measure of 90, then it isnot a right angle; true.

45. Converse: If an angle has measure less than 90, thenit is acute; true. Inverse: If an angle is not acute, thenits measure is not less than 90; true. Contrapositive: Ifan angle’s measure is not less than 90, then it is notacute; true.

47. Sample answer: In Alaska, if there are more hours ofdaylight than darkness, then it is summer; true. InAlaska, if there are more hours of darkness thandaylight, then it is winter; true.

Page 80, Practice Quiz 1

1. 3.

4. p q r q � r p � (q � r )

T T T T T

T T F F T

T F T F T

T F F F T

F T T T T

F T F F F

F F T F F

F F F F F

p q �p �p � q

T T F F

T F F F

F T T T

F F T F

W X

Y

p q r �q �r �q � �r p � (�q � �r)

T T T F F F F

T T F F T T T

T F T T F T T

T F F T T T T

F T T F F F F

F T F F T T F

F F T T F T F

F F F T T T F

p q r �p �r q � �r �p � (q � �r )

T T T F F F F

T T F F T T T

T F T F F F F

T F F F T F F

F T T T F F T

F T F T T T T

F F T T F F T

F F F T T F T

p q �p �q �p � �q

T T F F F

T F F T F

F T T F F

F F T T T

q r �r q � �r

T T F F

T F T T

F T F F

F F T F

Chapter 2 Additional Answers 123B

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5. Converse: If two angles have a common vertex, then the angles are adjacent. False; �ABD is notadjacent to �ABC.

Inverse: If two angles are not adjacent, then they do not have a common vertex. False, �ABCand �DBE have a common vertexand are not adjacent.

Contrapositive: If two angles do not have a commonvertex, then they are not adjacent; true.


33. Sample answer: Doctors and nurses use charts toassist in determining medications and their doses forpatients. Answers should include the following.

• Doctors need to note a patient’s symptoms todetermine which medication to prescribe, thendetermine how much to prescribe based on weight,age, severity of the illness, and so on.

• Doctors use what is known to be true about diseasesand when symptoms appear, then deduce that thepatient has a particular illness.

39. 40.

41.

42.


6. Sometimes; if the planes have a common intersection,then their intersection is one line.

10. Since P is the midpoint of Q�R� and S�T�, PPQ � PPR �

�12

�QR and PS � PT � �12

�ST by the definition of

midpoint. We are given Q�R� �S�T� so QR � ST by thedefinition of congruent segments. By the MultiplicationProperty, �

12

�QR � �12

�ST. So, by substitution, PQ � PT.16. Sometimes; the three points cannot be on the same line.17. Always; if two points lie in a plane, then the entire line

containing those points lies in that plane.18. Never; the intersection of a line and a plane can be a

point, but the intersection of two planes is a line.19. Sometimes; the three points cannot be on the same line.20. Always; one plane contains at least three points, so it

must contain two.21. Sometimes; � and m could be skew, so they would not

lie in the same plane.22. Postulate 2.1; through any two points, there is exactly

one line.23. Postulate 2.5; if two points lie in a plane, then the

entire line containing those points lies in that plane.24. Postulate 2.2; through any three points not on the

same line, there is exactly one plane.25. Postulate 2.5; if two points lie in a plane, then the

entire line containing those points lies in the plane.26. Postulate 2.1; through any two points, there is exactly

one line.27. Postulate 2.2; through any three points not on the

same line, there is exactly one plane.38.


10. Given: 25 � �7(y � 3) � 5yProve: �2 � yProof:Statements (Reasons)

1. 25 � �7(y � 3) � 5y (Given)2. 25 � �7y � 21 � 5y (Dist. Prop.)3. 25 � �2y � 21 (Substitution)4. 4 � �2y (Subt. Prop.)5. �2 � y (Div. Prop.)

11. Given: Rectangle ABCD,AD � 3, AB � 10

Prove: AC � BDProof:Statements (Reasons)

1. Rectangle ABCD, AD � 3, AB � 10 (Given)2. Draw segments AC and DB. (Two points determine

a line.)3. �ABC and �BCD are right triangles. (Def. of rt. �)

4. AC � �32 � 1�02�, DB � �32 � 1�02� (Pythag. Th.) 5. AC � BD (Substitution)

A

D10

10

3 3B

C

Animal

Arthropod

Butterfly

p q r �q �q � r p � (�q � r )

T T T F F T

T T F F F T

T F T T T T

T F F T F T

F T T F F F

F T F F F F

F F T T T T

F F F T F F

p q r q � r p � (q � r )

T T T T T

T T F T T

T F T T T

T F F F F

F T T T F

F T F T F

F F T T F

F F F F F

p r �p �p � r

T T F T

T F F F

F T T T

F F T T

q r q � r

T T T

T F F

F T F

F F F

AB

D E

C

A

BD

C

123C Chapter 2 Additional Answers

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12. Given: c2 � a2 � b2

Prove: a � �c2 � b�2�Proof:Statements (Reasons)

1. c2 � a2 � b2 (Given)2. c2 � b2 � a2 (Subt. Prop.)3. a2 � c2 � b2 (Reflexive Prop.)

4. �a2� � �c2 � b�2� (Square Root Prop.)

5. a � �c2 � b�2� (Square Root Prop.)

26. Given: 4 � �12

�a � �72

� � a

Prove: a � �1


1. 4 � �12

�a � �72

� � a (Given)

2. 2�4 � �12

�a� � 2��72

� � a� (Mult. Prop.)

3. 8 � a � 7 � 2a (Dist. Prop.)4. 1 � a � �2a (Subt. Prop.)5. 1 � �1a (Add. Prop.)6. �1 � a (Div. Prop.)7. a � �1 (Symmetric Prop.)

27. Given: �2y � �32

� � 8

Prove: y � ��143�


1. �2y � �32

� � 8 (Given)

2. 2��2y � �32

�� 2(8) (Mult. Prop.)

3. �4y � 3 � 16 (Dist. Prop.)4. �4y � 13 (Subt. Prop.)

5. y � ��143� (Div. Prop.)

28. Given: ��12

�m � 9

Prove: m � �18


1. ��12

�m � 9 (Given)

2. �2��12

�m� � �2(9) (Mult. Prop.)

3. m � �18 (Substitution)

29. Given: 5 � �23

�z � 1

Prove: z � 6


1. 5 � �23

�z � 1 (Given)

2. 3�5 � �23

�z� � 3(1) (Mult. Prop.)

3. 15 � 2x � 3 (Dist. Prop.)4. 15 � 2x � 15 � 3 � 15 (Subt. Prop.)5. �2x � �12 (Substitution)

6. ��

22x

� � ��122

� (Div. Prop.)

7. x � 6 (Substitution)

30. Given: XZ � ZY, XZ � 4x � 1, and ZY � 6x � 13

Prove: x � 7


1. XZ � ZY, XZ � 4x � 1, and ZY � 6x � 13 (Given)

2. 4x � 1 � 6x � 13 (Substitution)

3. 4x � 1 � 4x � 6x � 13 � 4x (Subt. Prop.)

4. 1 � 2x � 13 (Substitution)

5. 1 � 13 � 2x � 13 � 13 (Add. Prop.)

6. 14 � 2x (Substitution)

7. �124� � �

22x� (Div. Prop.)

8. 7 � x (Substitution)

9. x � 7 (Symmetric Prop.)

31. Given: m�ACB � m�ABCProve: m�XCA � m�YBA


1. m�ACB � m�ABC (Given)2. m�XCA � m�ACB � 180,

m�YBA � m�ABC � 180 (Def. of supp. �)3. m�XCA � m�ACB � m�YBA � m�ABC

(Substitution)4. m�XCA � m�ACB � m�YBA � m�ACB

(Substitution)5. m�XCA � m�YBA (Subt. Prop.)

X C B Y

A

X Z

Y

4x � 1

6x � 13

Chapter 2 Additional Answers 123D

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, Practice Quiz 2

5. Given: 2(n � 3) � 5 � 3(n � 1)Prove: n � 2


1. 2(n � 3) � 5 � 3(n � 1) (Given)2. 2n � 6 � 5 � 3n � 3 (Dist. Prop.)3. 2n � 1 � 3n � 3 (Substitution)4. 2n � 1 � 2n � 3n � 3 � 2n (Subt. Prop.)5. �1 � n � 3 (Substitution)6. �1 � 3 � n � 3 � 3 (Add. Prop.)7. 2 � n (Substitution)8. n � 2 (Symmetric Prop.)


8. Given: A�P� � C�P�B�P� � D�P�

Prove: A�B� � C�D�Proof:Statements (Reasons)

1. A�P� � C�P� and B�P� � D�P� (Given)2. AP � CP and BP � DP (Def. of � segs.)3. AP � PB � AB (Seg. Add. Post.)4. CP � DP � AB (Substitution)5. CP � PD � CD (Seg. Add. Post.)6. AB � CD (Transitive Prop.)7. A�B� � C�D� (Def. of � segs.)

9. Given: H�I� � T�U� and H�J� � T�V�

Prove: I�J� � U�V�


1. H�I� � T�U� and H�J� � T�V� (Given)2. HI � TU and HJ � TV (Def. of � segs.)3. HI � IJ � HJ (Seg. Add. Post.)4. TU � IJ � TV (Substitution)5. TU � UV � TV (Seg. Add. Post.)6. TU � IJ � TU � UV (Substitution)7. TU � TU (Reflexive Prop.)8. IJ � UV (Subt. Prop.)9. I�J� � U�V� (Def. of � segs.)

19. Given: X�Y� � W�Z� and W�Z� � A�B�Prove: X�Y� � A�B�


1. X�Y� � W�Z� and W�Z� � A�B� (Given)2. XY � WZ and WZ � AB (Def. of � segs.)3. XY � AB (Transitive Prop.)4. X�Y� � A�B� (Def. of � segs.)

20. Given: A�B� � A�C� and P�C� � Q�B�Prove: A�P� � A�Q�


1. A�B� � A�C� and P�C� � Q�B� (Given)2. AB � AC, PC � QB (Def. of � segs.)3. AB � AQ � QB, AC � AP � PC (Seg. Add. Post.)4. AQ � QB� AP � PC (Substitution)5. AQ � QB � AP � QB (Substitution)6. QB � QB (Reflexive Prop.)7. AP � AQ (Subt. Prop.)8. A�P� � A�Q� (Def. of � segs.)

22. Given: L�M� � P�N� and X�M� � X�N�

Prove: L�X� � P�X�Proof:Statements (Reasons)

1. L�M� � P�N� and X�M� � X�N� (Given)2. LM � PN and XM � XN (Def. of � segs.)3. LM � LX � XM, PN � PX � XN (Seg. Add. Post.)4. LX � XM � PX � XN (Substitution)5. LX � XN � PX � XN (Substitution)6. XN � XN (Reflexive Prop.)7. LX � PX (Subt. Prop.)8. L�X� � P�X� (Def. of � segs.)

23. Given: AB � BCProve: AC � 2BC


1. AB � BC (Given)2. AC � AB � BC (Seg. Add. Post.)3. AC � BC � BC (Substitution)4. AC � 2BC (Substitution)

24. Given: A�B�Prove: A�B� � A�B�Proof:Statements (Reasons)

1. A�B� (Given)2. AB � AB (Reflexive Prop.)3. A�B� � A�B� (Def. of � segs.)

25. Given: A�B� � D�E�, C is the midpoint of B�D�.

Prove: A�C� � C�E�


1. A�B� � D�E�, C is the midpoint of B�D�. (Given)2. BC � CD (Def. of midpoint)3. AB � DE (Def. of � segs.)4. AB � BC � CD � DE (Add. Prop.)5. AB � BC � AC, CD � DE � CE (Seg. Add. Post.)6. AC � CE (Substitution)7. A�C� � C�E� (Def. of � segs.)

A B C D E

A B

A B C

ML P

NX

C

A

P Q

B

AB W

ZX Y

H

I

JT U V

A

CP D

B

123E Chapter 2 Additional Answers

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26. Given: A�B� � E�F� and B�C� � D�E�Prove: A�C� � D�F�


1. A�B� � E�F� and B�C� � D�E� (Given)2. AB � EF and BC � DE (Def. of � segs.)3. AB � BC � DE � EF (Add. Prop.)4. AC � AB � BC, DF � DE � EF (Seg. Add. Post.)5. AC � DF (Substitution)6. A�C� � D�F� (Def. of � segs.)

27. Sample answers: L�N� � Q�O� and L�M� � M�N� � R�S� � S�T� � Q�P� � P�O�


25. Given: �AProve: �A � �AProof:Statements (Reasons)

1. �A is an angle. (Given)2. m�A � m�A (Reflexive Prop)3. �A � �A (Def. of � angles)

26. Given: �1 � �2,�2 � �3

Prove: �1 � �3


1. �1 � �2, �2 � �3 (Given)2. m�1 � m �2, m �2 � m �3 (Def. of � angles)3. m�1 � m�3 (Trans. Prop.)4. �1 � �3 (Def. of � angles)

33. Given: � ⊥ mProve: �2, �3, �4 are rt. �


1. � ⊥ m (Given)2. �1 is a right angle. (Def. of ⊥ )3. m�1 � 90 (Def. of rt. �)4. �1 � �4 (Vert. � are �)5. m�1 � m�4 (Def. of � �)6. m�4 � 90 (Substitution)7. �1 and �2 form a linear pair; �3 and �4 form a

linear pair. (Def. of linear pair)8. m�1 � m�2 � 180, m�4 � m�3 � 180 (Linear

pairs are supplementary.)9. 90 � m�2 � 180, 90 � m�3 � 180 (Substitution)

10. m�2 � 90, m�3 � 90 (Subt. Prop.)11. �2, �3, �4 are rt. �. (Def. of rt. � (steps 6, 10))

34. Given: �1 and �2 are rt. �.Prove: �1 � �2Proof:Statements (Reasons)

1. �1 and �2 are rt. �. (Given)2. m�1 � 90, m�2 � 90 (Def. of rt. �)3. m�1 � m�2 (Substitution)4. �1 � �2 (Def. of � angles)

35. Given: � ⊥ mProve: �1 � �2


1. � ⊥ m (Given)2. �1 and �2 are rt. �. (⊥ lines intersect to form 4 rt. �.)3. �1 � �2 (All rt. � are �.)

36. Given: �1 � �2, �1 and �2 are supplementary.

Prove: �1 and �2 are rt. �.


1. �1 � �2, �1 and �2 are supplementary. (Given )2. m�1 � m�2 � 180 (Def. of supplementary �)3. m�1 � m�2 (Def. of � angle)4. m�1 � m�1 � 180 (Substitution)5. 2(m�1) � 180 (Add. Prop.)6. m�1 � 90 (Div. Prop.)7. m�2 � 90 (Substitution (steps 3, 6))8. �1 and �2 are rt. �. (Def. of rt. �)

37. Given: �ABD � �CBD, �ABD and�CBD form a linear pair. Prove: �ABD and �CBD are rt. �.


1. �ABD � �CBD, �ABD and �CBD form a linearpair. (Given)

2. �ABD and �CBD are supplementary. (Linear pairsare supplementary.)

3. �ABD and �CBD are rt. �. (If � are � and suppl.,they are rt. �.)

A CB

D

1 2

1 23 4

�

m

1 2

1 23 4

�

m

1 2 3

A

A FEBDC

Chapter 2 Additional Answers 123F

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38. Given: �ABD � �YXZProve: �CBD � �WXZ


1. �ABD � �YXZ; �ABD and �CBD form a linearpair; �YXZ and �WXZ form a linear pair. (Given;from the figure)

2. m�ABD � m�CBD � 180, m�YXZ � m�WXZ �180 (Linear pairs are supplementary.)

3. m�ABD � m�CBD � m�YXZ � m�WXZ (Subst.)4. m�ABD � m�YXZ (Def. of � �)5. m�YXZ � m�CBD � m�YXZ � m�WXZ (Subst.)6. m�YXZ � m�YXZ (Reflexive Prop.)7. m�CBD � m�WXZ (Subt. Prop.)8. �CBD � �WXZ (Def. of � �)

39. Given: m�RSW � m�TSUProve: m�RST � m�WSU


1. m�RSW � m�TSU (Given)2. m�RSW � m�RST � m�TSW, m�TSU �

m�TSW � m�WSU (Angle Addition Postulate)3. m�RST � m�TSW � m�TSW � m�WSU

(Substitution)4. m�TSW � m�TSW (Reflexive Prop.)5. m�RST � m�WSU (Subt. Prop.)

48. Given: AC � BDProve: AB � CD


1. AC � BD (Given)2. AB � BC � AC, BC � CD � BD (Segment Addition

Postulate)3. BC � BC (Reflexive Prop.)4. AB � BC � BC � CD (Substitution (2 and 3))5. AB � CD (Subt. Prop.)

Page 115-120, Chapter 2 Study Guide and Review

53. Given: BC � EC, CA � CDProve: BA � DE


1. BC � EC, CA � CD (Given)2. BC � CA � EC � CA (Add. Prop.)3. BC � CA � EC � CD (Substitution)4. BC � CA � BA, EC � CD � DE (Seg. Add. Post.)5. BA � DE (Substitution)

54. Given: AB � CDProve: AC � BD


1. AB � CD (Given)2. BC � BC (Reflexive Prop.)3. AB � BC � CD � BC (Add. Prop.)4. AB � BC � AC, CD � BC � BD (Seg. Add. Post.)5. AC � BD (Substitution)

Page 123, Chapter 2 Standardized Test Practice

15. Given: �1 and �3 are vertical angles.m�1 � 3x � 5, m�3 � 2x � 8

Prove: m�1 � 14


a. �1 and �3 are vertical angles; m�1 � 3x � 5,m�3 � 2x � 8 (Given)

b. �1 � �3 (Vert. � are �.)c. m�1 � m�3 (Def. of � �)d. 3x � 5 � 2x � 8 (Substitution)e. x � 5 � 8 (Subt. Prop.)f. x � 3 (Subt. Prop.)g. m�1 � 3(3) � 5 (Substitution)h. m�1 � 14 (Substitution)

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