chapter 2 resource masters - math problem solving · ©glencoe/mcgraw-hill iv glencoe geometry...

Chapter 2Resource Masters

Geometry

Consumable WorkbooksMany of the worksheets contained in the Chapter Resource Masters bookletsare available as consumable workbooks.

Study Guide and Intervention Workbook 0-07-860191-6Skills Practice Workbook 0-07-860192-4Practice Workbook 0-07-860193-2Reading to Learn Mathematics Workbook 0-07-861061-3

ANSWERS FOR WORKBOOKS The answers for Chapter 2 of these workbookscan be found in the back of this Chapter Resource Masters booklet.

Copyright © by The McGraw-Hill Companies, Inc. All rights reserved.Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teachers, and families without charge; and be used solely in conjunction with Glencoe’s Geometry. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher.

Send all inquiries to:The McGraw-Hill Companies8787 Orion PlaceColumbus, OH 43240-4027

ISBN: 0-07-860179-7 GeometryChapter 2 Resource Masters

1 2 3 4 5 6 7 8 9 10 009 11 10 09 08 07 06 05 04 03

© Glencoe/McGraw-Hill iii Glencoe Geometry

Contents

Vocabulary Builder . . . . . . . . . . . . . . . . vii

Proof Builder . . . . . . . . . . . . . . . . . . . . . . ix

Lesson 2-1Study Guide and Intervention . . . . . . . . . 57–58Skills Practice . . . . . . . . . . . . . . . . . . . . . . . . 59Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60Reading to Learn Mathematics . . . . . . . . . . . 61Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 62







Lesson 2-8Study Guide and Intervention . . . . . . . . 99–100Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 101Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 102Reading to Learn Mathematics . . . . . . . . . . 103Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 104

Chapter 2 AssessmentChapter 2 Test, Form 1 . . . . . . . . . . . . 105–106Chapter 2 Test, Form 2A . . . . . . . . . . . 107–108Chapter 2 Test, Form 2B . . . . . . . . . . . 109–110Chapter 2 Test, Form 2C . . . . . . . . . . . 111–112Chapter 2 Test, Form 2D . . . . . . . . . . . 113–114Chapter 2 Test, Form 3 . . . . . . . . . . . . 115–116Chapter 2 Open-Ended Assessment . . . . . . 117Chapter 2 Vocabulary Test/Review . . . . . . . 118Chapter 2 Quizzes 1 & 2 . . . . . . . . . . . . . . . 119Chapter 2 Quizzes 3 & 4 . . . . . . . . . . . . . . . 120Chapter 2 Mid-Chapter Test . . . . . . . . . . . . 121Chapter 2 Cumulative Review . . . . . . . . . . . 122Chapter 2 Standardized Test Practice . 123–124

Standardized Test Practice Student Recording Sheet . . . . . . . . . . . . . . A1

ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A35

© Glencoe/McGraw-Hill iv Glencoe Geometry

Teacher’s Guide to Using theChapter 2 Resource Masters

The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 2 Resource Masters includes the core materials neededfor Chapter 2. These materials include worksheets, extensions, and assessment options.The answers for these pages appear at the back of this booklet.

All of the materials found in this booklet are included for viewing and printing in theGeometry TeacherWorks CD-ROM.

Vocabulary Builder Pages vii–viiiinclude a student study tool that presentsup to twenty of the key vocabulary termsfrom the chapter. Students are to recorddefinitions and/or examples for each term.You may suggest that students highlight orstar the terms with which they are notfamiliar.

WHEN TO USE Give these pages tostudents before beginning Lesson 2-1.Encourage them to add these pages to theirGeometry Study Notebook. Remind them toadd definitions and examples as theycomplete each lesson.

Vocabulary Builder Pages ix–xinclude another student study tool thatpresents up to fourteen of the key theoremsand postulates from the chapter. Studentsare to write each theorem or postulate intheir own words, including illustrations ifthey choose to do so. You may suggest thatstudents highlight or star the theorems orpostulates with which they are not familiar.

WHEN TO USE Give these pages tostudents before beginning Lesson 2-1.Encourage them to add these pages to theirGeometry Study Notebook. Remind them toupdate it as they complete each lesson.

Study Guide and InterventionEach lesson in Geometry addresses twoobjectives. There is one Study Guide andIntervention master for each objective.

WHEN TO USE Use these masters asreteaching activities for students who needadditional reinforcement. These pages canalso be used in conjunction with the StudentEdition as an instructional tool for studentswho have been absent.

Skills Practice There is one master foreach lesson. These provide computationalpractice at a basic level.

WHEN TO USE These masters can be used with students who have weakermathematics backgrounds or needadditional reinforcement.

Practice There is one master for eachlesson. These problems more closely followthe structure of the Practice and Applysection of the Student Edition exercises.These exercises are of average difficulty.

WHEN TO USE These provide additionalpractice options or may be used ashomework for second day teaching of thelesson.

Reading to Learn MathematicsOne master is included for each lesson. Thefirst section of each master asks questionsabout the opening paragraph of the lessonin the Student Edition. Additionalquestions ask students to interpret thecontext of and relationships among termsin the lesson. Finally, students are asked tosummarize what they have learned usingvarious representation techniques.

WHEN TO USE This master can be usedas a study tool when presenting the lessonor as an informal reading assessment afterpresenting the lesson. It is also a helpfultool for ELL (English Language Learner)students.

© Glencoe/McGraw-Hill v Glencoe Geometry

Enrichment There is one extensionmaster for each lesson. These activities mayextend the concepts in the lesson, offer anhistorical or multicultural look at theconcepts, or widen students’ perspectives onthe mathematics they are learning. Theseare not written exclusively for honorsstudents, but are accessible for use with alllevels of students.

WHEN TO USE These may be used asextra credit, short-term projects, or asactivities for days when class periods areshortened.

Assessment OptionsThe assessment masters in the Chapter 2Resources Masters offer a wide range ofassessment tools for intermediate and finalassessment. The following lists describe eachassessment master and its intended use.

Chapter Assessment CHAPTER TESTS• Form 1 contains multiple-choice questions

and is intended for use with basic levelstudents.

• Forms 2A and 2B contain multiple-choicequestions aimed at the average levelstudent. These tests are similar in formatto offer comparable testing situations.

• Forms 2C and 2D are composed of free-response questions aimed at the averagelevel student. These tests are similar informat to offer comparable testingsituations. Grids with axes are providedfor questions assessing graphing skills.

• Form 3 is an advanced level test withfree-response questions. Grids withoutaxes are provided for questions assessinggraphing skills.

All of the above tests include a free-response Bonus question.

• The Open-Ended Assessment includesperformance assessment tasks that aresuitable for all students. A scoring rubricis included for evaluation guidelines.Sample answers are provided forassessment.

• A Vocabulary Test, suitable for allstudents, includes a list of the vocabularywords in the chapter and ten questionsassessing students’ knowledge of thoseterms. This can also be used in conjunc-tion with one of the chapter tests or as areview worksheet.

Intermediate Assessment• Four free-response quizzes are included

to offer assessment at appropriateintervals in the chapter.

• A Mid-Chapter Test provides an optionto assess the first half of the chapter. It iscomposed of both multiple-choice andfree-response questions.

Continuing Assessment• The Cumulative Review provides

students an opportunity to reinforce andretain skills as they proceed throughtheir study of Geometry. It can also beused as a test. This master includes free-response questions.

• The Standardized Test Practice offerscontinuing review of geometry conceptsin various formats, which may appear onthe standardized tests that they mayencounter. This practice includes multiple-choice, grid-in, and short-responsequestions. Bubble-in and grid-in answersections are provided on the master.

Answers• Page A1 is an answer sheet for the

Standardized Test Practice questionsthat appear in the Student Edition onpages 122–123. This improves students’familiarity with the answer formats theymay encounter in test taking.

• The answers for the lesson-by-lessonmasters are provided as reduced pageswith answers appearing in red.

• Full-size answer keys are provided forthe assessment masters in this booklet.

Reading to Learn MathematicsVocabulary Builder

NAME ______________________________________________ DATE ____________ PERIOD _____

22

© Glencoe/McGraw-Hill vii Glencoe Geometry

Voca

bula

ry B

uild

erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 2.As you study the chapter, complete each term’s definition or description. Rememberto add the page number where you found the term. Add these pages to yourGeometry Study Notebook to review vocabulary at the end of the chapter.

Vocabulary Term Found on Page Definition/Description/Example

biconditional

conjecture

kuhn·JEK·chur

conjunction

contrapositive

converse

counterexample

deductive reasoning

disjunction

if-then statement

(continued on the next page)

© Glencoe/McGraw-Hill viii Glencoe Geometry

Vocabulary Term Found on Page Definition/Description/Example

inductive reasoning

inverse

negation

paragraph proof

postulate

statement

theorem

truth table

truth value

two-column proof

Reading to Learn MathematicsVocabulary Builder (continued)

NAME ______________________________________________ DATE ____________ PERIOD _____

22

Learning to Read MathematicsProof Builder

NAME ______________________________________________ DATE ____________ PERIOD _____

22

© Glencoe/McGraw-Hill ix Glencoe Geometry

Proo

f Bu

ilderThis is a list of key theorems and postulates you will learn in Chapter 2. As you

study the chapter, write each theorem or postulate in your own words. Includeillustrations as appropriate. Remember to include the page number where youfound the theorem or postulate. Add this page to your Geometry Study Notebookso you can review the theorems and postulates at the end of the chapter.

Theorem or Postulate Found on Page Description/Illustration/Abbreviation

Theorem 2.1Midpoint Theorem

Theorem 2.2

Theorem 2.3Supplement Theorem

Theorem 2.4Complement Theorem

Theorem 2.5

Theorem 2.6

Theorem 2.7


© Glencoe/McGraw-Hill x Glencoe Geometry

Theorem or Postulate Found on Page Description/Illustration/Abbreviation

Theorem 2.8Vertical Angle Theorem

Theorem 2.9

Theorem 2.10

Theorem 2.11

Theorem 2.12

Theorem 2.13

Postulate 2.9Segment Addition Postulate

Learning to Read MathematicsProof Builder (continued)

NAME ______________________________________________ DATE ____________ PERIOD _____

22

Study Guide and InterventionInductive Reasoning and Conjecture

NAME ______________________________________________ DATE ____________ PERIOD _____

2-12-1

© Glencoe/McGraw-Hill 57 Glencoe Geometry

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

2. 1, 10, 100, 1000

3. 1, �65�, �

75�, �

85�

Make a conjecture based on the given information. Draw a figure to illustrateyour conjecture.

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

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

FERT

QP

DC

EBA

T W

12

P

R

A(–1, –1)

B(2, 2)

C(4, 4)


Find Counterexamples A conjecture is false if there is even one situation in which theconjecture is not true. The false example is called a counterexample.

Determine whether the conjecture is true or false.If it is false, give a counterexample.Given: A�B� � B�C�Conjecture: B is the midpoint of A�C�.Is it possible to draw a diagram with A�B� � B�C� such that B is not the midpoint? This diagram is a counterexample because point B is not on A�C�. The conjecture is false.

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

1. Given: Points A, B, and C are collinear. 2. Given: �R and �S are supplementary.Conjecture: AB � BC � AC �R and �T are supplementary.

Conjecture: �T and �S are congruent.

3. Given: �ABC and �DEF are 4. Given: D�E� ⊥ E�F�supplementary. Conjecture: �DEF is a right angle.Conjecture: �ABC and �DEF form a linear pair.

EB FA

DC

AC

B

A

C

3 cm3 cm

B

Study Guide and Intervention (continued)

Inductive Reasoning and Conjecture

NAME ______________________________________________ DATE ____________ PERIOD _____

2-12-1

ExampleExample

ExercisesExercises

Skills PracticeInductive Reasoning and Conjecture

NAME ______________________________________________ DATE ____________ PERIOD _____

2-12-1


Less

on

2-1

Make a conjecture about the next item in each sequence.

1.

2. �4, �1, 2, 5, 8 3. 6, �121�, 5, �

92�, 4 4. �2, 4, �8, 16, �32


5. Points A, B, and C are collinear, 6. Point P is the midpoint of N�Q�.and D is between B and C.

7. �1, �2, �3, and �4 form four 8. �3 � �4linear pairs.


9. Given: �ABC and �CBD form a linear pair.Conjecture: �ABC � �CBD

10. Given: A�B�, B�C�, and A�C� are congruent.Conjecture: A, B, and C are collinear.

11. Given: AB � BC � ACConjecture: AB � BC

12. Given: �1 is complementary to �2, and �1 is complementary to �3.Conjecture: �2 � �3

A CB

431 243

N QPA C D B


Make a conjecture about the next item in each sequence.

1.

2. 5, �10, 15, �20 3. �2, 1, ��12�, �

14�, ��

18� 4. 12, 6, 3, 1.5, 0.75


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

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


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

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

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

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.

D

A

C

B

S

P

R

Q

TSRA

CB

Practice Inductive Reasoning and Conjecture

NAME ______________________________________________ DATE ____________ PERIOD _____

2-12-1

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?

• What is a factor that might contribute to long-term changes in theweather?

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

inductive reasoning.

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

a. 5, 9, 13, 17 b. 1, �13�, �

19�, �2

17�

c. 0, 1, 3, 6, 10 d. 8, 3, �2, �7e. 1, 8, 27, 64 f. 1, �2, 4, �8g. 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.

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

c. The opposite of an integer is a negative integer.

d. The perfect squares (squares of whole numbers) alternate between odd and even.

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.


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?January?

3. Is the equation �k2� � k true when you 4. Is the statement 2x � x � x true whenreplace k with 1, 2, and 3? Is the you replace x with �

12�, 4, and 0.7? Is the

equation true for all integers? If statement true for all real numbers? possible, find a counterexample. If possible, find a counterexample.

5. Suppose you draw four points A, B, C, 6. Suppose you draw a circle, mark three and D and then draw A�B�, B�C�, C�D�, and points on it, and connect them. Will the

D�A�. Does this procedure give a angles of the triangle be acute? Explain

quadrilateral always or only sometimes? your answers with figures.

Explain your answers with figures.

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

2-12-1

ExampleExample

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 .

2. p or r

3. q or r

4. q and �r


Truth Tables One way to organize the truth values of statements is in a truth table. The truth tables for negation, conjunction, and disjunction are shown at the right.

Disjunction

p q p � q

T T TT F TF T TF F F

Conjunction

p q p � q

T T TT F FF T FF F F

Negation

p �p

T FF T


Logic

NAME ______________________________________________ DATE ____________ PERIOD _____

2-22-2

Construct a truthtable for the compoundstatement q or r. Use thedisjunction table.

q r q or r

T T TT F TF T TF F F

Construct a truth table for thecompound statement p and (q or r).Use the disjunction table for (q or r). Then use theconjunction table for p and (q or r).

p q r q or r p and (q or r )

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


ExercisesExercises

Contruct a truth table for each compound statement.

1. p or r 2. �p � q 3. q � �r

4. �p � � r 5. ( p and r) or q

Skills PracticeLogic

NAME ______________________________________________ DATE ____________ PERIOD _____

2-22-2


Less

on

2-2

Use the following statements to write a compound statement for each conjunctionand disjunction. Then find its truth value.p: �3 � 2 � �5q: Vertical angles are congruent.r: 2 � 8 � 10s: The sum of the measures of complementary angles is 90°.

1. p and q

2. p � r

3. p or s

4. r � s

5. p � �q

6. q � �r

Copy and complete each truth table.

7. 8.

Construct a truth table for each compound statement.

9. �q � r 10. �p � �r

p q �q p � �q

T T F

T F T

F T F

F F T

p q �p �p � q �(�p � q)

T T

T F

F T

F F


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

2. q � r

3. �p � q

4. �p � �r

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?

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

WorkWeekends

17

WorkAfter

School5

3

p q �p �p � q p � (�p � q)

T T

T F

F T

F F

p q �p �q �p � �q

T T

T F

F T

F F

Practice Logic

NAME ______________________________________________ DATE ____________ PERIOD _____

2-22-2

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?

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.b. Sacramento is the capital of California and Chicago is not the capital of Illinois.

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.


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.

1. 2.

H � A � L � T � W � O �

F � W � O � F � U � R �

E �

3. 4.

T � H � R � S � E � N �

E � O � N � D � M � O �

S � V � R � Y �

5. Do research to find more alphametic puzzles, or create your own puzzles. Challengeanother student to solve them.

SEND� MORE��

MONEY

THREETHREE

� ONE��

SEVEN

TWO� TWO��

FOUR

HALF� HALF��

WHOLE

8310� 347��

8657

FOUR� ONE��

F I VE

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

2-22-2

ExampleExample

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.

2. If x � 8 � 32, then x � 40.

3. If a polygon has four right angles, then the polygon is a rectangle.

Write each statement in if-then form.

4. All apes love bananas.

5. The sum of the measures of complementary angles is 90.

6. Collinear points 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.

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

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

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


Converse, Inverse, and Contrapositive If you change the hypothesis or conclusionof a conditional statement, you form a related conditional. This chart shows the threerelated conditionals, converse, inverse, and contrapositive, and how they are related to aconditional statement.

Symbols Formed by Example

Conditional p → q using the given hypothesis and conclusion If two angles are vertical angles, then they are congruent.

Converse q → p exchanging the hypothesis and conclusion If two angles are congruent, then they are vertical angles.

Inverse �p → �q replacing the hypothesis with its negation If two angles are not vertical angles,and replacing the conclusion with its negation then they are not congruent.

Contrapositive �q → �p negating the hypothesis, negating the If two angles are not congruent,conclusion, and switching them then they are not vertical angles.

Just as a conditional statement can be true or false, the related conditionals also can be trueor false. A conditional statement always has the same truth value as its contrapositive, andthe converse and inverse always have the same truth value.

Write the converse, inverse, and contrapositive of each conditional statement. Tellwhich statements are true and which statements are false.

1. If you live in San Diego, then you live in California.

2. If a polygon is a rectangle, then it is a square.

3. If two angles are complementary, then the sum of their measures is 90.


Conditional Statements

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1. If you purchase a computer and do not like it, then you can return it within 30 days.

2. If x � 8 � 4, then x � �4.

3. If the drama class raises $2000, then they will go on tour.


4. A polygon with four sides is a quadrilateral.

5. “Those who stand for nothing fall for anything.” (Alexander Hamilton)

6. An acute angle has a measure less than 90.

Determine the truth value of the following statement for each set of conditions.If you finish your homework by 5 P.M., then you go out to dinner.

7. You finish your homework by 5 P.M. and you go out to dinner.

8. You finish your homework by 4 P.M. and you go out to dinner.

9. You finish your homework by 5 P.M. and you do not go out to dinner.

10. 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 89 is divisible by 2, then 89 is an even number.



1. If 3x � 4 � �5, then x � �3.

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


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

4. Adjacent angles share a common vertex and a common 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.

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

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

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.

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.

10. Write the converse of your conditional statement.

Practice Conditional Statements

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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.

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.

b. If two integers are even, 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.

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.

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.

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.

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?


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.

3. People who live in Iowa like corn. 4. Staff members are allowed in the faculty lounge.

people in thecafeteria

staffmembers

people wholike corn

people wholive in Iowa

real numbers

rationalnumbers

animals withlong hair

dogs

animals withlong ears

rabbits

Enrichment

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Study Guide and InterventionDeductive Reasoning

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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.

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

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

A D

EF

H

J

C

GB

ExampleExample

ExercisesExercises


Law of Syllogism Another way to make a valid conclusion is to use the Law ofSyllogism. It is similar to the Transitive Property.

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

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

The two conditional statements below are true. Use the Law ofSyllogism to find a valid conclusion. State the conclusion.(1) If a number is a whole number, then the number is an integer.(2) If a number is an integer, then it is a rational number.

p: A number is a whole number.q: A number is an integer.r: A number is a rational number.

The two conditional statements are p → q and q → r. Using the Law of Syllogism, a validconclusion is p → r. A statement of p → r is “if a number is a whole number, then it is arational number.”

Determine whether you can use the Law of Syllogism to reach a valid conclusionfrom each set of statements.

1. If a dog eats Superdog Dog Food, he will be happy.Rover is happy.

2. If an angle is supplementary to an obtuse angle, then it is acute.If an angle is acute, then its measure is less than 90.

3. If the measure of �A is less than 90, then �A is acute.If �A is acute, then �A � �B.

4. If an angle is a right angle, then the measure of the angle is 90.If two lines are perpendicular, then they form a right angle.

5. If you study for the test, then you will receive a high grade.Your grade on the test is high.


Deductive Reasoning

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ExampleExample

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Determine whether the stated conclusion is valid based on the given information.If not, write invalid. Explain your reasoning.If the sum of the measures of two angles is 180, then the angles are supplementary.

1. Given: m�A � m�B is 180.Conclusion: �A and �B are supplementary.

2. Given: m�ABC is 95 and m�DEF is 90.Conclusion: �ABC and �DEF are supplementary.

3. Given: �1 and �2 are a linear pair.Conclusion: �1 and �2 are supplementary.

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.

4. If two angles are complementary, then the sum of their measures is 90.If the sum of the measures of two angles is 90, then both of the angles are acute.

5. If the heat wave continues, then air conditioning will be used more frequently.If air conditioning is used more frequently, then energy costs will be higher.

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.

6. (1) If it is Tuesday, then Marla tutors chemistry.(2) If Marla tutors chemistry, then she arrives home at 4 P.M.(3) If Marla arrives at home at 4 P.M., then it is Tuesday.

7. (1) If a marine animal is a starfish, then it lives in the intertidal zone of the ocean.(2) The intertidal zone is the least stable of the ocean zones.(3) If a marine animal is a starfish, then it lives in the least stable of the ocean zones.


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�

2. Given: A�B� � B�C�Conclusion: B divides A�C� into two congruent segments.

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.

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.

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.

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.

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?

Practice Deductive Reasoning

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Reading to Learn MathematicsDeductive Reasoning

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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?

.

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 a. s � u

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

3. converse of s → t c. �s → �u

4. disjunction of s and u d. �u → �s

5. Law of Detachment e. �t

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

7. inverse of s → u g. s � u

8. contrapositive of s → u h. t → s

9. Law of Syllogism i. t

10. negation of �t 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.

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.

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.

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?


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.Conclusion: Justin’s luggage will Conclusion: Justin has Tuff Cotesurvive airline travel. luggage.

3. (1) If you use Clear Line long 4. (1) If you read the book Beautiful distance service, you will have Braids, you will be able to makeclear reception. beautiful braids easily.

(2) Anna has clear long distance (2) Nancy read the book Beautifulreception. Braids.

Conclusion: Anna uses Clear Line Conclusion: Nancy can make long distance service. beautiful braids easily.

5. (1) If you buy a word processor, you 6. (1) Great swimmers wear AquaLinewill be able to write letters faster. swimwear.

(2) Tania bought a word processor. (2) Gina wears AquaLine swimwear.Conclusion: Tania will be able to Conclusion: Gina is a great swimmer.write letters faster.

7. Write an example of faulty logic thatyou have seen in an advertisement.

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

Enrichment

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Study Guide and InterventionPostulates and Paragraph Proofs

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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.

2. Noncollinear points R, S, and T are contained in exactly one plane.

3. Any two lines � and m intersect.

4. If points G and H are contained in plane M, then G�H� is perpendicular to plane M.

5. Planes R and S intersect in point T.

6. If points A, B, and C are noncollinear, then segments A�B�, B�C�, and C�A� are contained inexactly one plane.

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.

8. BE�� lies in plane Q.

QB

C

A

DE

F

G

ExampleExample

ExercisesExercises


Paragraph Proofs A statement that can be proved true is called a theorem. You canuse undefined terms, definitions, postulates, and already-proved theorems to prove otherstatements true.

A logical argument that uses deductive reasoning to reach a valid conclusion is called aproof. In one type of proof, a paragraph proof, you write a paragraph to explain why astatement is true.

In �ABC, B�D� is an angle bisector. Write a paragraph proof to show that �ABD � �CBD.By definition, an angle bisector divides an angle into two congruentangles. Since B�D� is an angle bisector, �ABC is divided into twocongruent angles. Thus, �ABD � �CBD.

1. Given that �A � �D and �D � �E, write a paragraph proof to show that �A � �E.

2. It is given that B�C� � E�F�, M is the midpoint of B�C�, and N is the midpoint of E�F�. Write a paragraph proof to show that BM � EN.

3. Given that S is the midpoint of Q�P�, T is the midpoint of P�R�,and P is the midpoint of S�T�, write a paragraph proof to show that QS � TR. Q

RTP

S

B

CM

EN

F

B

AD C


Postulates and Paragraph Proofs

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ExampleExample

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Determine the number of line segments that can be drawn connecting each pairof points.

1. 2.

Determine whether the following statements are always, sometimes, or never true.Explain.

3. Three collinear points determine a plane.

4. Two points A and B determine a line.

5. A plane contains at least three lines.

In the figure, DG�� and DP�� lie in plane J and H lies on DG��. State the postulate that can be used to show each statement is true.

6. G and P are collinear.

7. Points D, H, and P are coplanar.

8. PROOF In the figure at the right, point B is the midpoint of A�C� and point C is the midpoint of B�D�. Write a paragraph proof to prove thatAB � CD.

DCBA

J

PD

H

G


Determine the number of line segments that can be drawn connecting each pairof points.

1. 2.

Determine whether the following statements are always, sometimes, or never true.Explain.

3. The intersection of two planes contains at least two points.

4. If three planes have a point in common, then they have a whole line 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.

6. Line m and S�T� intersect at T.

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�.

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.

BE

C

D

A

AmT

QL

S

Practice Postulates and Paragraph Proofs

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Reading to Learn MathematicsPostulates and Paragraph Proofs

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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.

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.

a. A plane contains at least that do not lie on the same line.

b. If intersect, then the intersection is a line.

c. Through any not on the same line, there is exactly one plane.

d. A line contains at least .

e. If two lines , then their intersection is exactly one point.

f. Through any two points, there is one line.

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.

b. The midpoint of a segment divides the segment into two congruent segments.

c. There is exactly one plane that contains three collinear points.

d. If two lines intersect, their intersection is one point.

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

b. two planes that do not intersect

c. three planes that intersect in a point

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.

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

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Nancy Olivia Mario Kenji

Peach

Orange

Banana

Apple

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?

3. Mr. Guthrie, Mrs. Hakoi, Mr. Mirza, andMrs. Riva have jobs of doctor, accountant,teacher, and office manager. Mr. Mirzalives near the doctor and the teacher.Mrs. Riva is not the doctor or the officemanager. Mrs. Hakoi is not theaccountant or the office manager. Mr.Guthrie went to lunch with the doctor.Mrs. Riva’s son is a high school studentand is only seven years younger thanhis algebra teacher. Which person haseach occupation?

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?

4. Yvette, Lana, Boris, and Scott each havea dog. The breeds are collie, beagle,poodle, and terrier. Yvette and Boriswalked to the library with the studentwho has a collie. Boris does not have apoodle or terrier. Scott does not have acollie. Yvette is in math class with thestudent who has a terrier. Whichstudent has each breed of dog?

Study Guide and InterventionAlgebraic Proof

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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

Complete each proof.

ExampleExample

ExercisesExercises

1. Given: �4x

2� 6� � 9

Prove: x � 3

Statements Reasons

a. �4x

2� 6� � 9 a.

b. ��4x2� 6�� 2(9) b. Mult. Prop.

c. 4x � 6 � 18 c.

d. 4x � 6 � 6 � 18 � 6 d.

e. 4x � e. Substitution

f. �44x� � f. Div. Prop.

g. g. Substitution

2. Given: 4x � 8 � x � 2Prove: x � �2

Statements Reasons

a. 4x � 8 � x � 2 a.

b. 4x � 8 � x �x � 2 � x b.

c. 3x � 8 � 2 c. Substitution

d.d. Subtr. Prop.

e. e. Substitution

f. �33x� � �

�36� f.

g. g. Substitution


Geometric Proof Geometry deals with numbers as measures, so geometric proofs useproperties of numbers. Here are some of the algebraic properties used in proofs.

Property Segments Angles

Reflexive AB � AB m�A � m�A

Symmetric If AB � CD, then CD � AB. If m�A � m�B, then m�B � m�A.

Transitive If AB � CD and CD � EF, then AB � EF. If m�1� m�2 and m�2 � m�3, then m�1� m�3.

Write a two-column proof.Given: m�1 � m�2, m�2 � m�3Prove: m�1 � m�3Proof:

Statements Reasons

1. m�1 � m�2 1. Given2. m�2 � m�3 2. Given3. m�1 � m�3 3. Transitive Property

State the property that justifies each statement.

1. If m�1 � m�2, then m�2 � m�1.

2. If m�1� 90 and m�2 � m�1, then m�2� 90.

3. If AB � RS and RS � WY, then AB � WY.

4. If AB � CD, then �12�AB � �

12�CD.

5. If m�1 � m�2 � 110 and m�2 � m�3, then m�1 � m�3 � 110.

6. RS � RS

7. If AB � RS and TU � WY, then AB � TU � RS � WY.

8. If m�1 � m�2 and m�2 � m�3, then m�1 � m�3.

9. A formula for the area of a triangle is A � �

12�bh. Prove that bh is equal

to 2 times the area of the triangle.

DA

R

TS

CB

1 23


Algebraic Proof

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ExampleExample

ExercisesExercises

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State the property that justifies each statement.

1. If 80 � m�A, then m�A � 80.

2. If RS � TU and TU � YP, then RS � YP.

3. If 7x � 28, then x � 4.

4. If VR � TY � EN � TY, then VR � EN.

5. If m�1 � 30 and m�1 � m�2, then m�2 � 30.

Complete the following proof.

6. Given: 8x � 5 � 2x � 1Prove: x � 1Proof:

Statements Reasonsa. 8x � 5 � 2x � 1 a.

b. 8x � 5 � 2x � 2x � 1 � 2x b.

c. c. Substitution Property

d. d. Addition Property

e. 6x � 6 e.

f. �66x� � �

66� f.

g. g.

Write a two-column proof for the following.

7. If P�Q� � Q�S� and Q�S� � S�T�, then PQ � ST. QP

TS


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.

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.

A

D C

B

Practice Algebraic Proof

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Reading to Learn MathematicsAlgebraic Proof

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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?

Reading the Lesson1. Name the property illustrated by each statement.

a. If a � 4.75 and 4.75 � b, then a � b.b. If x � y, then x � 8 � y � 8.c. 5(12 � 19) � 5 � 12 � 5 � 19d. If x � 5, then x may be replaced with 5 in any equation or expression.e. If x � y, then 8x � 8y.f. If x � 23.45, then 23.45 � x.g. If 5x � 7, then x � �

75�.

h. If x � 12, then x � 3 � 9.

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.

2. 5n � 15 � 8n � 28 � 14 2.

3. 5n � 15 � 8n � 42 3.

4. 5n � 15 � 15 � 8n � 42 � 15 4.

5. 5n � 8n � 27 5.

6. 5n � 8n � 8n � 27 � 8n 6.

7. �3n � �27 7.

8. ��33n

� � ��237

� 8.

9. n � 9 9.

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.


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.

1. is the same size as 2. is a family descendant of

3. is in the same room as 4. is the identical twin of

5. is warmer than 6. is on the same line as

7. is a sister of 8. is the same weight as

9. Find two other examples of relations,and tell which properties are true foreach relation.

a � aIf a � b, then b � a.If a � b and b � c, then a � c.

Enrichment

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Study Guide and InterventionProving Segment Relationships

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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.

b. b. Seg. Add. Post.

c. AB � DE � AC c.

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.c. R is between c.

Q and S.d. d. Seg. Add. Post.e. PR � QS e.f. PQ � QR � f.

QR � RSg.PQ � QR � QR � g.

QR � RS � QRh. h. Substitution

P QR S


Segment Congruence Three properties of algebra—the Reflexive, Symmetric, andTransitive Properties of Equality—have counterparts as properties of geometry. Theseproperties can be proved as a theorem. As with other theorems, the properties can then beused to prove relationships among segments.

Segment Congruence Theorem Congruence 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�.

Write a two-column proof.Given: A�B� � D�E�; B�C� � E�F�Prove: A�C� � D�F�Statements Reasons1. A�B� � D�E� 1. Given2. AB � DE 2. Definition of congruence of segments3. B�C� � E�F� 3. Given4. BC � EF 4. Definition of congruence of segments5. AB � BC � DE � EF 5. Addition Property6. AB � BC � AC, DE � EF � DF 6. Segment Addition Postulate7. AC � DF 7. Substitution8. A�C� � D�F� 8. Definition of congruence of segments

Justify each statement with a property of congruence.

1. If D�E� � G�H�, then G�H� � D�E�.

2. If A�B� � R�S� and R�S� � W�Y�, then A�B� � W�Y�.

3. R�S� � R�S� .

4. Complete the proof.Given: P�R� � Q�S�Prove: P�Q� � R�S�

Statements Reasons

a. P�R� � Q�S� a.b. PR � QS b.c. PQ � QR � PR c.d. d. Segment Addition Postulatee. PQ � QR � QR � RS e.f. f. Subtraction Propertyg. g. Definition of congruence of segments

P QR S

DE FA

B C


Proving Segment Relationships

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ExampleExample

ExercisesExercises

Skills PracticeProving Segment Relationships

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Justify each statement with a property of equality, a property of congruence, or apostulate.

1. QA � QA

2. If A�B� � B�C� and B�C� � C�E�, then A�B� � C�E�.

3. If Q is between P and R, then PR � PQ � QR.

4. If AB � BC � EF � FG and AB � BC � AC, then EF � FG � AC.


5. Given: S�U� � L�R�T�U� � L�N�

Prove: S�T� � N�R�Proof:

Statements Reasons

a. S�U� � L�R�, T�U� � L�N� a.b. b. Definition of � segments

c. SU � ST � TU c.LR � LN � NR

d. ST � TU � LN � NR d.e. ST � LN � LN � NR e.f. ST � LN � LN � LN � NR � LN f.g. g. Substitution Property

h. S�T� � N�R� h.

6. Given: A�B� � C�D�Prove: C�D� � A�B�Proof:

Statements Reasons

a. a. Given

b. AB � CD b.c. CD � AB c.d. d. Definition of � segments

US T

RL N


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. Given

b. AB � DE b.

c. c. Definition of Midpoint

d. AC � AB � BC d.

DF � DE � EF

e. AB � BC � DE � EF e.

f. AB � BC � AB � EF f.

g. AB � BC � AB � AB � EF � AB g. Subtraction Property

h. BC � EF h.

i. i.

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.

Grayson Apex Redding Pine Bluff

G A R P

CA B

FD

E

Practice Proving Segment Relationships

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Reading to Learn MathematicsProving Segment Relationships

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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?

• 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?

Reading the Lesson1. If E is between Y and S, which of the following statements are always true?

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.

2. BC � CD 2.

3. D is the midpoint of C�E�. 3.

4. CD � DE 4.

5. BC � DE 5.

6. BC � CD � CD � DE 6.7. BC � CD � BD 7.

CD � DE � CE

8. BD � CE 8.

9. B�D� � C�E� 9.

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?

A

B EDC


Enrichment

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2-72-7

Geometry Crossword Puzzle

ACROSS

3. Points on the same line are��.

4. A point on a line and allpoints of the line to one side of it.

9. An angle whose measure isgreater than 90.

10. Two endpoints and all pointsbetween them.

16. A flat figure with no thicknessthat extends indefinitely in alldirections.

17. Segments of equal length are�� segments.

18. Two noncollinear rays with acommon endpoint.

19. If m�A � m�B � 180, then�A and �B are �� angles.

DOWN

1. The set of all points collinear to two pointsis a ��.

2. The point where the x- and y-axis meet.

5. An angle whose measure is less than 90.

6. If m�A � m�D � 90, then �A and �D are�� angles.

7. Lines that meet at a 90° angle are ��.

8. Two angles with a common side but no commoninterior points are ��.

10. An “angle” formed by opposite rays is a �� angle.

11. The middle point of a line segment.

12. Points that lie in the same plane are ��.

13. The four parts of a coordinate plane.

14. Two nonadjacent angles formed by twointersecting lines are �� angles.

15. In angle ABC, point B is the ��.

1 2

3 4 5

6 7

9

14

18

15

17

1110

8

1312

16

19

Study Guide and InterventionProving Angle Relationships

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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 � 10

m�8 � 12x � 12

F CJ

HA 11

1213

WU

X YZ

V58

67R

TP

Q

S

87


Congruent and Right Angles Three properties of angles can be proved as theorems.

Congruence of angles is reflexive, symmetric, and transitive.

Angles supplementary to the same angle Angles complementary to the same angle or to or to congruent angles are congruent. congruent angles are congruent.

If �1 and �2 are supplementary to �3, If �4 and �5 are complementary to �6, then �1 � �2. then �4 � �5.

Write a two-column proof.Given: �ABC and �CBD are complementary.

�DBE and �CBD form a right angle.Prove: �ABC � �DBE

Statements Reasons1. �ABC and �CBD are complementary. 1. Given

�DBE and �CBD form a right angle.2. �DBE and �CBD are complementary. 2. Complement Theorem3. �ABC � �DBE 3. Angles complementary to the same � are �.


B E

D

CA

ML

NK

S

TR

PQ

45

6A B

C

F

HJ

G

ED

1

2 3


Proving Angle Relationships

NAME ______________________________________________ DATE ____________ PERIOD _____

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ExampleExample

ExercisesExercises

1. Given: A�B� ⊥ B�C�;�1 and �3 are complementary.Prove: �2 � �3

Statements Reasons

a. A�B� ⊥ B�C� a.b. b. Definition of ⊥

c. m�1 � m�2 � c.m�ABC

d. �1 and �2 form d.a rt �.

e. �1 and �2 are e.compl.

f. f. Given

g. �2 � �3 g.

B C

EA1 2

3D

2. Given: �1 and �2 form a linear pair.m�1 � m�3 � 180Prove: �2 � �3

Statements Reasons

a. �1 and �2 form a. Givena linear pair.m�1 � m�3 � 180

b. b. Suppl.Theorem

c. �1 is suppl. c.to �3.

d. d.� suppl. to thesame � are �.

BR

T

L

1 23

P

Skills PracticeProving Angle Relationships

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1. m�2 � 57 2. m�5 � 22 3. m�1 � 38

4. m�13 � 4x � 11, 5. �9 and �10 are 6. m�2 � 4x � 26,m�14 � 3x � 1 complementary. m�3 � 3x � 4

�7 � �9, m�8 � 41

Determine whether the following statements are always, sometimes, or never true.

7. Two angles that are supplementary form a linear pair.

8. Two angles that are vertical are adjacent.

9. Copy and complete the following proof.Given: �QPS � �TPRProve: �QPR � �TPSProof:

Statements Reasons

a. a.b. m�QPS � m�TPR b.c. m�QPS � m�QPR � m�RPS c.

m�TPR � m�TPS � m�RPS

d. d. Substitution

e. e.f. f.

Q P

RS

T

237

8 910

13 14

12

651 2



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

Determine whether the following statements are always, sometimes, or never true.

4. Two angles that are supplementary are complementary.

5. Complementary angles are congruent.

6. Write a two-column proof.Given: �1 and �2 form a linear pair.

�2 and �3 are supplementary.Prove: �1 � �3

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? Olive Tree Lane

BartonRd

TryonSt

1 23

6

7

453

1 2

Practice Proving Angle Relationships

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2-82-8

Reading to Learn MathematicsProving Angle Relationships

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Less

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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.

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.b. If two angles form a linear pair, they are complementary.c. Two vertical angles are supplementary.d. Two adjacent angles form a linear pair.e. Two vertical angles form a linear pair.f. Complementary angles are congruent.g. Two angles that are congruent to the same angle are congruent to each other.h. Complementary angles are adjacent angles.

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?


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.

2. Construct the bisectors of �DEC and �BCE.

3. Label the intersection of the two bisectors as point P.

4. Construct the bisectors of �BDE and �DBC.

5. Label the intersection of the two previous bisectors as point Q.

6. Use a straightedge to draw line PQ, which bisects the hidden angle.

7. Another hidden angle is shown at right. Constructthe bisector using themethod above, or deviseyour own method.

AD

E

C

PQ

B

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

2-82-8

Chapter 2 Test, Form 122


Ass

essm

ents

Write the letter for the correct answer in the blank at the right of eachquestion.

1. Make a conjecture about the next term in this sequence: 92, 87, 82, 77, 72.A. �5 B. 62 C. 67 D. 77

2. Make a conjecture given that M is the midpoint of B�C�.A. BM � BC B. BM � MC C. MC � BC D. M bisects �C

3. Given: a � b � 8 and a � 2Conjecture: b � 5Which of the following would be a counterexample?A. b � 3 B. b � 5 C. b � 6 D. b � a

4. If p is true and q is false, what is the truth value of p or q?A. true B. false C. 0 D. 1

For Questions 5 and 6, use this truth table.

5. Which would be the values in the �p column?A. F T F T B. F F T TC. T F F T D. T T F F

6. Which would be the values in the �p � q column?A. F F T F B. T T T FC. T T T T D. T F T T

7. Identify the hypothesis of the statement If x � 4 � 5, then x � 1.A. If x � 1, then x � 4 � 5. B. If x � 4 � 5, then x � 1.C. x � 4 � 5 D. x � 1

8. Identify the converse of the statement If cats fly, then ducks bark.A. If cats don’t fly, then ducks don’t bark.B. If ducks don’t bark, then cats don’t fly.C. If cats bark, then ducks fly.D. If ducks bark, then cats fly.

9. Identify the inverse of the statement If a triangle has 3 equal sides, then it isequilateral.A. If a triangle does not have 3 equal sides, then it is not equilateral.B. If a triangle is equilateral, then it has 3 equal sides.C. If a triangle is not equilateral, then it does not have 3 equal sides.D. If a triangle has 2 equal sides, then it is isosceles.

10. Which of the following illustrates the Law of Detachment?A. [(p → q) � (q → r)] → (p → r) B. [(p → q) � (q → r)] → (p → r)C. [(p → q) � q] → p D. [(p → q) � p] → q

p q �p �p � q

T T

T F

F T

F F

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

NAME DATE PERIOD

SCORE


Chapter 2 Test, Form 1 (continued)22

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

11. Which of the following illustrates the Law of Syllogism?A. [(p → q) � (q ( r)] → (p → r) B. [(p → q) � (q → r)] → (p → r)C. [(p → q) � q] → p D. [(p → q) � p] → q

12. Which best describes the statement A plane contains at least 3 points not onthe same line?A. always true B. sometimes trueC. never true D. cannot tell

13. Which is a type of proof where you write a paragraph to explain why aconjecture for a given situation is true?A. argument B. explanationC. paragraph proof D. two-column proof

For Questions 14–16, choose the property that justifies the statement.

14. If 3x � 6, then x � 2.A. Addition B. Subtraction C. Multiplication D. Division

15. If m�A � 10 and m�B � 10, then m�A � m�B.A. Reflexive B. Symmetric C. Substitution D. Equality

16. If P�S� � W�X�, then PS � WX.A. Reflexive B. SymmetricC. Definition of congruent segments D. Transitive

17. If A, B and N are collinear and AB � BN � AN, which point is between theother two?A. A B. B C. N D. cannot tell

18. Find x.A. 25 B. 35C. 55 D. 125

19. If m�ABD � 56, find m�DBC.A. 124 B. 56C. 44 D. 34

20. If a right angle is trisected, what is the measure of each of the smallerangles?A. 30 B. 45 C. 60 D. 90

Bonus If m�1 is twice m�2, find m�1.

1 2

A

BC

D

55� (x � 30)�

B:

NAME DATE PERIOD

Chapter 2 Test, Form 2A22


Ass

essm

ents


1. Make a conjecture about the next object in this sequence.

A. B. C. D.

2. Given: |n| is a positive number. Conjecture: n is a negative number.Which of the following would be a counterexample?A. �10 B. 0 C. �1 D. 10

3. If p is true and q is false, what is the truth value of p and q?A. true B. false C. 0 D. 1


4. Which would be the values in the �q column?A. F F T T B. T T F FC. F T F T D. T F T F

5. Which would be the values in the p � �q column?A. F T F F B. F T T FC. T T F T D. T F T T

6. Identify the conclusion of the statement Jack will go to school if today is Monday.A. Jack will go to school B. Jack will not go to schoolC. today is Monday D. today is not Monday

7. Identify the inverse of the following statement.If x � 2, then x � 3 � 5.A. If x � 3 � 5, then x � 2. B. If x � 3 � 5, then x � 2.C. If x � 2, then x � 3 � 5. D. x � 2 and x � 3 � 5.

8. Identify the contrapositive of the following statement.If x � 2, then x � 3 � 5.A. If x � 3 � 5, then x � 2. B. If x � 3 � 5, then x � 2.C. If x � 2, then x � 3 � 5. D. x � 2 and x � 3 � 5.

9. Which law can be used to determine that statement (3) is a valid conclusionto statements (1) and (2)?(1) If an angle is acute, then it cannot be obtuse.(2) �A is acute.(3) �A cannot be obtuse.A. Law of Detachment B. Law of SyllogismC. Law of Converse D. Statement (3) does not follow.

p q �q p � �q

T T

T F

F T

F F

1.

2.

3.

4.

5.

6.

7.

8.

9.

NAME DATE PERIOD

SCORE


Chapter 2 Test, Form 2A (continued)22

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

10. Which law can be used to determine that statement (3) is a valid conclusionof statements (1) and (2)?(1) If a figure has 4 right angles, then the figure is a rectangle.(2) A rectangle has 2 pairs of parallel sides.(3) If a figure has 4 right angles, then the figure has 2 pair of parallel sides.A. Law of Detachment B. Law of SyllogismC. Law of Converse D. Statement (3) does not follow.

11. Which best describes the statement If two planes intersect, then theirintersection is a point?A. always true B. sometimes true C. never true D. cannot tell

12. Which of the following is an essential part of a good proof?A. an if-then statement B. a postulateC. using the contrapositive D. a system of deductive reasoning

13. Choose the property that justifies the following statement.If x � 2 and x � y � 3, then 2 � y � 3.A. Reflexive B. Symmetric C. Transitive D. Substitution

14. Choose the property that justifies the statement m�A � m�A.A. Reflexive B. Symmetric C. Transitive D. Substitution

15. Choose the property that justifies the statement If G�H� � F�D�, then F�D� � G�H�.A. Reflexive B. SymmetricC. Transitive D. Definition of congruent segments

16. If XY � 6, YZ � 4, and XZ � 2, which point is between the other two?A. X B. Y C. Z D. cannot tell

For Questions 17 and 18, use the figure at the right.

17. If m�BFC � 70, find m�EFD.A. 10 B. 20C. 35 D. 70

18. If m�AFB � 5x � 10 and m�BFC � 3x � 20, find x.A. 10 B. 15 C. 21.25 D. 23.3�


19. If �ABC � �EFG, and m�ABC � 72, find m�GFH.A. 18 B. 72C. 90 D. 108

20. If m�ABJ � 28, �ABC � �DBJ, find m�JBC.A. 90 B. 56 C. 45 D. 34

Bonus Angles A and B are vertical angles, m�A � 9x � 10,and m�B � x2 � 6x. Find x.

A

B D

CJ E

F H

G

A B

DE

C

F

B:

NAME DATE PERIOD

Chapter 2 Test, Form 2B22


Ass

essm

ents


1. Make a conjecture based on the information A�B� bisects X�Y� at Z.A. AZ � ZB B. �AZX � �AZYC. AB � XY D. XZ � YZ

2. Given: n is a real number.Conjecture: n � |n |Which of the following would be a counterexample?A. �2 B. 0 C. 5 D. 10

3. If p is false and q is true, what is the truth value of p or q?A. true B. false C. 0 D. 1


4. What would be the values in the p � q column?A. T F F T B. T F T FC. T F F F D. T T T F

5. What would be the values in the p � q column?A. T F F T B. T F T FC. T F F F D. T T T F

6. Identify the conclusion of the statement Sue will watch the Rose Bowl if todayis January 1st.A. today is January 1st B. today is not January 1stC. Sue will watch the Rose Bowl D. Sue will not watch the Rose Bowl

7. Identify the inverse of the statement.If x � 5, then x � 8 � 13.A. If x � 8 �13, then x � 5. B. x � 5 and x � 8 � 13.C. If x � 8 � 13, then x5. D. If x � 5, then x � 8 � 13.

8. Identify the contrapositive of the following statement.If x � 5, then x � 8 � 13.A. If x � 8 �13, then x � 5. B. x � 5 and x � 8 � 13.C. If x � 8 �13, then x � 5. D. If x � 5, then x � 8 � 13.

9. Which law can be used to determine that statement (3) is a valid conclusionto statements (1) and (2)?(1) All dogs like biscuits.(2) Sammy is a dog.(3) Sammy likes biscuits.A. Law of Detachment B. Law of SyllogismC. Law of Converse D. Statement (3) does not follow.

p q p � q p � q

T T

T F

F T

F F

1.

2.

3.

4.

5.

6.

7.

8.

9.

NAME DATE PERIOD

SCORE


Chapter 2 Test, Form 2B (continued)22

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

10. Which law can be used to determine that statement (3) is a valid conclusionof statements (1) and (2)?(1) All sparrows fly. (2) All robins fly. (3) All sparrows are robins.A. Law of Detachment B. Law of SyllogismC. Law of Converse D. Statement (3) does not follow.

11. Which best describes the statement If two points lie in a plane, then the entireline containing those points lies in that plane?A. always true B. sometimes true C. never true D. cannot tell

12. Which of the following is an essential part of a good proof?A. an if-then statement B. using the contrapositiveC. listing the given information D. defining all the terms

13. Choose the property that justifies the statement.If x � y and y � z, then x � z.A. Conditional B. Transitive C. Symmetric D. Reflexive

14. Choose the property that justifies the following statement.

If 3AB � CD, then AB � �13�CD.

A. Addition B. Subtraction C. Division D. Substitution

15. Choose the property that justifies the statement.If G�H� � F�D� and F�D� � C�B�, then G�H� � C�B�.A. Reflexive B. SymmetricC. Transitive D. Def. of � segments

16. If XY � 6, YZ � 8, and XZ � 2, which point is between the other two?A. X B. Y C. Z D. cannot tell


17. Find m�AFE if m�BFC � 55.A. 20 B. 35C. 45 D. 55

18. If m�1 � 5x � 10 and m�2 � 3x � 20, find x.A. 10 B. 15 C. 21.25 D. 23.3�


19. If m�ABC � 34, find m�CBD.A. 90 B. 56C. 45 D. 34

20. For �ABC � �EFG, and m�ABC � 41, find m�GFH.A. 59 B. 49 C. 41 D. 39

Bonus Twenty-two students were asked about drink preferences.Twelve liked grape juice, 15 liked orange juice, and 10 liked both. How many did not like either?

A

B

C G

D

E

F H

AB

D

E

CF

21

B:

NAME DATE PERIOD

Chapter 2 Test, Form 2C22


Ass

essm

ents

1. Make a conjecture about the next term in this sequence:�11, �7, �3, 1, 5.

2. Given: XY � YZConjecture: Y is a midpoint of X�Z�.Find a counterexample.

3. What is the truth value of the following statement?�16� � �4 and 2 � 2.

4. Suppose p is true and q is false. What is the truth value of thedisjunction �p � �q?

5. Write the statement All dogs have four feet in if-then form.

6. Identify the hypothesis of the statement If you live in Chicago,then you live in Illinois.

7. Write the converse of the statement If two lines areperpendicular to the same line, then they are parallel.

8. Use the Law of Detachment to write a valid conclusion for thegiven information.(1) If two angles are supplementary, then their measures have

a sum of 180.(2) �X and �Y are supplementary.

9. Use the Law of Syllogism to write a valid conclusion for thegiven information.(1) If today is January 1, then it is New Year’s Day.(2) If today is New Year’s Day, then school is closed.

10. Name the operation that transforms 3x � 6 � 5x � 8 to 3x � 5x � 14, then find x.

11. Complete the proof by supplying the missing information.

If 2x � 7 � 4, then x � �121�.

Statements Reasons1. 2x � 7 � 4 1. Given

2. 2x � 7 � 7 � 4 � 7 2. Addition Property

3. 2x � 11 3. Substitution

4. �22x� � �

121� 4.

5. x � �121� 5. Substitution

12. If m�1 � x � 50 and m�2 � 3x � 20,find m�1. 1 2

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

NAME DATE PERIOD

SCORE


Chapter 2 Test, Form 2C (continued)22

For Questions 13 and 14, complete the proof below bysupplying the reasons for each location.

Given: A�C� bisects �BAD.A�C� bisects �BCD.�1 � �2

Prove: �3 � �4

Statements Reasons

1. A�C� bisects �BAD. 1. Given

2. A�C� bisects �BCD. 2. Given

3. �1 � �2 3. Given

4. �1 � �3 and �2 � �4 4.

5. �3 � �4 5.

For Questions 15–20, state the definition, property,postulate, or theorem that justifies each statement.

15. If M is the midpoint of A�B�, then A�M� � M�B�.

16. If �A � �B and �B � �C, then �A � �C.

17. If m�A � m�B � 90 and m�B � 20, then m�A � 20 � 90.

18. If �X and �Y are complementary, �Z and �Q arecomplementary, and �X � �Z, then �Y � �Q.

19. If P�R� � Q�T�, then PR � QT.

20. AB � BC � AC

Bonus Write the contrapositive of the statement A rhombus is a parallelogram.

B CA

(Question 14)

(Question 13)

2 4B D

C

A

1 3

13.

14.

15.

16.

17.

18.

19.

20.

B:

NAME DATE PERIOD

Chapter 2 Test, Form 2D22


Ass

essm

ents

1. Make a conjecture about the next term in this sequence:5, �10, 20, �40.

2. Given: MN � NPConjecture: N is a midpoint of M�P�.Find a counterexample.

3. What is the truth value of the following statement?�25� � �5 or 3 � 3.

4. Suppose p is true and q is false. What is the truth value of theconjunction �p � �q?

5. Write the statement All chickens have two wings in if-then form.

6. Identify the conclusion of the statement If you live in Chicago,then you live in Illinois.

7. Write the inverse of the statement If two lines are parallel to athird line, then they are parallel to each other.

8. Use the Law of Detachment to write a valid conclusion for thegiven information.(1) If a football team wins the Big 10 championship, then they

will play in the Rose Bowl.(2) Ohio State wins the Big 10 championship.

9 Use the Law of Syllogism to write a valid conclusion for thegiven information.(1) If today is Thursday, then ER is on television.(2) If ER is on television, then Amy will stay home.

10. Name the operation that transforms 4x � 2 � 7x � 7 to 4x � 7x � 9, then find x.

11. Complete the proof by supplying the missing information.If �3

x� � 1 � �4, then x � �15.

Statements Reasons

1. �3x

� � 1 � �4 1. Given

2. �3x

� � 1 � 1 � �4 � 1 2. Subtraction Property

3. �3x

� � �5 3. Substitution

4. ��3x

��3 � (�5)(3) 4.5. x � �15 5. Substitution

12. If m�1 � 5x � 20 and m�2 � 3x � 80,find m�1.

12

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

NAME DATE PERIOD

SCORE


Chapter 2 Test, Form 2D (continued)22

For Questions 13 and 14, complete the proof below bysupplying the reasons for each location.

Given: B�D� bisects �ABC.�1 and �3 are complementary.�2 and �4 are complementary.

Prove: �1 � �2Statements Reasons

1. B�D� bisects �ABC. 1. Given

2. �1 and �3 are complementary. 2. Given

3. �2 and �4 are complementary. 3. Given

4. �3 � �4 4.

5. �1 � �2 5.

For Questions 15–20, name the definition, property,postulate, or theorem that justifies each statement.

15. If X is the midpoint of Z�W�, then Z�X� � X�W�.

16. If AB � CD, then CD � AB.

17. If PQ � RS, then PQ � AB � RS � AB.

18. If A�B� � C�D� and C�D� � E�F�, then A�B� � E�F�.

19. If two angles form a linear pair, then they are supplementaryangles.

20. If m�A � m�B, then �A � �B.

Bonus If the ratio of m�1 to m�2 is 5 to 4, find m�2.

21

(Question 14)

(Question 13)

2

4

B

D CA1

3

13.

14.

15.

16.

17.

18.

19.

20.

B:

NAME DATE PERIOD

Chapter 2 Test, Form 322


Ass

essm

ents

1. Make a conjecture given that m�A � m�B and m�B � m�C.

2. Given: A�B� || C�D� and B�D� || A�C�Conjecture: ABDC is a rectangle.Find a counterexample.

3. What is the truth value of the statement A square has 4 congruent sides and a rectangle has 4 parallel sides?

4. Suppose p, q and r are all false. What is the truth value of ( p � �q) � �r?

5. In a class of 30 students, 22 like watching football, 17 likewatching basketball, and 12 like watching both. Make a Venndiagram of these data. How many of these students do not liketo watch either football or basketball?

6. Write the statement An elephant is a mammal in if-then form.

7. Write the contrapositive of the statement If two angles aresupplements of the same angle, then they are congruent.

8. Use the Law of Detachment to write a valid conclusion thatfollows from statements (1) and (2).(1) The swim team has practice on Saturday.(2) Matt is on the swim team.

9. Use the Law of Syllogism to write a valid conclusion thatfollows from statements (1) and (2).(1) If x � 6 � 10, then x � 4.(2) If x � 4, then x2 � 16.

10. Name the theorem that can be used to state that A�B� � B�C�, A�D� � D�B�, andB�E� � E�C�, given B is the midpoint of A�C�, D the midpoint of A�B�, and E the midpoint of B�C�.

A D B E C

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Basketball5

Football10

12

3

NAME DATE PERIOD

SCORE


Chapter 2 Test, Form 3 (continued)22

For Questions 11–14, complete the proofs below bysupplying the reasons for each location.

Given: 3 � 2(4 � x) � 11 � 6xProve: x � �4Statements Reasons1. 3 � 2(4 � x) � 11 � 6x 1. Given2. 3 � 8 � 2x � 11 � 6x 2.3. �5 � 2x � 11 � 6x 3. Addition of like terms4. 2x � 16 � 6x 4.5. �4x � 16 5. Subtraction Property6. x � �4 6. Division Property

Given: A�B� ⊥ B�D��EFG and �CBDare complementary.

Prove: �EFG � �ABCStatements Reasons1. A�B� ⊥ B�D� 1. Given2. �EFG and �CBD are

complementary. 2. Given3. �ABD is a right angle. 3. ⊥ lines form right angles.4. m�ABD � 90 4. Def. of right angle5. �ABC � �CBD � �ABD 5.6. m�ABC � m�CBD � 90 6. Substitution7. �ABC and �CBD are 7. Def. of complementary �

complementary.8. �EFG � �ABC 8.

For Questions 15–20, name the definition, property,postulate, or theorem that justifies each statement.

15. Points A, C, and E are coplanar.

16. If A�B� � C�D�, then AB � EF � CD � EF.

17. If A�B� � X�Y�, then X�Y� � A�B�.

18. If x( y � z) � a, then xy � xz � a.

19. If two angles are vertical, then they are congruent.

20. If �1 and �2 are right angles, then �1 � �2.

Bonus If m�ABD � 6y � 2x, m�DBC � 5x � 8,m�ADB � 9x � 4y � 3, and m�CDB � 10y � 4x � 1, find x and y.

CA D

B

(Question 14)

(Question 13)

A

B

C

D

E

F G

(Question 12)

(Question 11) 11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

NAME DATE PERIOD

Chapter 2 Open-Ended Assessment22


Ass

essm

ents

Demonstrate your knowledge by giving a clear, concise solution toeach problem. Be sure to include all relevant drawings and justifyyour answers. You may show your solution in more than one way orinvestigate beyond the requirements of the problem.

1. Make a truth table to prove that an if-then statement is equivalent to itscontrapositive and its inverse is equivalent to its converse.

2. Write three statements that illustrate the Law of Syllogism.

3. Write three statements that illustrate the Law of Detachment.

4. Write an example of the Transitive Property and the SubstitutionProperty that illustrates the difference between them.

5. Draw a pair of vertical angles with measures of 40. Label the verticalangles with algebraic expressions involving x. Solve an equation using theexpressions and show that each angle measures 40.

6. If two lines intersect, their intersection is a point. Must two linesintersect? Draw a diagram to illustrate what the lines would look like ifthey do not intersect.

7. Your woodworking teacher suggests you build a stool with only 3 legsbecause he says it will not rock if it has only 3 legs. Explain why this istrue. Use a postulate from this chapter to support your answer.

8. a. Write a true if-then statement for which the converse is false.

b. Write the converse, inverse, and contrapositive of your sentence.

c. Give the truth value of each statement you wrote for part b.

NAME DATE PERIOD

SCORE


Chapter 2 Vocabulary Test/Review22

Write whether each sentence is true or false. If false, replacethe underlined word or number to make a true sentence.

1. A is a statement that has been proved.

2. A is a statement that describes a fundamentalrelationship between the basic terms of geometry.

3. If p implies q is true and p is true, then q is also true by the.

4. If p implies q and q implies r are true, then p implies r is alsotrue by the .

5. The phrase immediately following the word then is called theof an if-then statement.

6. The phrase immediately following the word if is called theof an if-then statement.

7. A false example is called a .

8. A is an educated guess based on knowninformation.

9. Statements with the same truth values are said to be .

10. uses facts, rules, definitions, or propertiesto reach logical conclusions.

Define each term.

11. conditional statement

12. truth table

13. conjunction

Inductive reasoning

equivalentlogically

conjecture

counterexample

conclusion

hypothesis

Law of Syllogism

Law of Detachment

theorem

postulate 1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

compound statementconclusionconditional statementconjectureconjunctioncontrapositiveconversecounterexample

deductive argumentdeductive reasoningdisjunctionformal proofhypothesisif-then statementinductive reasoninginformal proof

inverseLaw of DetachmentLaw of Syllogismlogically equivalentnegationparagraph proofpostulate

proofrelated conditionalsstatementtheoremtruth tabletruth valuetwo-column proof

NAME DATE PERIOD

SCORE

Chapter 2 Quiz (Lessons 2–1 and 2–3)

22


Ass

essm

ents

NAME DATE PERIOD

SCORE

1.

2.

3.

4.

5.

1. Make a conjecture given that �ABC is an equilateral triangle.

2. Given: �A and �B are complementary.Conjecture: m�A � 45Find a counterexample.

3. What is the truth value of (p � q) � r?p: (�4)2 0q: An isosceles triangle has two congruent sides.r: Two angles, whose measure have a sum of 90, are

supplements.

4. Suppose p and q are both false. What is the truth value of ( p � �q) � �p?

5. STANDARDIZED TEST PRACTICE What is the next term inthe sequence 1, 4, 9, 16, 25?A. 6 B. 36C. 40 D. 50


22

1.

2.

3.

4.

5.

1. Identify the conclusion of the following statement.Either x � 2 or x � �2 if x2 � 4.

For Questions 2 and 3, complete the converse of thestatement If two angles are supplementary and congruent,then they are right angles.

If , then they are .

4. Use the Law of Detachment to tell whether the followingreasoning is valid. Write valid or not valid.If you drive at more than 65 miles per hour you will alwaysget a ticket.Joe got a ticket.Therefore, Joe drove over 65 miles per hour.

5. Use the Law of Syllogism to write a valid conclusion from thisset of statements.(1) All isosceles trapezoids are trapezoids.(2) All trapezoids are quadrilaterals.

(Question 3)(Question 2)

NAME DATE PERIOD

SCORE



22

1.

2.

3.

4.

5.

1. Determine the number of line segments that can be drawn connecting each pair of points.

2. Complete the following statement.If AB � BC and A, B, and C are collinear, then B is the

of A�C�.

For Questions 3–5, name the definition, property, postulate,or theorem that justifies each statement.

3. If x � 2, then 2 � x.

4. If x � 3 � y, then x � y � 3.

5. If m�A � m�B and m�B � m�C, then m�A � m�C.

NAME DATE PERIOD

SCORE


22

1.

2.

3.

4.

5.

For Questions 1–4, name the definition, property, postulate,or theorem that justifies each statement.

1. If D�E� � F�G�, then F�G� � D�E�.

2. If XY � WZ, then XY � TU � WZ � TU.

3. If m�1 � m�2 � 180 and m�2 � m�3 � 180, then �1 � �3.

4. If �1 and �2 are vertical angles, then �1 ��2.

5. STANDARDIZED TEST PRACTICE If m�A � 5x � 12,m�B � 2x � 18, �A and �C are supplementary, �B and �Care supplementary, find x.A. 5 B. 6C. 10 D. 24

NAME DATE PERIOD

SCORE

Chapter 2 Mid-Chapter Test (Lessons 2–1 through 2–4)

22


Ass

essm

ents

1. Make a conjecture given that points A, B, and C are collinear and AC � CB � AB.A. C is between A and B. B. A is between B and C.C. B is between A and C. D. �ABC is equilateral.

2. What is the truth value of (�p � q) � r if p is true, q is false, and r is true?A. true B. false C. 0 D. cannot tell

3. What is the truth value of (�p � q) � r if p is true, q is false, and r is true?A. true B. false C. 0 D. cannot tell

For Questions 4 and 5, use the statement If a ray bisects an angle then itdivides the angle into two congruent angles and the given choices.

A. If a ray divides an angle into two congruent angles, then it bisects the angle.B. A ray bisects an angle if and only if it divides it into two congruent angles.C. If a ray does not bisect an angle, then it does not divide the angle into two

congruent angles.D. If a ray does not divide an angle into two congruent angles, then it does not

bisect the angle.

4. Which choice is the inverse of the given statement?

5. Which choice is the contrapositive of the given statement?

6.

7.

8.

9.

10.

NAME DATE PERIOD

SCORE

1.

2.

3.

4.

5.

Part II

6. Given: 2a2 � 72. Conjecture: a � 6Write a counterexample.

7. Write the statement All right angles are congruent in if-then form.

8. Use the Law of Detachment to write a valid conclusion forstatements (1) and (2).(1) All fish can swim.(2) Charlie is a fish.

9. Use the Law of Syllogism to write a valid conclusion forstatements (1) and (2).(1) If the Giants score a touchdown they will win.(2) If the Giants win they will play in the Super Bowl.

10. Suppose p is false and q is true. What is the truth value of (�p � �q) � q?

Part I Write the letter for the correct answer in the blank at the right of each question.


Chapter 2 Cumulative Review(Chapters 1–2)

22

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

1. Draw and label a figure so that AB�� is the intersection of planeR and plane T. (Lesson 1-1)

For Questions 2 and 3, use the Distance Formula to findthe distance between each pair of points to the nearesthundredth. (Lesson 1-3)

2. A(1, �4), B(5, 3) 3. C(�6, 8), D(4, �1)

For Questions 4–7, use the figure.

4. Name the vertex of �3. (Lesson 1-4)

5. Name the sides of �2. (Lesson 1-4)

6. Name the angle that forms a linear pair with �4. (Lesson 1-5)

7. Name two acute adjacent angles. (Lesson 1-5)

8. If the perimeter of an n-gon is 4.25 inches, find the perimeterwhen the length of each side is multiplied by 6. (Lesson 1-6)

9. Make a conjecture based on the statement AB�� and CD�� areparallel. Draw a figure to illustrate your conjecture. (Lesson 2-1)

10. Suppose p and r are true and q is false. What is the truthvalue of the conjunction (�p � �q) � r? (Lesson 2-2)

11. Write the converse of the statement If a ray bisects an angle, itextends from the vertex of the angle. (Lesson 2-3)

12. Use the Law of Syllogism to determine whether a validconclusion can be reached from the following set of statements.(1) If two angles are congruent, they have the same measure.(2) �A and �B are congruent. (Lesson 2-4)

For Questions 13–15, complete the proof of the statement If x � 3 � 15x � 53, then x � 4. (Lesson 2-6)

Statements Reasons

1. x � 3 � 15x � 53 1. Given

2. x � x � 3 � 15x � x � 53 2. Subtraction Property

3. 3. Substitution Property

4. 3 � 53 � 14x � 53 � 53 4.5. 56 � 14x 5. Substitution Property

6. 6. Division Property

7. 4 � x 7. Substitution Property

8. x � 4 8. Symmetric Property

(Question 15)

(Question 14)

(Question 13)

JD

EF G

H12 3

4

A BC D

AB R

T

NAME DATE PERIOD

SCORE

Standardized Test Practice (Chapters 1–2)


1. Find JL if JK � 17 � x, KL � 2x � 7, and K is the midpoint of J�L�.(Lesson 1-3)

A. 8 B. 9 C. 16 D. 18

For Questions 2–4, use the figure at the right.

2. What is another name for �DFE? (Lesson 1-4)

E. �1 F. �2G. �3 H. �4

3. Classify �1 if m�1 � 115. (Lesson 1-4)

A. right B. acute C. obtuse D. scalene

4. What can be assumed from the figure? (Lesson 1-5)

E. �1 � �3 F. �2 � �4 G. B�F� � F�E� H. C�F� ⊥ B�E�

5. Find the perimeter of a regular octagon if one of its sides is x � 6and another side is 14 � x. (Lesson 1-6)

A. 4 B. 40 C. 8 D. 80

6. If p → q is the conditional, then its converse is . (Lesson 2-3)

E. q → p F. �q → p G. �q → �p H. q → �p

7. Which statement is always true? (Lesson 2-5)

A. x � 2 B. x � x C. x � y D. x � 0

8. If �1 � �2 and �2 � �3, then which is a valid conclusion? (Lesson 2-6)

I m�1 � m�2II �1 � �3III m�1 � m�2 � m�3

E. I, II, and III F. II only G. I and II H. I and III

For Questions 9 and 10, name the property that justifies thegiven statement.

9. If AB � CD and CD � 11, then AB � 11. (Lesson 2-7)

A. Transitive B. Symmetric C. Congruence D. Reflexive

10. If �XYZ � �PQR, then �PQR � �XYZ. (Lesson 2-8)

E. Transitive F. Symmetric G. Congruence H. Reflexive

?

FD

C

B EA

432 1

NAME DATE PERIOD

SCORE

Part 1: Multiple Choice

Instructions: Fill in the appropriate oval for the best answer.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10. E F G H

A B C D

E F G H

A B C D

E F G H

A B C D

E F G H

A B C D

E F G H

A B C D

Ass

essm

ents

22


Standardized Test Practice (continued)

11. Find the measure of Q�R� if Q is between pointsP and R, PR � 42, PQ � 8x, and QR � 4x.(Lesson 1-2)

12. Use the Distance Formula to find the distancein units between H(4, �1) and K(�8, 4).(Lesson 1-3)

13. If �1 and �2 are complementary, m�1 � 6x � 15and m�2 � 2x � 21, find m�2. (Lesson 1-4)

14. Make a conjecture about the next number inthe sequence �2, 4, �8, 16, �32. (Lesson 2-1)

NAME DATE PERIOD

Part 2: Grid In

Instructions: Enter your answer by writing each digit of the answer in a column boxand then shading in the appropriate oval that corresponds to that entry.

Part 3: Short Response

Instructions: Show your work or explain in words how you found your answer.

11. 12.

13. 14.

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

For Questions 15 and 16, use the results of a survey of 250 members of a local health club, shown in the Venn diagram.

15. How many members enjoy swimming or tennis? (Lesson 2-2)

16. How many members do not enjoy any of the activities? (Lesson 2-2)

17. The owner of Pop’s Pizza and Games says that to win theradio/CD player, you must first win 4500 credits. Each timeyou play the Racetrack game, you win 30 credits. How manytimes must you play the Racetrack game to win enoughcredits for the radio/CD player? (Lesson 2-4)

18. If B is in the interior of �DEF, m�DEB � 27.2, and m�DEF � 92.5, find m�BEF. (Lesson 2-7)

Tennis36Swimming

62

Hiking45

Exercise Preference

50

21

519 12 15.

16.

17.

18.

1 4 1 3

3 6 4

22

Standardized Test PracticeStudent Record Sheet (Use with pages 122–123 of the Student Edition.)

22

© Glencoe/McGraw-Hill A1 Glencoe Geometry

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

NAME DATE PERIOD

Part 1 Multiple ChoicePart 1 Multiple Choice

Part 2 Short Response/Grid InPart 2 Short Response/Grid In

Part 3 Open-EndedPart 3 Open-Ended

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.


Stu

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ab

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um

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in

th

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1,3

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27,8

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nu

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ice

that

eac

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um

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is a

pow

er o

f 3.

13

927

8130

3132

3334

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ject

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nex

t nu

mbe

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e 35

or 2

43.

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a co

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and

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or t

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all

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ab

out

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is �

80.

2.1,

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is 1

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are

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(�1,

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,B(2

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C(4

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and

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ple

men

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.

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and

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are

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and

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are

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FE

RT

QP

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A

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12

P

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x

y OA

( –1,

–1)

B( 2

, 2)

C( 4

, 4)

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her

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ne

situ

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con

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not

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e.T

he

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e is

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mid

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poss

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to

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a d

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ith

A �B�

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C�su

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B i

s

not

th

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idpo

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Th

is d

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cau

se

poin

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ot o

n A �

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.

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form

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AC

B

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3 cm

3 cm

B

Stu

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)

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Exam

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Exer

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Answers (Lesson 2-1)


An

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Skil

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2-1

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ab

out

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nex

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seq

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1. 2.�

4,�

1,2,

5,8

113.

6,�1 21 �

,5,�

9 2� ,4

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�2,

4,�

8,16

,�32

64

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give

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ject

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8.S

amp

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ers

are

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oin

ts A

,B,a

nd

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e co

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ear,

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is t

he

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N�Q�

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d D

is b

etw

een

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d C

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and

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by t

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.

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iven

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BC

and

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BD

form

a l

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gle

.

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xam

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tary

to

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and

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is c

ompl

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to

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ure

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��

3

tru

e

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B

43

12

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the

nex

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�10

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�20

253.

�2,

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5

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give

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fig

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to

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amp

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nsw

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5.�

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Cis

a r

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t an

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t S

is b

etw

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Ran

d T

.

BA

��

⊥B

C��

�R

S�

ST

�R

T

7.P

,Q,R

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non

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r 8.

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is a

par

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logr

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and

P �Q�

�Q�

R��

R�S�

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ents

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B�

CD

and

BC

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D.

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ine

wh

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er e

ach

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is

tru

e or

fal

se.G

ive

a co

un

tere

xam

ple

for

any

fals

e co

nje

ctu

re.

9.G

iven

:S,T

,an

d U

are

coll

inea

r an

d S

T�

TU

.C

onje

ctu

re:T

is t

he

mid

poin

t of

S �U�

.

tru

e

10.G

iven

:�1

and

�2

are

adja

cen

t an

gles

.C

onje

ctu

re:�

1 an

d �

2 fo

rm a

lin

ear

pair

.

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e;�

1 an

d �

2 co

uld

eac

h m

easu

re 6

0°.

11.G

iven

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and

J�K�fo

rm a

rig

ht

angl

e an

d in

ters

ect

at P

.C

onje

ctu

re:G �

H�⊥

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ue

12.A

LLER

GIE

SE

ach

spri

ng,R

ache

l sta

rts

snee

zing

whe

n th

e pe

ar t

rees

on

her

stre

et b

loss

om.

She

rea

sons

tha

t sh

e is

alle

rgic

to

pear

tre

es.F

ind

a co

unte

rexa

mpl

e to

Rac

hel’s

con

ject

ure.

Sam

ple

an

swer

:R

ach

el c

ou

ld b

e al

lerg

ic t

o o

ther

typ

es o

f p

lan

ts t

hat

blo

sso

m w

hen

th

e p

ear

tree

s b

loss

om

.D

A

C

B

SP

RQ

TS

RA

CB

Pra

ctic

e (

Ave

rag

e)

Ind

uct

ive

Rea

son

ing

an

d C

on

ject

ure

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

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____

____

__P

ER

IOD

____

_

2-1

2-1



Readin

g t

o L

earn

Math

em

ati

csIn

du

ctiv

e R

easo

nin

g a

nd

Co

nje

ctu

re

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-1

2-1

©G

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Gle

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Geo

met

ry

Lesson 2-1

Pre-

Act

ivit

yH

ow c

an i

nd

uct

ive

reas

onin

g h

elp

pre

dic

t w

eath

er c

ond

itio

ns?

Rea

d th

e in

trod

uct

ion

to

Les

son

2-1

at

the

top

of p

age

62 i

n y

our

text

book

.

•W

hat

kin

d of

wea

ther

pat

tern

s do

you

th

ink

met

eoro

logi

sts

look

at

toh

elp

pred

ict

the

wea

ther

? S

amp

le a

nsw

er:

pat

tern

s o

f h

igh

an

dlo

w t

emp

erat

ure

s,in

clu

din

g h

eat

spel

ls a

nd

co

ld s

pel

ls;

pat

tern

s o

f p

reci

pit

atio

n,i

ncl

ud

ing

wet

sp

ells

an

d d

ry s

pel

ls•

Wh

at i

s a

fact

or t

hat

mig

ht

con

trib

ute

to

lon

g-te

rm c

han

ges

in t

he

wea

ther

? S

amp

le a

nsw

er:

glo

bal

war

min

g d

ue

to h

igh

usa

ge

of

foss

il fu

els

Rea

din

g t

he

Less

on

1.E

xpla

in i

n y

our

own

wor

ds t

he

rela

tion

ship

bet

wee

n a

con

ject

ure

,a c

oun

tere

xam

ple,

and

indu

ctiv

e re

ason

ing.

Sam

ple

an

swer

:A

co

nje

ctu

re is

an

ed

uca

ted

gu

ess

bas

ed o

n s

pec

ific

exam

ple

s o

r in

form

atio

n.A

co

un

tere

xam

ple

is a

n e

xam

ple

th

at s

ho

ws

that

a c

on

ject

ure

is f

alse

.In

du

ctiv

e re

aso

nin

g is

th

e p

roce

ss o

f m

akin

g a

con

ject

ure

bas

ed o

n s

pec

ific

exa

mp

les

or

info

rmat

ion

.

2.M

ake

a co

nje

ctu

re a

bou

t th

e n

ext

item

in

eac

h s

equ

ence

.

a.5,

9,13

,17

21b

.1,

�1 3� ,�1 9� ,

� 21 7�� 81 1�

c.0,

1,3,

6,10

15d

.8,

3,�

2,�

7�

12e.

1,8,

27,6

412

5f.

1,�

2,4,

�8

16g.

h.

3.S

tate

wh

eth

er e

ach

con

ject

ure

is

tru

eor

fal

se.I

f th

e co

nje

ctu

re i

s fa

lse,

give

aco

un

tere

xam

ple.

a.T

he

sum

of

two

odd

inte

gers

is

even

.

tru

eb

.T

he

prod

uct

of

an o

dd i

nte

ger

and

an e

ven

in

tege

r is

odd

.Fa

lse;

sam

ple

an

swer

:5

�8

�40

,wh

ich

is e

ven

.c.

Th

e op

posi

te o

f an

in

tege

r is

a n

egat

ive

inte

ger.

Fals

e;sa

mp

le a

nsw

er:T

he

op

po

site

of

the

inte

ger

�5

is 5

,wh

ich

is a

po

siti

ve in

teg

er.

d.

Th

e pe

rfec

t sq

uar

es (

squ

ares

of

wh

ole

nu

mbe

rs)

alte

rnat

e be

twee

n o

dd a

nd

even

.tr

ue

Hel

pin

g Y

ou

Rem

emb

er4.

Wri

te a

sho

rt s

ente

nce

that

can

hel

p yo

u re

mem

ber

why

it o

nly

take

s on

e co

unte

rexa

mpl

eto

pro

ve t

hat

a c

onje

ctu

re i

s fa

lse.

Sam

ple

an

swer

:Tru

e m

ean

s al

way

str

ue.

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Co

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sW

hen

you

mak

e a

con

clu

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aft

er e

xam

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g se

vera

l sp

ecif

icca

ses,

you

hav

e u

sed

ind

uct

ive

reas

onin

g.H

owev

er,y

ou m

ust

be

cau

tiou

s w

hen

usi

ng

this

for

m o

f re

ason

ing.

By

fin

din

g on

ly o

ne

cou

nte

rexa

mp

le,y

ou d

ispr

ove

the

con

clu

sion

.

Is t

he

stat

emen

t �1 x�

�1

tru

e w

hen

you

rep

lace

xw

ith

1,

2,an

d 3

? Is

th

e st

atem

ent

tru

e fo

r al

l re

als?

If

pos

sib

le,f

ind

a

cou

nte

rexa

mp

le.

�1 1��

1,�1 2�

�1,

and

�1 3��

1.B

ut

wh

en x

��1 2� ,

then

�1 x��

2.T

his

cou

nte

rexa

mpl

e

show

s th

at t

he

stat

emen

t is

not

alw

ays

tru

e.

An

swer

eac

h q

ues

tion

.

1.T

he

cold

est

day

of t

he

year

in

Ch

icag

o2.

Su

ppos

e Jo

hn

mis

ses

the

sch

ool

bus

occu

rred

in

Jan

uar

y fo

r fi

ve s

trai

ght

fou

r T

ues

days

in

a r

ow.C

an y

ouye

ars.

Is i

t sa

fe t

o co

ncl

ude

th

at t

he

safe

ly c

oncl

ude

th

at J

ohn

mis

ses

the

cold

est

day

in C

hic

ago

is a

lway

s in

sc

hoo

l bu

s ev

ery

Tu

esda

y?n

oJa

nu

ary?

no

3.Is

th

e eq

uat

ion

�k2 �

�k

tru

e w

hen

you

4.Is

th

e st

atem

ent

2x�

x�

xtr

ue

wh

enre

plac

e k

wit

h 1

,2,a

nd

3? I

s th

eyo

u r

epla

ce x

wit

h �1 2� ,

4,an

d 0.

7? I

s th

eeq

uat

ion

tru

e fo

r al

l in

tege

rs?

If

stat

emen

t tr

ue

for

all

real

nu

mbe

rs?

poss

ible

,fin

d a

cou

nte

rexa

mpl

e.If

pos

sibl

e,fi

nd

a co

un

tere

xam

ple.

It is

tru

e fo

r 1,

2,an

d 3

.It

is n

ot

It is

tru

e fo

r al

l rea

l nu

mb

ers.

tru

e fo

r n

egat

ive

inte

ger

s.S

amp

le a

nsw

er:

�2

5.S

upp

ose

you

dra

w f

our

poin

ts A

,B,C

,6.

Su

ppos

e yo

u d

raw

a c

ircl

e,m

ark

thre

e an

d D

and

then

dra

w A �

B�,B�

C�,C�

D�,a

nd

poin

ts o

n i

t,an

d co

nn

ect

them

.Wil

l th

e

D�A�

.Doe

s th

is p

roce

dure

giv

e a

angl

es o

f th

e tr

ian

gle

be a

cute

? E

xpla

in

quad

rila

tera

l al

way

s or

on

ly s

omet

imes

? yo

ur

answ

ers

wit

h f

igu

res.

Exp

lain

you

r an

swer

s w

ith

fig

ure

s.n

o,o

nly

so

met

imes

on

ly s

om

etim

esE

xam

ple

:C

ou

nte

rexa

mp

le:

Exa

mp

le:

Co

un

tere

xam

ple

:A

A

B

BC

C

D

D

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-1

2-1

Exam

ple

Exam

ple



An

swer

s

Stu

dy G

uid

e a

nd I

nte

rven

tion

Lo

gic

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-2

2-2

©G

lenc

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w-H

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Gle

ncoe

Geo

met

ry

Lesson 2-2

Det

erm

ine

Tru

th V

alu

esA

sta

tem

ent

is a

ny

sen

ten

ce t

hat

is

eith

er t

rue

or f

alse

.Th

etr

uth

or

fals

ity

of a

sta

tem

ent

is i

ts t

ruth

val

ue.

A s

tate

men

t ca

n b

e re

pres

ente

d by

usi

ng

ale

tter

.For

exa

mpl

e,

Sta

tem

ent

p:C

hic

ago

is a

cit

y in

Ill

inoi

s.T

he

tru

th v

alu

e of

sta

tem

ent

pis

tru

e.

Sev

eral

sta

tem

ents

can

be

join

ed i

n a

com

pou

nd

sta

tem

ent.

Sta

tem

ent

pan

d st

atem

ent

qjo

ined

S

tate

men

t p

and

stat

emen

t q

join

ed

Neg

atio

n:

not p

is t

he n

egat

ion

ofby

the

wor

d an

dis

a c

on

jun

ctio

n.

by t

he w

ord

oris

a d

isju

nct

ion

.th

e st

atem

ent

p.

Sym

bols

: p

�q

(Rea

d: p

and

q)

Sym

bols

: p

�q

(Rea

d: p

or

q)S

ymbo

ls:

�p

(Rea

d: n

ot p

)

The

con

junc

tion

p�

qis

tru

e on

lyT

he d

isju

nctio

n p

�q

is t

rue

if p

is

The

sta

tem

ents

pan

d �

pha

ve

whe

n bo

th p

and

qar

e tr

ue.

true

, if

qis

tru

e, o

r if

both

are

tru

e.op

posi

te t

ruth

val

ues.

Wri

te a

com

pou

nd

stat

emen

t fo

r ea

ch c

onju

nct

ion

.Th

enfi

nd

its

tru

th v

alu

e.p:

An

ele

phan

t is

a m

amm

al.

q:A

squ

are

has

fou

r ri

ght

angl

es.

a.p

�q

Join

the

sta

tem

ents

wit

h an

d:A

n el

epha

ntis

a m

amm

al a

nd a

squ

are

has

four

rig

htan

gles

.Bot

h pa

rts

of t

he s

tate

men

t ar

etr

ue s

o th

e co

mpo

und

stat

emen

t is

tru

e.

b.

�p

�q

�p

is t

he

stat

emen

t “A

n e

leph

ant

is n

ot a

mam

mal

.”Jo

in �

pan

d q

wit

h t

he

wor

dan

d:A

n e

leph

ant

is n

ot a

mam

mal

an

d a

squ

are

has

fou

r ri

ght

angl

es.T

he

firs

tpa

rt o

f th

e co

mpo

un

d st

atem

ent,

�p,

isfa

lse.

Th

eref

ore

the

com

pou

nd

stat

emen

tis

fal

se.

Wri

te a

com

pou

nd

stat

emen

t fo

r ea

ch d

isju

nct

ion

.Th

enfi

nd

its

tru

th v

alu

e.p:

A d

iam

eter

of

a ci

rcle

is

twic

e th

e ra

diu

s.q:

A r

ecta

ngl

e h

as f

our

equ

al s

ides

.

a.p

�q

Join

th

e st

atem

ents

pan

d q

wit

h t

he

wor

d or

:A d

iam

eter

of

a ci

rcle

is

twic

eth

e ra

diu

s or

a r

ecta

ngl

e h

as f

our

equ

alsi

des.

Th

e fi

rst

part

of

the

com

pou

nd

stat

emen

t,p,

is t

rue,

so t

he

com

pou

nd

stat

emen

t is

tru

e.

b.

�p

�q

Join

�p

and

qw

ith

th

e w

ord

or:A

diam

eter

of

a ci

rcle

is

not

tw

ice

the

radi

us

or a

rec

tan

gle

has

fou

r eq

ual

side

s.N

eith

er p

art

of t

he

disj

un

ctio

n i

str

ue,

so t

he

com

pou

nd

stat

emen

t is

fal

se.

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Wri

te a

com

pou

nd

sta

tem

ent

for

each

con

jun

ctio

n a

nd

dis

jun

ctio

n.

Th

en f

ind

its

tru

th v

alu

e.p:

10 �

8 �

18q:

Sep

tem

ber

has

30

days

.r:

A r

ecta

ngl

e h

as f

our

side

s.

1.p

and

q10

�8

�18

an

d S

epte

mb

er h

as 3

0 d

ays;

tru

e.

2.p

or r

10 �

8 �

18 o

r a

rect

ang

le h

as f

ou

r si

des

;tr

ue.

3.q

or r

Sep

tem

ber

has

30

day

s o

r a

rect

ang

le h

as f

ou

r si

des

;tr

ue.

4.q

and

�r

Sep

tem

ber

has

30

day

s an

d a

rec

tan

gle

do

es n

ot

hav

e fo

ur

sid

es;

fals

e.

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Geo

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Tru

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able

sO

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way

to

orga

niz

e th

e tr

uth

val

ues

of

stat

emen

ts i

s in

a

tru

th t

able

.Th

e tr

uth

tab

les

for

neg

atio

n,c

onju

nct

ion

,an

d di

sju

nct

ion

ar

e sh

own

at

the

righ

t.

Dis

jun

ctio

n

pq

p�

q

TT

TT

FT

FT

TF

FF

Co

nju

nct

ion

pq

p�

q

TT

TT

FF

FT

FF

FF

Neg

atio

n

p�

p

TF

FT

Stu

dy G

uid

e a

nd I

nte

rven

tion

(con

tinued

)

Lo

gic

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-2

2-2

Con

stru

ct a

tru

thta

ble

for

th

e co

mp

oun

dst

atem

ent

qor

r.U

se t

he

dis

jun

ctio

n t

able

.

qr

qo

r r

TT

TT

FT

FT

TF

FF

Con

stru

ct a

tru

th t

able

for

th

eco

mp

oun

d s

tate

men

t p

and

(q

or r

).U

se t

he

disj

un

ctio

n t

able

for

(q

or r

).T

hen

use

th

eco

nju

nct

ion

tab

le f

or p

and

(qor

r).

pq

rq

or

rp

and

(q

or

r)

TT

TT

TT

TF

TT

TF

TT

TT

FF

FF

FT

TT

FF

TF

TF

FF

TT

FF

FF

FF

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Con

tru

ct a

tru

th t

able

for

eac

h c

omp

oun

d s

tate

men

t.

1.p

or r

2.�

p�

q3.

q�

�r

4.�

p�

�r

5.(p

and

r) o

r q

pq

rp

and

r(p

and

r)

or

q

TT

TT

TT

TF

FT

TF

TT

TT

FF

FF

FT

TF

TF

TF

FT

FF

TF

FF

FF

FF

pr

�p

�r

�p

��

r

TT

FF

FT

FF

TF

FT

TF

FF

FT

TT

qr

�r

q�

�r

TT

FF

TF

TT

FT

FF

FF

TF

p�

pq

�p

�q

TF

TT

TF

FF

FT

TT

FT

FT

pr

po

r r

TT

TT

FT

FT

TF

FF



Skil

ls P

ract

ice

Lo

gic

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-2

2-2

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ncoe

Geo

met

ry

Lesson 2-2

Use

th

e fo

llow

ing

stat

emen

ts t

o w

rite

a c

omp

oun

d s

tate

men

t fo

r ea

ch c

onju

nct

ion

and

dis

jun

ctio

n.T

hen

fin

d i

ts t

ruth

val

ue.

p:�

3 �

2 �

�5

q:V

erti

cal

angl

es a

re c

ongr

uen

t.r:

2 �

8 �

10s:

Th

e su

m o

f th

e m

easu

res

of c

omp

lem

enta

ry a

ngl

es i

s 90

°.

1.p

and

q�

3 �

2 �

�5

and

ver

tica

l an

gle

s ar

e co

ng

ruen

t;tr

ue.

2.p

�r

�3

�2

��

5 an

d 2

�8

�10

;fa

lse.

3.p

or s

�3

�2

��

5 o

r th

e su

m o

f th

e m

easu

res

of

com

ple

men

tary

an

gle

sis

90°

;tr

ue.

4.r

�s

2 �

8 �

10 o

r th

e su

m o

f th

e m

easu

res

of

com

ple

men

tary

an

gle

s is

90°;

tru

e.

5.p

��

q�

3 �

2 �

�5

and

ver

tica

l an

gle

s ar

e n

ot

con

gru

ent;

fals

e.

6.q

��

rV

erti

cal a

ng

les

are

con

gru

ent

or

2 �

8 �

10;

tru

e.

Cop

y an

d c

omp

lete

eac

h t

ruth

tab

le.

7.8.

Con

stru

ct a

tru

th t

able

for

eac

h c

omp

oun

d s

tate

men

t.

9.�

q�

r10

.�p

��

r

pr

�p

�r

�p

��

r

TT

FF

F

TF

FT

T

FT

TF

T

FF

TT

T

qr

�q

�q

�r

TT

FF

TF

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FT

TT

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pq

�q

p�

�q

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FT

TF

TT

FT

FF

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pq

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p�

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TT

FF

TT

FF

FT

FT

TT

FF

FT

FT

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lenc

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cGra

w-H

ill66

Gle

ncoe

Geo

met

ry

Use

th

e fo

llow

ing

stat

emen

ts t

o w

rite

a c

omp

oun

d s

tate

men

t fo

r ea

ch c

onju

nct

ion

and

dis

jun

ctio

n.T

hen

fin

d i

ts t

ruth

val

ue.

p:6

0 se

con

ds

�1

min

ute

q:C

ongr

uen

t su

pp

lem

enta

ry a

ngl

es e

ach

hav

e a

mea

sure

of

90.

r:�

12 �

11 �

�1

1.p

�q

60 s

eco

nd

s �

1 m

inu

te a

nd

co

ng

ruen

t su

pp

lem

enta

ry a

ng

les

each

hav

e a

mea

sure

of

90;

tru

e.

2.q

�r

Co

ng

ruen

t su

pp

lem

enta

ry a

ng

les

each

hav

e a

mea

sure

of

90 o

r�

12 �

11 �

�1;

tru

e.

3.�

p�

q60

sec

on

ds

1

min

ute

or

con

gru

ent

sup

ple

men

tary

an

gle

s ea

chh

ave

a m

easu

re o

f 90

;tr

ue.

4.�

p�

�r

60 s

eco

nd

s

1 m

inu

te a

nd

�12

�11

�

1;fa

lse.

Cop

y an

d c

omp

lete

eac

h t

ruth

tab

le.

5.6.

Con

stru

ct a

tru

th t

able

for

eac

h c

omp

oun

d s

tate

men

t.

7.q

�(p

��

q)8.

�q

�(�

p�

q)

SCH

OO

LF

or E

xerc

ises

9 a

nd

10,

use

th

e fo

llow

ing

info

rmat

ion

.T

he

Ven

n d

iagr

am s

how

s th

e n

um

ber

of s

tude

nts

in

th

e ba

nd

wh

o w

ork

afte

r sc

hoo

l or

on

th

e w

eeke

nds

.

9.H

ow m

any

stu

den

ts w

ork

afte

r sc

hoo

l an

d on

wee

ken

ds?

3

10.H

ow m

any

stu

den

ts w

ork

afte

r sc

hoo

l or

on

wee

ken

ds?

25

Wor

kW

eeke

nds

17

Wor

kAf

ter

Scho

ol5

3

pq

�p

�q

�p

�q

�q

�(�

p�

q)

TT

FF

TF

TF

FT

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FT

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TT

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pq

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p�

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(p�

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)

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F

pq

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�p

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(�p

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)

TT

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pq

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�p

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q

TT

FF

FT

FF

TT

FT

TF

TF

FT

TT

Pra

ctic

e (

Ave

rag

e)

Lo

gic

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-2

2-2



An

swer

s

Readin

g t

o L

earn

Math

em

ati

csL

og

ic

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-2

2-2

©G

lenc

oe/M

cGra

w-H

ill67

Gle

ncoe

Geo

met

ry

Lesson 2-2

Pre-

Act

ivit

yH

ow d

oes

logi

c ap

ply

to

sch

ool?

Rea

d th

e in

trod

uct

ion

to

Les

son

2-2

at

the

top

of p

age

67 i

n y

our

text

book

.

How

can

you

use

log

ic t

o h

elp

you

an

swer

a m

ult

iple

-ch

oice

qu

esti

on o

n a

stan

dard

ized

tes

t if

you

are

not

sur

e of

the

cor

rect

ans

wer

?S

ampl

e an

swer

:E

limin

ate

the

choi

ces

that

you

kno

w a

re w

rong

.The

n ch

oose

the

one

you

thin

k is

mos

t lik

ely

corr

ect

from

the

one

s th

at a

re le

ft.

Rea

din

g t

he

Less

on

1.S

upp

ly o

ne

or t

wo

wor

ds t

o co

mpl

ete

each

sen

ten

ce.

a.T

wo

or m

ore

stat

emen

ts c

an b

e jo

ined

to

form

a

stat

emen

t.b

.A

sta

tem

ent

that

is

form

ed b

y jo

inin

g tw

o st

atem

ents

wit

h t

he

wor

d or

is c

alle

d a

.c.

Th

e tr

uth

or

fals

ity

of a

sta

tem

ent

is c

alle

d it

s

.d

.A

sta

tem

ent

that

is

form

ed b

y jo

inin

g tw

o st

atem

ents

wit

h t

he

wor

d an

dis

cal

led

a

.e.

A s

tate

men

t th

at h

as t

he

oppo

site

tru

th v

alu

e an

d th

e op

posi

te m

ean

ing

from

a g

iven

st

atem

ent

is c

alle

d th

e

of t

he

stat

emen

t.

2.U

se t

rue

or f

alse

to c

ompl

ete

each

sen

ten

ce.

a.If

a s

tate

men

t is

tru

e,th

en i

ts n

egat

ion

is

.

b.

If a

sta

tem

ent

is f

alse

,th

en i

ts n

egat

ion

is

.c.

If t

wo

stat

emen

ts a

re b

oth

tru

e,th

en t

hei

r co

nju

nct

ion

is

and

thei

r di

sju

nct

ion

is

.d

.If

tw

o st

atem

ents

are

bot

h f

alse

,th

en t

hei

r co

nju

nct

ion

is

and

thei

r di

sju

nct

ion

is

.e.

If o

ne

stat

emen

t is

tru

e an

d an

oth

er i

s fa

lse,

then

th

eir

con

jun

ctio

n i

s

and

thei

r di

sju

nct

ion

is

.

3.C

onsi

der

the

foll

owin

g st

atem

ents

:p:

Ch

icag

o is

th

e ca

pita

l of

Ill

inoi

s.q:

Sac

ram

ento

is

the

capi

tal

of C

alif

orn

ia.

Wri

te e

ach

sta

tem

ent

sym

boli

call

y an

d th

en f

ind

its

tru

th v

alu

e.a.

Sac

ram

ento

is

not

th

e ca

pita

l of

Cal

ifor

nia

.�

q;

fals

eb

.S

acra

men

to i

s th

e ca

pita

l of

Cal

ifor

nia

an

d C

hic

ago

is n

ot t

he

capi

tal

of I

llin

ois.

q�

�p

;tr

ue

Hel

pin

g Y

ou

Rem

emb

er4.

Pre

fixe

s ca

n o

ften

hel

p yo

u t

o re

mem

ber

the

mea

nin

g of

wor

ds o

r to

dis

tin

guis

h b

etw

een

sim

ilar

wor

ds.U

se y

our

dict

ion

ary

to f

ind

the

mea

nin

gs o

f th

e pr

efix

es c

onan

d d

isan

dex

plai

n h

ow t

hes

e m

ean

ings

can

hel

p yo

u r

emem

ber

the

diff

eren

ce b

etw

een

aco

nju

nct

ion

and

a d

isju

nct

ion

.S

amp

le a

nsw

er:

Co

nm

ean

s to

get

her

and

dis

mea

ns

apar

t,so

a c

on

jun

ctio

nis

an

an

d(o

r b

oth

to

get

her

) st

atem

ent

and

a d

isju

nct

ion

is a

n o

rst

atem

ent.

tru

efa

lse

fals

efa

lse

tru

etr

ue

tru

efa

lse

neg

atio

n

con

jun

ctio

n

tru

th v

alu

ed

isju

nct

ion

com

po

un

d

©G

lenc

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cGra

w-H

ill68

Gle

ncoe

Geo

met

ry

Let

ter

Pu

zzle

sA

n a

lph

amet

icis

a c

ompu

tati

on p

uzz

le u

sin

g le

tter

s in

stea

d of

digi

ts.E

ach

let

ter

repr

esen

ts o

ne

of t

he

digi

ts 0

–9,a

nd

two

diff

eren

t le

tter

s ca

nn

ot r

epre

sen

t th

e sa

me

digi

t.S

ome

alph

amet

icpu

zzle

s h

ave

mor

e th

an o

ne

answ

er.

Sol

ve t

he

alp

ham

etic

pu

zzle

at

the

righ

t.

Sin

ce R

�E

�E

,th

e va

lue

of R

mu

st b

e 0.

Not

ice

that

th

eth

ousa

nds

dig

it m

ust

be

the

sam

e in

th

e fi

rst

adde

nd

and

the

sum

.Sin

ce t

he

valu

e of

I i

s 9

or l

ess,

O m

ust

be

4 or

les

s.U

se

tria

l an

d er

ror

to f

ind

valu

es t

hat

wor

k.

F �

8,O

�3,

U �

1,R

�0

N �

4,E

�7,

I �

6,an

d V

�5.

Can

you

fin

d ot

her

sol

uti

ons

to t

his

pu

zzle

?

Fin

d a

val

ue

for

each

let

ter

in e

ach

alp

ham

etic

.S

amp

le a

nsw

ers

are

sho

wn

1.2.

H �

A �

L �

T �

W �

O �

F �

W �

O �

F �

U �

R �

E �

3.4.

T �

H �

R �

S �

E �

N �

E �

O �

N �

D �

M �

O �

S �

V �

R �

Y �

5.D

o re

sear

ch t

o fi

nd

mor

e al

pham

etic

pu

zzle

s,or

cre

ate

you

r ow

n p

uzz

les.

Ch

alle

nge

anot

her

stu

den

t to

sol

ve t

hem

.S

ee s

tud

ents

’wo

rk.

28

08

01

71

57

65

92

34

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t

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ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-2

2-2

Exam

ple

Exam

ple



Stu

dy G

uid

e a

nd I

nte

rven

tion

Co

nd

itio

nal

Sta

tem

ents

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-3

2-3

©G

lenc

oe/M

cGra

w-H

ill69

Gle

ncoe

Geo

met

ry

Lesson 2-3

If-t

hen

Sta

tem

ents

An

if-

then

sta

tem

ent

is a

sta

tem

ent

such

as

“If

you

are

rea

din

gth

is p

age,

then

you

are

stu

dyin

g m

ath

.”A

sta

tem

ent

that

can

be

wri

tten

in

if-

then

for

m i

sca

lled

a c

ond

itio

nal

sta

tem

ent.

Th

e ph

rase

im

med

iate

ly f

ollo

win

g th

e w

ord

ifis

th

eh

ypot

hes

is.T

he

phra

se i

mm

edia

tely

fol

low

ing

the

wor

d th

enis

th

e co

ncl

usi

on.

A c

ondi

tion

al s

tate

men

t ca

n b

e re

pres

ente

d in

sym

bols

as

p→

q,w

hic

h i

s re

ad “

pim

plie

s q”

or “

if p

,th

en q

.”

Iden

tify

th

e h

ypot

hes

is a

nd

con

clu

sion

of

the

stat

emen

t.

If �

X�

�R

and

�R

��

S,t

hen

�X

��

S.

hypo

thes

isco

nclu

sion

Iden

tify

th

e h

ypot

hes

is a

nd

con

clu

sion

.W

rite

th

e st

atem

ent

in i

f-th

en f

orm

.

You

rec

eive

a f

ree

pizz

a w

ith

12

cou

pon

s.

If y

ou h

ave

12 c

oupo

ns,

then

you

rec

eive

a f

ree

pizz

a.hy

poth

esis

conc

lusi

on

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Iden

tify

th

e h

ypot

hes

is a

nd

con

clu

sion

of

each

sta

tem

ent.

1.If

it

is S

atu

rday

,th

en t

her

e is

no

sch

ool.

H:

it is

Sat

urd

ay;

C:

ther

e is

no

sch

oo

l

2.If

x�

8 �

32,t

hen

x�

40.

H:

x�

8 �

32;

C:

x�

40

3.If

a p

olyg

on h

as f

our

righ

t an

gles

,th

en t

he

poly

gon

is

a re

ctan

gle.

H:

a p

oly

go

n h

as f

ou

r ri

gh

t an

gle

s;C

:th

e p

oly

go

n is

a r

ecta

ng

le

Wri

te e

ach

sta

tem

ent

in i

f-th

en f

orm

.

4.A

ll a

pes

love

ban

anas

.If

an

an

imal

is a

n a

pe,

then

it lo

ves

ban

anas

.

5.T

he

sum

of

the

mea

sure

s of

com

plem

enta

ry a

ngl

es i

s 90

.If

tw

o a

ng

les

are

com

ple

men

tary

,th

en t

he

sum

of

thei

r m

easu

res

is 9

0.

6.C

olli

nea

r po

ints

lie

on

th

e sa

me

lin

e.If

po

ints

are

co

llin

ear,

then

th

ey li

e o

n t

he

sam

e lin

e.

Det

erm

ine

the

tru

th v

alu

e of

th

e fo

llow

ing

stat

emen

t fo

r ea

ch s

et o

f co

nd

itio

ns.

If i

t d

oes

not

rai

n t

his

Sat

urd

ay,w

e w

ill

hav

e a

picn

ic.

7.It

rai

ns

this

Sat

urd

ay,a

nd

we

hav

e a

picn

ic.

tru

e

8.It

rai

ns

this

Sat

urd

ay,a

nd

we

don

’t h

ave

a pi

cnic

.tr

ue

9.It

doe

sn’t

rain

th

is S

atu

rday

,an

d w

e h

ave

a pi

cnic

.tr

ue

10.I

t do

esn

’t ra

in t

his

Sat

urd

ay,a

nd

we

don

’t h

ave

a pi

cnic

.fa

lse

©G

lenc

oe/M

cGra

w-H

ill70

Gle

ncoe

Geo

met

ry

Co

nve

rse,

Inve

rse,

an

d C

on

trap

osi

tive

If y

ou c

han

ge t

he

hyp

oth

esis

or

con

clu

sion

of a

con

diti

onal

sta

tem

ent,

you

for

m a

rel

ated

con

dit

ion

al.T

his

ch

art

show

s th

e th

ree

rela

ted

con

diti

onal

s,co

nve

rse,

inve

rse,

and

con

trap

osit

ive,

and

how

th

ey a

re r

elat

ed t

o a

con

diti

onal

sta

tem

ent.

Sym

bo

lsF

orm

ed b

yE

xam

ple

Co

nd

itio

nal

p→

qus

ing

the

give

n hy

poth

esis

and

con

clus

ion

If tw

o an

gles

are

ver

tical

ang

les,

th

en t

hey

are

cong

ruen

t.

Co

nver

seq

→p

exch

angi

ng t

he h

ypot

hesi

s an

d co

nclu

sion

If tw

o an

gles

are

con

grue

nt,

then

th

ey a

re v

ertic

al a

ngle

s.

Inve

rse

�p

→�

qre

plac

ing

the

hypo

thes

is w

ith it

s ne

gatio

n If

two

angl

es a

re n

ot v

ertic

al a

ngle

s,an

d re

plac

ing

the

conc

lusi

on w

ith it

s ne

gatio

nth

en t

hey

are

not

cong

ruen

t.

Co

ntr

apo

siti

ve�

q→

�p

nega

ting

the

hypo

thes

is,

nega

ting

the

If tw

o an

gles

are

not

con

grue

nt,

conc

lusi

on,

and

switc

hing

the

mth

en t

hey

are

not

vert

ical

ang

les.

Just

as

a co

ndi

tion

al s

tate

men

t ca

n b

e tr

ue

or f

alse

,th

e re

late

d co

ndi

tion

als

also

can

be

tru

eor

fal

se.A

con

diti

onal

sta

tem

ent

alw

ays

has

th

e sa

me

tru

th v

alu

e as

its

con

trap

osit

ive,

and

the

con

vers

e an

d in

vers

e al

way

s h

ave

the

sam

e tr

uth

val

ue.

Wri

te t

he

con

vers

e,in

vers

e,an

d c

ontr

apos

itiv

e of

eac

h c

ond

itio

nal

sta

tem

ent.

Tel

lw

hic

h s

tate

men

ts a

re t

rue

and

wh

ich

sta

tem

ents

are

fa

lse.

1.If

you

liv

e in

San

Die

go,t

hen

you

liv

e in

Cal

ifor

nia

.

Co

nver

se:

If y

ou

live

in C

alifo

rnia

,th

en y

ou

live

in S

an D

ieg

o.I

nver

se:

If y

ou

do

no

t liv

e in

San

Die

go

,th

en y

ou

do

no

t liv

e in

Cal

iforn

ia.

Co

ntr

apo

siti

ve:

If y

ou

do

no

t liv

e in

Cal

iforn

ia,t

hen

yo

u d

o n

ot

live

in S

an D

ieg

o.T

rue:

stat

emen

t an

d c

on

trap

osi

tive

;fa

lse:

conv

erse

an

d in

vers

e

2.If

a p

olyg

on i

s a

rect

angl

e,th

en i

t is

a s

quar

e.

Co

nver

se:

If a

po

lyg

on

is a

sq

uar

e,th

en it

is a

rec

tan

gle

.Inv

erse

:If

a p

oly

go

n is

no

t a

rect

ang

le,t

hen

it is

no

t a

squ

are.

Co

ntr

apo

siti

ve:

If a

po

lyg

on

is n

ot

a sq

uar

e,th

en it

is n

ot

a re

ctan

gle

.Tru

e:co

nver

se

and

inve

rse;

fals

e:st

atem

ent

and

co

ntr

apo

siti

ve

3.If

tw

o an

gles

are

com

plem

enta

ry,t

hen

th

e su

m o

f th

eir

mea

sure

s is

90.

Co

nver

se:

If t

he

sum

of

the

mea

sure

s o

f tw

o a

ng

les

is 9

0,th

en t

he

ang

les

are

com

ple

men

tary

.Inv

erse

:If

tw

o a

ng

les

are

no

t co

mp

lem

enta

ry,

then

th

e su

m o

f th

eir

mea

sure

s is

no

t 90

.Co

ntr

apo

siti

ve:

if t

he

sum

of

the

mea

sure

s o

f tw

o a

ng

le is

no

t 90

,th

en t

he

ang

les

are

no

tco

mp

lem

enta

ry.T

rue:

all;

fals

e:n

on

e

Stu

dy G

uid

e a

nd I

nte

rven

tion

(con

tinued

)

Co

nd

itio

nal

Sta

tem

ents

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-3

2-3

Exer

cises

Exer

cises



An

swer

s

Skil

ls P

ract

ice

Co

nd

itio

nal

Sta

tem

ents

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-3

2-3

©G

lenc

oe/M

cGra

w-H

ill71

Gle

ncoe

Geo

met

ry

Lesson 2-3

Iden

tify

th

e h

ypot

hes

is a

nd

con

clu

sion

of

each

sta

tem

ent.

1.If

you

pu

rch

ase

a co

mpu

ter

and

do n

ot l

ike

it,t

hen

you

can

ret

urn

it

wit

hin

30

days

.H

:yo

u p

urc

has

e a

com

pu

ter

and

do

no

t lik

e it

;C

:yo

u c

an r

etu

rn it

wit

hin

30

day

s

2.If

x�

8 �

4,th

en x

��

4.H

:x

�8

�4;

C:

x�

�4

3.If

th

e dr

ama

clas

s ra

ises

$20

00,t

hen

th

ey w

ill

go o

n t

our.

H:

the

dra

ma

clas

s ra

ises

$20

00;

C:

they

will

go

on

to

ur

Wri

te e

ach

sta

tem

ent

in i

f-th

en f

orm

.

4.A

pol

ygon

wit

h f

our

side

s is

a q

uad

rila

tera

l.If

a p

oly

go

n h

as f

ou

r si

des

,th

en it

is a

qu

adri

late

ral.

5.“T

hos

e w

ho

stan

d fo

r n

oth

ing

fall

for

an

yth

ing.

”(A

lexa

nd

er H

amil

ton

)If

yo

u s

tan

d f

or

no

thin

g,t

hen

yo

u w

ill f

all f

or

anyt

hin

g.

6.A

n a

cute

an

gle

has

a m

easu

re l

ess

than

90.

If a

n a

ng

le is

acu

te,t

hen

its

mea

sure

is le

ss t

han

90.

Det

erm

ine

the

tru

th v

alu

e of

th

e fo

llow

ing

stat

emen

t fo

r ea

ch s

et o

f co

nd

itio

ns.

If y

ou f

inis

h y

our

hom

ewor

k b

y 5

P.M

.,th

en y

ou g

o ou

t to

din

ner

.

7.Yo

u f

inis

h y

our

hom

ewor

k by

5 P

.M.a

nd

you

go

out

to d

inn

er.

tru

e

8.Yo

u f

inis

h y

our

hom

ewor

k by

4 P

.M.a

nd

you

go

out

to d

inn

er.

tru

e

9.Yo

u f

inis

h y

our

hom

ewor

k by

5 P

.M.a

nd

you

do

not

go

out

to d

inn

er.

fals

e

10.W

rite

th

e co

nve

rse,

inve

rse,

and

con

trap

osit

ive

of t

he

con

diti

onal

sta

tem

ent.

Det

erm

ine

wh

eth

er e

ach

sta

tem

ent

is t

rue

or f

alse

.If

a st

atem

ent

is f

alse

,fin

d a

cou

nte

rexa

mpl

e.If

89

is d

ivis

ible

by

2,th

en 8

9 is

an

eve

n n

um

ber.

Co

nver

se:

If 8

9 is

an

eve

n n

um

ber

,th

en 8

9 is

div

isib

le b

y 2;

tru

e.In

vers

e:If

89

is n

ot

div

isib

le b

y 2,

then

89

is n

ot

an e

ven

nu

mb

er;

tru

e.C

on

trap

osi

tive

:If

89

is n

ot

an e

ven

nu

mb

er,t

hen

89

is n

ot

div

isib

le b

y 2;

tru

e.

©G

lenc

oe/M

cGra

w-H

ill72

Gle

ncoe

Geo

met

ry

Iden

tify

th

e h

ypot

hes

is a

nd

con

clu

sion

of

each

sta

tem

ent.

1.If

3x

�4

��

5,th

en x

��

3.H

:3x

�4

��

5;C

:x

��

3

2.If

you

tak

e a

clas

s in

tel

evis

ion

bro

adca

stin

g,th

en y

ou w

ill

film

a s

port

ing

even

t.H

:yo

u t

ake

a cl

ass

in t

elev

isio

n b

road

cast

ing

;C

:yo

u w

ill f

ilm a

sp

ort

ing

eve

nt

Wri

te e

ach

sta

tem

ent

in i

f-th

en f

orm

.

3.“T

hos

e w

ho

do n

ot r

emem

ber

the

past

are

con

dem

ned

to

repe

at i

t.”

(Geo

rge

San

taya

na)

If y

ou

do

no

t re

mem

ber

th

e p

ast,

then

yo

u a

re c

on

dem

ned

to

rep

eat

it.

4.A

djac

ent

angl

es s

har

e a

com

mon

ver

tex

and

a co

mm

on s

ide.

If t

wo

an

gle

s ar

e ad

jace

nt,

then

th

ey s

har

e a

com

mo

n v

erte

x an

d a

com

mo

n s

ide.

Det

erm

ine

the

tru

th v

alu

e of

th

e fo

llow

ing

stat

emen

t fo

r ea

ch s

et o

f co

nd

itio

ns.

If D

VD

pla

yers

are

on

sa

le f

or l

ess

tha

n $

100,

then

you

bu

y on

e.

5.D

VD

pla

yers

are

on

sal

e fo

r $9

5 an

d yo

u b

uy

one.

tru

e

6.D

VD

pla

yers

are

on

sal

e fo

r $1

00 a

nd

you

do

not

bu

y on

e.tr

ue

7.D

VD

pla

yers

are

not

on

sal

e fo

r u

nde

r $1

00 a

nd

you

do

not

bu

y on

e.tr

ue

8.W

rite

th

e co

nve

rse,

inve

rse,

and

con

trap

osit

ive

of t

he

con

diti

onal

sta

tem

ent.

Det

erm

ine

wh

eth

er e

ach

sta

tem

ent

is t

rue

or f

alse

.If

a st

atem

ent

is f

alse

,fin

d a

cou

nte

rexa

mpl

e.If

(�

8)2

�0,

then

�8

�0.

Co

nver

se:

If �

8 �

0,th

en (

�8)

2�

0;tr

ue.

Inve

rse:

If (

�8)

2�

0,th

en �

8 �

0;tr

ue.

Co

ntr

apo

siti

ve:

If �

8 �

0,th

en (

�8)

2�

0;fa

lse.

SUM

MER

CA

MP

For

Exe

rcis

es 9

an

d 1

0,u

se t

he

foll

owin

g in

form

atio

n.

Old

er c

ampe

rs w

ho

atte

nd

Woo

dlan

d Fa

lls

Cam

p ar

e ex

pect

ed t

o w

ork.

Cam

pers

wh

o ar

eju

nio

rs w

ait

on t

able

s.

9.W

rite

a c

ondi

tion

al s

tate

men

t in

if-

then

for

m.

Sam

ple

an

swer

:If

yo

u a

re a

jun

ior,

then

yo

u w

ait

on

tab

les.

10.W

rite

th

e co

nve

rse

of y

our

con

diti

onal

sta

tem

ent.

If y

ou

wai

t o

n t

able

s,th

en y

ou

are

a ju

nio

r.

Pra

ctic

e (

Ave

rag

e)

Co

nd

itio

nal

Sta

tem

ents

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-3

2-3



Readin

g t

o L

earn

Math

em

ati

csC

on

dit

ion

al S

tate

men

ts

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-3

2-3

©G

lenc

oe/M

cGra

w-H

ill73

Gle

ncoe

Geo

met

ry

Lesson 2-3

Pre-

Act

ivit

yH

ow a

re c

ond

itio

nal

sta

tem

ents

use

d i

n a

dve

rtis

emen

ts?

Rea

d th

e in

trod

uct

ion

to

Les

son

2-3

at

the

top

of p

age

75 i

n y

our

text

book

.

Doe

s th

e se

con

d ad

vert

isin

g st

atem

ent

in t

he

intr

odu

ctio

n m

ean

th

at y

ouw

ill

not

get

a f

ree

phon

e if

you

sig

n a

con

trac

t fo

r on

ly s

ix m

onth

s of

serv

ice?

Exp

lain

you

r an

swer

.N

o;

it o

nly

tel

ls y

ou

wh

at h

app

ens

ifyo

u s

ign

up

fo

r o

ne

year

.

Rea

din

g t

he

Less

on

1.Id

enti

fy t

he

hyp

oth

esis

an

d co

ncl

usi

on o

f ea

ch s

tate

men

t.a.

If y

ou a

re a

reg

iste

red

vote

r,th

en y

ou a

re a

t le

ast

18 y

ears

old

. Hyp

oth

esis

:yo

uar

e a

reg

iste

red

vo

ter;

Co

ncl

usi

on

:yo

u a

re a

t le

ast

18 y

ears

old

b.

If t

wo

inte

gers

are

eve

n,t

hei

r pr

odu

ct i

s ev

en.

Hyp

oth

esis

:tw

o in

teg

ers

are

even

;C

on

clu

sio

n:

thei

r p

rod

uct

is e

ven

2.C

ompl

ete

each

sen

ten

ce.

a.T

he

stat

emen

t th

at i

s fo

rmed

by

repl

acin

g bo

th t

he

hyp

oth

esis

an

d th

e co

ncl

usi

on o

f a

con

diti

onal

wit

h t

hei

r n

egat

ion

s is

th

e .

b.

Th

e st

atem

ent

that

is

form

ed b

y ex

chan

gin

g th

e h

ypot

hes

is a

nd

con

clu

sion

of

a co

ndi

tion

al i

s th

e .

3.C

onsi

der

the

foll

owin

g st

atem

ent:

You

liv

e in

Nor

th A

mer

ica

if y

ou l

ive

in t

he

Un

ited

Sta

tes.

a.W

rite

th

is c

ondi

tion

al s

tate

men

t in

if-

then

for

m a

nd

give

its

tru

th v

alu

e.If

th

est

atem

ent

is f

alse

,giv

e a

cou

nte

rexa

mpl

e.If

yo

u li

ve in

th

e U

nit

ed S

tate

s,th

enyo

u li

ve in

No

rth

Am

eric

a;fa

lse:

You

live

in H

awai

i.b

.W

rite

th

e in

vers

e of

th

e gi

ven

con

diti

onal

sta

tem

ent

in i

f-th

en f

orm

an

d gi

ve i

ts t

ruth

valu

e.If

th

e st

atem

ent

is f

alse

,giv

e a

cou

nte

rexa

mpl

e.If

yo

u d

o n

ot

live

in t

he

Un

ited

Sta

tes,

then

yo

u d

o n

ot

live

in N

ort

h A

mer

ica;

fals

e;sa

mp

lean

swer

:Yo

u li

ve in

Mex

ico

.c.

Wri

te t

he

con

trap

osit

ive

of t

he

give

n c

ondi

tion

al s

tate

men

t in

if-

then

for

m a

nd

give

its

tru

th v

alu

e.If

th

e st

atem

ent

is f

alse

,giv

e a

cou

nte

rexa

mpl

e.If

yo

u d

o n

ot

live

in N

ort

h A

mer

ica,

then

yo

u d

o n

ot

live

in t

he

Un

ited

Sta

tes;

fals

e:Yo

uliv

e in

Haw

aii.

d.

Wri

te t

he

con

vers

e of

th

e gi

ven

con

diti

onal

sta

tem

ent

in i

f-th

en f

orm

an

d gi

ve i

tstr

uth

val

ue.

If t

he

stat

emen

t is

fal

se,g

ive

a co

un

tere

xam

ple.

If y

ou

live

in N

ort

hA

mer

ica,

then

yo

u li

ve in

th

e U

nit

ed S

tate

s;fa

lse;

sam

ple

an

swer

:Yo

uliv

e in

Can

ada.

Hel

pin

g Y

ou

Rem

emb

er4.

Wh

en w

orki

ng

wit

h a

con

diti

onal

sta

tem

ent

and

its

thre

e re

late

d co

ndi

tion

als,

wh

at i

san

eas

y w

ay t

o re

mem

ber

wh

ich

sta

tem

ents

are

log

ical

ly e

quiv

alen

t to

eac

h o

ther

?S

amp

le a

nsw

er:T

he

two

sta

tem

ents

wh

ose

nam

es c

on

tain

ver

se(t

he

conv

erse

an

d t

he

inve

rse)

are

a lo

gic

ally

eq

uiv

alen

t p

air.

Th

e o

ther

tw

o(t

he

ori

gin

al c

on

dit

ion

al a

nd

th

e co

ntr

apo

siti

ve)

are

the

oth

er lo

gic

ally

equ

ival

ent

pai

r.

conv

erse

inve

rse

©G

lenc

oe/M

cGra

w-H

ill74

Gle

ncoe

Geo

met

ry

Ven

n D

iag

ram

s

A t

ype

of d

raw

ing

call

ed a

Ven

n d

iagr

amca

n b

e u

sefu

l in

exp

lain

ing

con

diti

onal

stat

emen

ts.A

Ven

n d

iagr

am u

ses

circ

les

to r

epre

sen

t se

ts o

f ob

ject

s.

Con

side

r th

e st

atem

ent

“All

rab

bits

hav

e lo

ng

ears

.”T

o m

ake

a V

enn

dia

gram

for

th

isst

atem

ent,

a la

rge

circ

le i

s dr

awn

to

repr

esen

t al

l an

imal

s w

ith

lon

g ea

rs.T

hen

asm

alle

r ci

rcle

is

draw

n i

nsi

de t

he

firs

t to

rep

rese

nt

all

rabb

its.

Th

e V

enn

dia

gram

show

s th

at e

very

rab

bit

is i

ncl

ude

d in

th

e gr

oup

of l

ong-

eare

d an

imal

s.

Th

e se

t of

rab

bits

is

call

ed a

su

bse

t of

th

e se

t of

lon

g-ea

red

anim

als.

Th

e V

enn

dia

gram

can

als

o ex

plai

n h

ow t

o w

rite

th

est

atem

ent,

“All

rab

bits

hav

e lo

ng

ears

,”in

if-

then

for

m.E

very

rabb

it i

s in

th

e gr

oup

of l

ong-

eare

d an

imal

s,so

if

an a

nim

al i

sa

rabb

it,t

hen

it

has

lon

g ea

rs.

For

eac

h s

tate

men

t,d

raw

a V

enn

dia

gram

.Th

en w

rite

th

e se

nte

nce

in

if-

then

for

m.

1.E

very

dog

has

lon

g h

air.

2.A

ll r

atio

nal

nu

mbe

rs a

re r

eal.

If a

n a

nim

al is

a d

og

,th

en it

If a

nu

mb

er is

rat

ion

al,t

hen

has

lon

g h

air.

it is

rea

l.

3.P

eopl

e w

ho

live

in

Iow

a li

ke c

orn

.4.

Sta

ff m

embe

rs a

re a

llow

ed i

n t

he

facu

lty

lou

nge

.

If a

per

son

live

s in

Iow

a,th

en t

he

per

son

like

s co

rn.

If a

per

son

is a

sta

ff m

emb

er,

then

th

e p

erso

n is

allo

wed

inth

e fa

cult

y ca

fete

ria.

peop

le in

the

cafe

teria staf

fm

embe

rs

peop

le w

holik

e co

rn

peop

le w

holiv

e in

Iow

a

real

num

bers

ratio

nal

num

bers

anim

als

with

long

hai

r

dogs

anim

als

with

long

ear

s

rabb

itsEn

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-3

2-3



An

swer

s

Stu

dy G

uid

e a

nd I

nte

rven

tion

Ded

uct

ive

Rea

son

ing

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-4

2-4

©G

lenc

oe/M

cGra

w-H

ill75

Gle

ncoe

Geo

met

ry

Lesson 2-4

Law

of

Det

ach

men

tD

edu

ctiv

e re

ason

ing

is t

he

proc

ess

of u

sin

g fa

cts,

rule

s,de

fin

itio

ns,

or p

rope

rtie

s to

rea

ch c

oncl

usi

ons.

On

e fo

rm o

f de

duct

ive

reas

onin

g th

at d

raw

sco

ncl

usi

ons

from

a t

rue

con

diti

onal

p→

qan

d a

tru

e st

atem

ent

pis

cal

led

the

Law

of

Det

ach

men

t.

Law

of

Det

ach

men

tIf

p→

qis

tru

e an

d p

is t

rue,

the

n q

is t

rue.

Sym

bo

ls[(

p→

q)]

�p]

→q

Th

e st

atem

ent

If t

wo

an

gles

are

su

pp

lem

enta

ry t

o th

e sa

me

an

gle,

then

th

ey a

re c

ong

ruen

tis

a t

rue

con

dit

ion

al.D

eter

min

e w

het

her

eac

h c

oncl

usi

onis

val

id b

ased

on

th

e gi

ven

in

form

atio

n.E

xpla

in y

our

reas

onin

g.

a.G

iven

:�A

and

�C

are

sup

ple

men

tary

to

�B

.C

oncl

usi

on:�

Ais

con

gru

ent

to �

C.

Th

e st

atem

ent

�A

an

d �

C a

re s

upp

lem

enta

ry t

o �

Bis

th

e h

ypot

hes

is o

f th

e co

ndi

tion

al.T

her

efor

e,by

th

e L

awof

Det

ach

men

t,th

e co

ncl

usi

on i

s tr

ue.

b.

Giv

en:�

Ais

con

gru

ent

to �

C.

Con

clu

sion

:�A

and

�C

are

sup

ple

men

tary

to

�B

.T

he

stat

emen

t �

A i

s co

ngr

uen

t to

�C

is n

ot t

he

hyp

oth

esis

of

th

e co

ndi

tion

al,s

o th

e L

aw o

f D

etac

hm

ent

can

not

be

use

d.T

he

con

clu

sion

is

not

val

id.

Det

erm

ine

wh

eth

er e

ach

con

clu

sion

is

vali

d b

ased

on

th

e tr

ue

con

dit

ion

al g

iven

.If

not

,wri

te i

nva

lid

.Exp

lain

you

r re

ason

ing.

If t

wo

angl

es a

re c

ompl

emen

tary

to

the

sam

e an

gle,

then

th

e an

gles

are

con

gru

ent.

1.G

iven

:�A

and

�C

are

com

plem

enta

ry t

o �

B.

Con

clu

sion

:�A

is c

ongr

uen

t to

�C

.

Th

e g

iven

sta

tem

ent

is t

he

hyp

oth

esis

of

the

con

dit

ion

al s

tate

men

t.S

ince

th

e co

nd

itio

nal

is t

rue,

the

con

clu

sio

n�

A�

�C

is t

rue.

2.G

iven

:�A

��

CC

oncl

usi

on:�

Aan

d �

Car

e co

mpl

emen

ts o

f �

B.

Th

e g

iven

sta

tem

ent

is n

ot

the

hyp

oth

esis

of

the

con

dit

ion

al.

Th

eref

ore

,th

e co

ncl

usi

on

is in

valid

.

3.G

iven

:�E

and

�F

are

com

plem

enta

ry t

o �

G.

Con

clu

sion

:�E

and

�F

are

vert

ical

an

gles

.

Wh

ile t

he

giv

en s

tate

men

t is

th

e hy

po

thes

is o

f th

e co

nd

itio

nal

sta

tem

ent,

the

stat

emen

t th

at �

Ean

d �

Far

e ve

rtic

al a

ng

les

is n

ot

the

con

clu

sio

n o

fth

e co

nd

itio

nal

.Th

e co

ncl

usi

on

is in

valid

.

ADE

F

H J

C

GB

Exam

ple

Exam

ple

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-H

ill76

Gle

ncoe

Geo

met

ry

Law

of

Syllo

gis

mA

not

her

way

to

mak

e a

vali

d co

ncl

usi

on i

s to

use

th

e L

aw o

fS

yllo

gism

.It

is s

imil

ar t

o th

e T

ran

siti

ve P

rope

rty.

Law

of

Syl

log

ism

If p

→q

is t

rue

and

q→

ris

tru

e, t

hen

p→

ris

als

o tr

ue.

Sym

bo

ls[(

p→

q)]

�(q

→r)

] →

(p→

r)

Th

e tw

o co

nd

itio

nal

sta

tem

ents

bel

ow a

re t

rue.

Use

th

e L

aw o

fS

yllo

gism

to

fin

d a

val

id c

oncl

usi

on.S

tate

th

e co

ncl

usi

on.

(1)

If a

nu

mbe

r is

a w

hol

e n

um

ber,

then

th

e n

um

ber

is a

n i

nte

ger.

(2)

If a

nu

mbe

r is

an

in

tege

r,th

en i

t is

a r

atio

nal

nu

mbe

r.

p:A

nu

mbe

r is

a w

hol

e n

um

ber.

q:A

nu

mbe

r is

an

in

tege

r.r:

A n

um

ber

is a

rat

ion

al n

um

ber.

Th

e tw

o co

ndi

tion

al s

tate

men

ts a

re p

→q

and

q→

r.U

sin

g th

e L

aw o

f S

yllo

gism

,a v

alid

con

clu

sion

is

p→

r.A

sta

tem

ent

of p

→r

is “

if a

nu

mbe

r is

a w

hol

e n

um

ber,

then

it

is a

rati

onal

nu

mbe

r.”

Det

erm

ine

wh

eth

er y

ou c

an u

se t

he

Law

of

Syl

logi

sm t

o re

ach

a v

alid

con

clu

sion

from

eac

h s

et o

f st

atem

ents

.

1.If

a d

og e

ats

Su

perd

og D

og F

ood,

he

wil

l be

hap

py.

Rov

er i

s h

appy

.

No

co

ncl

usi

on

is p

oss

ible

.

2.If

an

an

gle

is s

upp

lem

enta

ry t

o an

obt

use

an

gle,

then

it

is a

cute

.If

an

an

gle

is a

cute

,th

en i

ts m

easu

re i

s le

ss t

han

90.

A c

on

clu

sio

n is

po

ssib

le:

If a

n a

ng

le is

su

pp

lem

enta

ry t

o a

n o

btu

sean

gle

,th

en it

s m

easu

re is

less

th

an 9

0.

3.If

th

e m

easu

re o

f �

Ais

les

s th

an 9

0,th

en �

Ais

acu

te.

If �

Ais

acu

te,t

hen

�A

��

B.

A c

on

clu

sio

n is

po

ssib

le:

If t

he

mea

sure

of

�A

is le

ss 9

0,th

en �

A�

�B

.

4.If

an

an

gle

is a

rig

ht

angl

e,th

en t

he

mea

sure

of

the

angl

e is

90.

If t

wo

lin

es a

re p

erpe

ndi

cula

r,th

en t

hey

for

m a

rig

ht

angl

e.

A c

on

clu

sio

n is

po

ssib

le:

If t

wo

lin

es a

re p

erp

end

icu

lar,

then

th

e an

gle

they

fo

rm h

as m

easu

re 9

0.

5.If

you

stu

dy f

or t

he

test

,th

en y

ou w

ill

rece

ive

a h

igh

gra

de.

You

r gr

ade

on t

he

test

is

hig

h.

No

co

ncl

usi

on

is p

oss

ible

.

Stu

dy G

uid

e a

nd I

nte

rven

tion

(con

tinued

)

Ded

uct

ive

Rea

son

ing

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-4

2-4

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Ded

uct

ive

Rea

son

ing

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-4

2-4

©G

lenc

oe/M

cGra

w-H

ill77

Gle

ncoe

Geo

met

ry

Lesson 2-4

Det

erm

ine

wh

eth

er t

he

stat

ed c

oncl

usi

on i

s va

lid

bas

ed o

n t

he

give

n i

nfo

rmat

ion

.If

not

,wri

te i

nva

lid

.Exp

lain

you

r re

ason

ing.

If t

he

sum

of

the

mea

sure

s of

tw

o a

ngl

es i

s 18

0,th

en t

he

an

gles

are

su

pp

lem

enta

ry.

1.G

iven

:m�

A�

m�

Bis

180

.C

oncl

usi

on:�

Aan

d �

Bar

e su

pple

men

tary

.V

alid

;th

e co

ncl

usi

on

fo

llow

s by

th

e L

aw o

f D

etac

hm

ent.

2.G

iven

:m�

AB

Cis

95

and

m�

DE

Fis

90.

Con

clu

sion

:�A

BC

and

�D

EF

are

supp

lem

enta

ry.

Inva

lid;

the

giv

en in

form

atio

n s

ho

ws

that

th

e su

m o

f th

e an

gle

mea

sure

sis

185

,no

t 18

0.Yo

u c

ann

ot

use

�p

and

p→

qto

co

ncl

ud

e q

.

3.G

iven

:�1

and

�2

are

a li

nea

r pa

ir.

Con

clu

sion

:�1

and

�2

are

supp

lem

enta

ry.

Val

id;

sin

ce t

he

sum

of

the

mea

sure

s o

f th

e an

gle

s o

f a

linea

r p

air

is 1

80,

the

ang

les

are

sup

ple

men

tary

.

Use

th

e L

aw o

f S

yllo

gism

to

det

erm

ine

wh

eth

er a

val

id c

oncl

usi

on c

an b

e re

ach

edfr

om e

ach

set

of

stat

emen

ts.I

f a

vali

d c

oncl

usi

on i

s p

ossi

ble

,wri

te i

t.

4.If

tw

o an

gles

are

com

plem

enta

ry,t

hen

th

e su

m o

f th

eir

mea

sure

s is

90.

If t

he

sum

of

the

mea

sure

s of

tw

o an

gles

is

90,t

hen

bot

h o

f th

e an

gles

are

acu

te.

If t

wo

an

gle

s ar

e co

mp

lem

enta

ry,t

hen

bo

th o

f th

e an

gle

s ar

e ac

ute

.

5.If

th

e h

eat

wav

e co

nti

nu

es,t

hen

air

con

diti

onin

g w

ill

be u

sed

mor

e fr

equ

entl

y.If

air

con

diti

onin

g is

use

d m

ore

freq

uen

tly,

then

en

ergy

cos

ts w

ill

be h

igh

er.

If t

he

hea

t w

ave

con

tin

ues

,th

en e

ner

gy

cost

s w

ill b

e h

igh

er.

Det

erm

ine

wh

eth

er s

tate

men

t (3

) fo

llow

s fr

om s

tate

men

ts (

1) a

nd

(2)

by

the

Law

of D

etac

hm

ent

or t

he

Law

of

Syl

logi

sm.I

f it

doe

s,st

ate

wh

ich

law

was

use

d.I

f it

doe

s n

ot,w

rite

in

vali

d.

6.(1

) If

it

is T

ues

day,

then

Mar

la t

uto

rs c

hem

istr

y.(2

) If

Mar

la t

uto

rs c

hem

istr

y,th

en s

he

arri

ves

hom

e at

4 P

.M.

(3)

If M

arla

arr

ives

at

hom

e at

4 P

.M.,

then

it

is T

ues

day.

inva

lid

7.(1

) If

a m

arin

e an

imal

is

a st

arfi

sh,t

hen

it

live

s in

th

e in

tert

idal

zon

e of

th

e oc

ean

.(2

) Th

e in

tert

idal

zon

e is

th

e le

ast

stab

le o

f th

e oc

ean

zon

es.

(3)

If a

mar

ine

anim

al i

s a

star

fish

,th

en i

t li

ves

in t

he

leas

t st

able

of

the

ocea

n z

ones

.ye

s;L

aw o

f S

yllo

gis

m

©G

lenc

oe/M

cGra

w-H

ill78

Gle

ncoe

Geo

met

ry

Det

erm

ine

wh

eth

er t

he

stat

ed c

oncl

usi

on i

s va

lid

bas

ed o

n t

he

give

n i

nfo

rmat

ion

.If

not

,wri

te i

nva

lid

.Exp

lain

you

r re

ason

ing.

If a

poi

nt

is t

he

mid

poi

nt

of a

seg

men

t,th

en i

t d

ivid

es t

he

segm

ent

into

tw

oco

ng

ruen

t se

gmen

ts.

1.G

iven

:Ris

th

e m

idpo

int

of Q �

S�.

Con

clu

sion

:Q �R�

�R�

S�V

alid

;si

nce

Ris

th

e m

idp

oin

t o

f Q�

S�,t

he

Law

of

Det

ach

men

t in

dic

ates

that

it d

ivid

es Q�

S�in

to t

wo

co

ng

ruen

t se

gm

ents

.

2.G

iven

:A�B�

�B�

C�C

oncl

usi

on:B

divi

des

A�C�

into

tw

o co

ngr

uen

t se

gmen

ts.

Inva

lid;

the

po

ints

A,B

,an

d C

may

no

t b

e co

llin

ear,

and

if t

hey

are

no

t,th

en B

will

no

t b

e th

e m

idp

oin

t o

f A�

C�.

Use

th

e L

aw o

f S

yllo

gism

to

det

erm

ine

wh

eth

er a

val

id c

oncl

usi

on c

an b

e re

ach

edfr

om e

ach

set

of

stat

emen

ts.I

f a

vali

d c

oncl

usi

on i

s p

ossi

ble

,wri

te i

t.

3.If

tw

o an

gles

for

m a

lin

ear

pair

,th

en t

he

two

angl

es a

re s

upp

lem

enta

ry.

If t

wo

angl

es a

re s

upp

lem

enta

ry,t

hen

th

e su

m o

f th

eir

mea

sure

s is

180

.If

tw

o a

ng

les

form

a li

nea

r p

air,

then

th

e su

m o

f th

eir

mea

sure

s is

180

.

4.If

a h

urr

ican

e is

Cat

egor

y 5,

then

win

ds a

re g

reat

er t

han

155

mil

es p

er h

our.

If w

inds

are

gre

ater

tha

n 15

5 m

iles

per

hour

,the

n tr

ees,

shru

bs,a

nd s

igns

are

blo

wn

dow

n.If

a h

urr

ican

e is

Cat

ego

ry 5

,th

en t

rees

,sh

rub

s,an

d s

ign

s ar

e bl

own

dow

n.

Det

erm

ine

wh

eth

er s

tate

men

t (3

) fo

llow

s fr

om s

tate

men

ts (

1) a

nd

(2)

by

the

Law

of D

etac

hm

ent

or t

he

Law

of

Syl

logi

sm.I

f it

doe

s,st

ate

wh

ich

law

was

use

d.I

f it

doe

s n

ot,w

rite

in

vali

d.

5.(1

) If

a w

hol

e n

um

ber

is e

ven

,th

en i

ts s

quar

e is

div

isib

le b

y 4.

(2) T

he

nu

mbe

r I

am t

hin

kin

g of

is

an e

ven

wh

ole

nu

mbe

r.(3

) Th

e sq

uar

e of

th

e n

um

ber

I am

th

inki

ng

of i

s di

visi

ble

by 4

.ye

s;L

aw o

f D

etac

hm

ent

6.(1

) If

th

e fo

otba

ll t

eam

win

s it

s h

omec

omin

g ga

me,

then

Con

rad

wil

l at

ten

d th

e sc

hoo

lda

nce

th

e fo

llow

ing

Fri

day.

(2)

Con

rad

atte

nds

th

e sc

hoo

l da

nce

on

Fri

day.

(3) T

he

foot

ball

tea

m w

on t

he

hom

ecom

ing

gam

e.in

valid

7.B

IOLO

GY

If a

n o

rgan

ism

is

a pa

rasi

te,t

hen

it

surv

ives

by

livi

ng

on o

r in

a h

ost

orga

nis

m.I

f a

para

site

liv

es i

n o

r on

a h

ost

orga

nis

m,t

hen

it

har

ms

its

hos

t.W

hat

con

clu

sion

can

you

dra

w i

f a

viru

s is

a p

aras

ite?

If a

vir

us

is a

par

asit

e,th

en it

har

ms

its

ho

st.

Pra

ctic

e (

Ave

rag

e)

Ded

uct

ive

Rea

son

ing

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-4

2-4



An

swer

s

Readin

g t

o L

earn

Math

em

ati

csD

edu

ctiv

e R

easo

nin

g

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-4

2-4

©G

lenc

oe/M

cGra

w-H

ill79

Gle

ncoe

Geo

met

ry

Lesson 2-4

Pre-

Act

ivit

yH

ow d

oes

ded

uct

ive

reas

onin

g ap

ply

to

hea

lth

?

Rea

d th

e in

trod

uct

ion

to

Les

son

2-4

at

the

top

of p

age

82 i

n y

our

text

book

.

Su

ppos

e a

doct

or w

ants

to

use

th

e do

se c

har

t in

you

r te

xtbo

ok t

o pr

escr

ibe

an a

nti

biot

ic,b

ut

the

only

sca

le i

n h

er o

ffic

e gi

ves

wei

ghts

in

pou

nds

.How

can

sh

e u

se t

he

fact

th

at 1

kil

ogra

m i

s ab

out

2.2

pou

nds

to

dete

rmin

e th

eco

rrec

t do

se f

or a

pat

ien

t?S

amp

le a

nsw

er:T

he

do

cto

r ca

n d

ivid

eth

e p

atie

nt’s

wei

gh

t in

po

un

ds

by 2

.2 t

o f

ind

th

e eq

uiv

alen

tm

ass

in k

ilog

ram

s.S

he

can

th

en u

se t

he

do

se c

har

t.

Rea

din

g t

he

Less

on

If s

,t,a

nd

uar

e th

ree

stat

emen

ts,m

atch

eac

h d

escr

ipti

on f

rom

th

e li

st o

n t

he

left

wit

h a

sym

bol

ic s

tate

men

t fr

om t

he

list

on

th

e ri

ght.

1.n

egat

ion

of

te

a.s

�u

2.co

nju

nct

ion

of

san

d u

gb

.[(

s→

t) �

s] →

t

3.co

nve

rse

of s

→t

hc.

�s

→�

u

4.di

sju

nct

ion

of

san

d u

ad

.�

u→

�s

5.L

aw o

f D

etac

hm

ent

be.

�t

6.co

ntr

apos

itiv

e of

s→

tj

f.[(

u→

t) �

(t→

s)]

→(u

→s)

7.in

vers

e of

s→

uc

g.s

�u

8.co

ntr

apos

itiv

e of

s→

ud

h.

t→

s

9.L

aw o

f S

yllo

gism

fi.

t

10.n

egat

ion

of

�t

ij.

�t

→�

s

11.D

eter

min

e w

het

her

sta

tem

ent

(3)

foll

ows

from

sta

tem

ents

(1)

an

d (2

) by

th

e L

aw o

fD

etac

hm

ent

or t

he

Law

of

Syl

logi

sm.I

f it

doe

s,st

ate

wh

ich

law

was

use

d.If

it

does

not

,w

rite

in

vali

d.

a.(1

) E

very

squ

are

is a

par

alle

logr

am.

(2)

Eve

ry p

aral

lelo

gram

is

a po

lygo

n.

(3)

Eve

ry s

quar

e is

a p

olyg

on.

yes;

Law

of

Syl

log

ism

b.

(1)I

f tw

o li

nes

th

at l

ie i

n t

he

sam

e pl

ane

do n

ot i

nte

rsec

t,th

ey a

re p

aral

lel.

(2)

Lin

es �

and

mli

e in

pla

ne

Uan

d do

not

in

ters

ect.

(3)

Lin

es �

and

mar

e pa

rall

el.

yes;

Law

of

Det

ach

men

tc.

(1)

Per

pen

dicu

lar

lin

es i

nte

rsec

t to

for

m f

our

righ

t an

gles

.(2

) �

A,�

B,�

C,a

nd

�D

are

fou

r ri

ght

angl

es.

(3)

�A

,�B

,�C

,an

d �

Dar

e fo

rmed

by

inte

rsec

tin

g pe

rpen

dicu

lar

lin

es.

inva

lid

Hel

pin

g Y

ou

Rem

emb

er12

.A g

ood

way

to

rem

embe

r so

met

hin

g is

to

expl

ain

it

to s

omeo

ne

else

.Su

ppos

e th

at a

clas

smat

e is

hav

ing

trou

ble

rem

embe

rin

g w

hat

th

e L

aw o

f D

etac

hm

ent

mea

ns?

Sam

ple

an

swer

:Th

e w

ord

det

ach

mea

ns

to t

ake

som

eth

ing

off

of

ano

ther

thin

g.T

he

Law

of

Det

ach

men

t sa

ys t

hat

wh

en a

co

nd

itio

nal

an

d it

shy

po

thes

is a

re b

oth

tru

e,yo

u c

an d

etac

h t

he

con

clu

sio

n a

nd

fee

lco

nfi

den

t th

at it

to

o is

a t

rue

stat

emen

t.

©G

lenc

oe/M

cGra

w-H

ill80

Gle

ncoe

Geo

met

ry

Valid

an

d F

ault

y A

rgu

men

tsC

onsi

der

the

stat

emen

ts a

t th

e ri

ght.

Wh

at c

oncl

usi

ons

can

you

mak

e?

Fro

m s

tate

men

ts 1

an

d 3,

it i

s co

rrec

t to

con

clu

de t

hat

Boo

tspu

rrs

if i

t is

hap

py.H

owev

er,i

t is

fau

lty

to c

oncl

ude

fro

m o

nly

st

atem

ents

2 a

nd

3 th

at B

oots

is

hap

py.T

he

if-t

hen

for

m o

f st

atem

ent

3 is

If

a ca

t is

hap

py,t

hen

it

purr

s.

Adv

erti

sers

oft

en u

se f

ault

y lo

gic

in s

ubt

le w

ays

to h

elp

sell

thei

r pr

odu

cts.

By

stu

dyin

g th

e ar

gum

ents

,you

can

dec

ide

wh

eth

er t

he

argu

men

t is

val

id o

r fa

ult

y.

Dec

ide

if e

ach

arg

um

ent

is v

alid

or

fau

lty.

1.(1

)If

you

bu

y T

uff

Cot

e lu

ggag

e,it

2.(1

)If

you

bu

y T

uff

Cot

e lu

ggag

e,it

wil

l su

rviv

e ai

rlin

e tr

avel

.w

ill

surv

ive

airl

ine

trav

el.

(2)

Just

in b

uys

Tu

ff C

ote

lugg

age.

(2)

Just

in’s

lu

ggag

e su

rviv

ed a

irli

ne

trav

el.

Con

clu

sion

:Ju

stin

’s l

ugg

age

wil

lC

oncl

usi

on:J

ust

in h

as T

uff

Cot

esu

rviv

e ai

rlin

e tr

avel

.va

lidlu

ggag

e.fa

ult

y

3.(1

)If

you

use

Cle

ar L

ine

lon

g4.

(1)

If y

ou r

ead

the

book

Bea

uti

ful

dist

ance

ser

vice

,you

wil

l h

ave

B

raid

s,yo

u w

ill

be a

ble

to m

ake

clea

r re

cept

ion

.be

auti

ful

brai

ds e

asil

y.(2

)A

nn

a h

as c

lear

lon

g di

stan

ce

(2)

Nan

cy r

ead

the

book

Bea

uti

ful

rece

ptio

n.

Bra

ids.

Con

clu

sion

:An

na

use

s C

lear

Lin

eC

oncl

usi

on:N

ancy

can

mak

e lo

ng

dist

ance

ser

vice

.fa

ult

ybe

auti

ful

brai

ds e

asil

y.va

lid

5.(1

)If

you

bu

y a

wor

d pr

oces

sor,

you

6.(1

)G

reat

sw

imm

ers

wea

r A

quaL

ine

wil

l be

abl

e to

wri

te l

ette

rs f

aste

r.sw

imw

ear.

(2)

Tan

ia b

ough

t a

wor

d pr

oces

sor.

(2)

Gin

a w

ears

Aqu

aLin

e sw

imw

ear.

Con

clu

sion

:Tan

ia w

ill

be a

ble

to

Con

clu

sion

:Gin

a is

a g

reat

sw

imm

er.

wri

te l

ette

rs f

aste

r.va

lidfa

ult

y

7.W

rite

an

exa

mpl

e of

fau

lty

logi

c th

atyo

u h

ave

seen

in

an

adv

erti

sem

ent.

See

stu

den

ts’w

ork

.

(1)

Boo

ts i

s a

cat.

(2)

Boo

ts i

s pu

rrin

g.(3

)A

cat

pu

rrs

if i

t is

hap

py.

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-4

2-4



Stu

dy G

uid

e a

nd I

nte

rven

tion

Po

stu

late

s an

d P

arag

rap

h P

roo

fs

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-5

2-5

©G

lenc

oe/M

cGra

w-H

ill81

Gle

ncoe

Geo

met

ry

Lesson 2-5

Poin

ts, L

ines

, an

d P

lan

esIn

geo

met

ry,a

pos

tula

teis

a s

tate

men

t th

at i

s ac

cept

ed a

str

ue.

Pos

tula

tes

desc

ribe

fu

nda

men

tal

rela

tion

ship

s in

geo

met

ry.

Po

stu

late

:T

hrou

gh a

ny t

wo

poin

ts,

ther

e is

exa

ctly

one

line

.P

ost

ula

te:

Thr

ough

any

thr

ee p

oint

s no

t on

the

sam

e lin

e, t

here

is e

xact

ly o

ne p

lane

.P

ost

ula

te:

Alin

e co

ntai

ns a

t le

ast

two

poin

ts.

Po

stu

late

:A

plan

e co

ntai

ns a

t le

ast

thre

e po

ints

not

on

the

sam

e lin

e.P

ost

ula

te:

If tw

o po

ints

lie

in a

pla

ne,

then

the

line

con

tain

ing

thos

e po

ints

lies

in t

he p

lane

.P

ost

ula

te:

If tw

o lin

es in

ters

ect,

then

the

ir in

ters

ectio

n is

exa

ctly

one

poi

nt.

Po

stu

late

:If

two

plan

es in

ters

ect,

then

the

ir in

ters

ectio

n is

a li

ne.

Det

erm

ine

wh

eth

er e

ach

sta

tem

ent

is a

lwa

ys,

som

etim

es,o

r n

ever

tru

e.a.

Th

ere

is e

xact

ly o

ne

pla

ne

that

con

tain

s p

oin

ts A

,B,a

nd

C.

Som

etim

es;i

f A

,B,a

nd

Car

e co

llin

ear,

they

are

con

tain

ed i

n m

any

plan

es.I

f th

ey a

ren

onco

llin

ear,

then

th

ey a

re c

onta

ined

in

exa

ctly

on

e pl

ane.

b.

Poi

nts

Ean

d F

are

con

tain

ed i

n e

xact

ly o

ne

lin

e.A

lway

s;th

e fi

rst

post

ula

te s

tate

s th

at t

her

e is

exa

ctly

on

e li

ne

thro

ugh

an

y tw

o po

ints

.

c.T

wo

lin

es i

nte

rsec

t in

tw

o d

isti

nct

poi

nts

Man

d N

.N

ever

;th

e in

ters

ecti

on o

f tw

o li

nes

is

one

poin

t.

Use

pos

tula

tes

to d

eter

min

e w

het

her

eac

h s

tate

men

t is

alw

ays

,som

etim

es,o

rn

ever

tru

e.

1.A

lin

e co

nta

ins

exac

tly

one

poin

t.n

ever

2.N

onco

llin

ear

poin

ts R

,S,a

nd

Tar

e co

nta

ined

in

exa

ctly

on

e pl

ane.

alw

ays

3.A

ny

two

lin

es �

and

min

ters

ect.

som

etim

es

4.If

poi

nts

Gan

d H

are

cont

aine

d in

pla

ne M

,the

n G�

H�is

per

pend

icul

ar t

o pl

ane

M.n

ever

5.P

lan

es R

and

Sin

ters

ect

in p

oin

t T

.nev

er

6.If

poi

nts

A,B

,an

d C

are

non

coll

inea

r,th

en s

egm

ents

A�B�

,B�C�

,an

d C�

A�ar

e co

nta

ined

in

exac

tly

one

plan

e.al

way

s

In t

he

figu

re,A�

C�an

d D�

E�ar

e in

pla

ne

Qan

d A�

C�|| D�

E�.

Sta

te t

he

pos

tula

te t

hat

can

be

use

d t

o sh

ow e

ach

st

atem

ent

is t

rue.

7.E

xact

ly o

ne

plan

e co

nta

ins

poin

ts F

,B,a

nd

E. T

hro

ug

h

any

thre

e p

oin

ts n

ot

on

th

e sa

me

line,

ther

e is

exac

tly

on

e p

lan

e.

8.B

E��

�li

es i

n p

lan

e Q

.If

two

po

ints

lie

in a

pla

ne,

then

th

e lin

e co

nta

inin

g t

ho

se p

oin

ts li

es in

th

e p

lan

e.

QB

C

A

DE

F

G

Exam

ple

Exam

ple

Exer

cises

Exer

cises

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met

ry

Para

gra

ph

Pro

ofs

A s

tate

men

t th

at c

an b

e pr

oved

tru

e is

cal

led

a th

eore

m.Y

ou c

anu

se u

nde

fin

ed t

erm

s,de

fin

itio

ns,

post

ula

tes,

and

alre

ady-

prov

ed t

heo

rem

s to

pro

ve o

ther

stat

emen

ts t

rue.

A l

ogic

al a

rgu

men

t th

at u

ses

dedu

ctiv

e re

ason

ing

to r

each

a v

alid

con

clu

sion

is

call

ed a

pro

of.I

n o

ne

type

of

proo

f,a

par

agra

ph

pro

of,y

ou w

rite

a p

arag

raph

to

expl

ain

wh

y a

stat

emen

t is

tru

e. In �

AB

C,B �

D�is

an

an

gle

bis

ecto

r.W

rite

a

par

agra

ph

pro

of t

o sh

ow t

hat

�A

BD

��

CB

D.

By

defi

nit

ion

,an

an

gle

bise

ctor

div

ides

an

an

gle

into

tw

o co

ngr

uen

tan

gles

.Sin

ce B �

D�is

an

an

gle

bise

ctor

,�A

BC

is d

ivid

ed i

nto

tw

oco

ngr

uen

t an

gles

.Th

us,

�A

BD

��

CB

D.

1.G

iven

th

at �

A�

�D

and

�D

��

E,w

rite

a p

arag

raph

pro

of t

o sh

ow t

hat

�A

��

E.

Sin

ce �

A�

�D

and

�D

��

E,t

hen

m�

A�

m�

Dan

d m

�D

�m

�E

byth

e d

ef.o

f co

ng

ruen

ce.S

o m

�A

�m

�E

by t

he

Tran

siti

ve P

rop

erty

,an

d�

A�

�E

by t

he

def

.of

con

gru

ence

.

2.It

is

give

n t

hat

B�C�

�E�

F�,M

is t

he

mid

poin

t of

B�C�

,an

d N

is t

he

mid

poin

t of

E �F�

.Wri

te a

par

agra

ph p

roof

to

show

th

at B

M�

EN

.

Mis

th

e m

idp

oin

t o

f B�

C�an

d N

is t

he

mid

po

int

of

E�F�,

so B

M�

�1 2� BC

and

EN

��1 2� E

F.B�

C��

E�F�

so B

C�

EF

by t

he

def

.of

con

gru

ence

,an

d �1 2� B

C�

�1 2� EF

by t

he

mu

lt.p

rop

.Th

us

BM

�E

Nby

th

e Tr

ansi

tive

Pro

per

ty.

3.G

iven

th

at S

is t

he

mid

poin

t of

Q�P�

,Tis

th

e m

idpo

int

of P�

R�,

and

Pis

th

e m

idpo

int

of S �

T�,w

rite

a p

arag

raph

pro

of t

o sh

ow

that

QS

�T

R.

Pis

th

e m

idp

oin

t o

f Q�

R�so

QP

�P

R.�

1 2� QP

��1 2� P

Rby

th

e M

ult

iplic

atio

n P

rop

erty

.San

d T

are

the

mid

po

ints

of

Q�P�

and

P�R�

,res

pec

tive

ly,s

o Q

S�

�1 2� QP

and

TR

��1 2� P

R.

Th

en T

R�

�1 2� QP

by s

ub

stit

uti

on

,an

d Q

S�

TR

by t

he

Tran

siti

ve P

rop

erty

.

Q

RT

PS

B

CM E

NF

B

AD

C

Stu

dy G

uid

e a

nd I

nte

rven

tion

(con

tinued

)

Po

stu

late

s an

d P

arag

rap

h P

roo

fs

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-5

2-5

Exam

ple

Exam

ple

Exer

cises

Exer

cises



An

swer

s

Skil

ls P

ract

ice

Po

stu

late

s an

d P

arag

rap

h P

roo

fs

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-5

2-5

©G

lenc

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ncoe

Geo

met

ry

Lesson 2-5

Det

erm

ine

the

nu

mb

er o

f li

ne

segm

ents

th

at c

an b

e d

raw

n c

onn

ecti

ng

each

pai

rof

poi

nts

.

1.6

2.15

Det

erm

ine

wh

eth

er t

he

foll

owin

g st

atem

ents

are

alw

ays

,som

etim

es,o

r n

ever

tru

e.E

xpla

in.

3.T

hre

e co

llin

ear

poin

ts d

eter

min

e a

plan

e.N

ever

;3

no

nco

llin

ear

po

ints

det

erm

ine

a p

lan

e.

4.T

wo

poin

ts A

and

Bde

term

ine

a li

ne.

Alw

ays;

thro

ug

h a

ny t

wo

po

ints

th

ere

is e

xact

ly o

ne

line.

5.A

pla

ne

con

tain

s at

lea

st t

hre

e li

nes

.A

lway

s;a

pla

ne

con

tain

s at

leas

t th

ree

po

ints

no

t o

n t

he

sam

e lin

e,an

dea

ch p

air

of

thes

e d

eter

min

es a

lin

e.

In t

he

figu

re,D

G��

�an

d D

P��

�li

e in

pla

ne

Jan

d H

lies

on

DG

�� .

Sta

te

the

pos

tula

te t

hat

can

be

use

d t

o sh

ow e

ach

sta

tem

ent

is t

rue.

6.G

and

Par

e co

llin

ear.

Po

stu

late

2.1

:th

rou

gh

any

tw

o p

oin

ts,t

her

e is

exa

ctly

on

e lin

e.

7.P

oin

ts D

,H,a

nd

Par

e co

plan

ar.

Po

stu

late

2.2

;Th

rou

gh

any

th

ree

po

ints

no

t o

n t

he

sam

e lin

e,th

ere

isex

actl

y o

ne

pla

ne.

8.PR

OO

FIn

th

e fi

gure

at

the

righ

t,po

int

Bis

th

e m

idpo

int

of A�

C�an

d po

int

Cis

th

e m

idpo

int

of B �

D�.W

rite

a p

arag

raph

pro

of t

o pr

ove

that

AB

�C

D.

Giv

en:

Bis

th

e m

idp

oin

t o

f A�

C�.

Cis

th

e m

idp

oin

t o

f B�

D�.

Pro

ve:

AB

�C

DP

roo

f:S

ince

Bis

th

e m

idp

oin

t o

f A�

C�an

d C

is t

he

mid

po

int

of

B�D�

,we

kno

w b

y th

e M

idp

oin

t Th

eore

m t

hat

A�B�

�B�

C�an

d B�

C��

C�D�

.Sin

ceco

ng

ruen

t se

gm

ents

hav

e eq

ual

mea

sure

s,A

B�

BC

and

BC

�C

D.T

hu

s,by

th

e Tr

ansi

tive

Pro

per

ty o

f E

qu

alit

y,A

B�

CD

.

DC

BAJ

PD

H

G

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Geo

met

ry

Det

erm

ine

the

nu

mb

er o

f li

ne

segm

ents

th

at c

an b

e d

raw

n c

onn

ecti

ng

each

pai

rof

poi

nts

.

1.21

2.28

Det

erm

ine

wh

eth

er t

he

foll

owin

g st

atem

ents

are

alw

ays

,som

etim

es,o

r n

ever

tru

e.E

xpla

in.

3.T

he

inte

rsec

tion

of

two

plan

es c

onta

ins

at l

east

tw

o po

ints

.A

lway

s;th

e in

ters

ecti

on

of

two

pla

nes

is a

lin

e,an

d a

lin

e co

nta

ins

atle

ast

two

po

ints

.

4.If

th

ree

plan

es h

ave

a po

int

in c

omm

on,t

hen

th

ey h

ave

a w

hol

e li

ne

in c

omm

on.

So

met

imes

;th

ey m

igh

t h

ave

on

ly t

hat

sin

gle

po

int

in c

om

mo

n.

In t

he

figu

re,l

ine

man

d T

Q��

�li

e in

pla

ne

A.S

tate

th

e p

ostu

late

th

at c

an b

e u

sed

to

show

th

at e

ach

sta

tem

ent

is t

rue.

5.L

,T,a

nd

lin

e m

lie

in t

he

sam

e pl

ane.

Po

stu

late

2.5

:If

tw

o p

oin

ts li

e in

a p

lan

e,th

en t

he

enti

re li

ne

con

tain

ing

th

ose

po

ints

lies

in t

hat

pla

ne.

6.L

ine

man

d S�

T�in

ters

ect

at T

.P

ost

ula

te 2

.6:

If t

wo

lin

es in

ters

ect,

then

th

eir

inte

rsec

tio

n is

exa

ctly

on

ep

oin

t.

7.In

th

e fi

gure

,Eis

th

e m

idpo

int

of A�

B�an

d C�

D�,a

nd

AB

�C

D.W

rite

a

para

grap

h p

roof

to

prov

e th

at A �

E��

E �D�

.

Giv

en:

Eis

th

e m

idp

oin

t o

f A�

B�an

d C�

D�A

B�

CD

Pro

ve:

A�E�

�E�

D�P

roo

f:S

ince

Eis

th

e m

idp

oin

t o

f A�

B�an

d C�

D�,w

e kn

ow

by

the

Mid

po

int

Th

eore

m,t

hat

A�E�

�E�

B�an

d C�

E��

E�D�

.By

the

def

init

ion

of

con

gru

ent

seg

men

ts,A

E�

EB

��1 2� A

Ban

d C

E�

ED

��1 2� C

D.S

ince

AB

�C

D,

�1 2� AB

��1 2� C

Dby

th

e M

ult

iplic

atio

n P

rop

erty

.So

AE

�E

D,a

nd

by

the

def

init

ion

of

con

gru

ent

seg

men

ts, A�

E��

E�D�

.

8.LO

GIC

Poi

nts

A,B

,an

d C

are

not

col

lin

ear.

Poi

nts

B,C

,an

d D

are

not

col

lin

ear.

Poi

nts

A,B

,C,a

nd

Dar

e n

ot c

opla

nar

.Des

crib

e tw

o pl

anes

th

at i

nte

rsec

t in

lin

e B

C.

the

pla

ne

that

co

nta

ins

A,B

,an

d C

and

th

e p

lan

e th

at c

on

tain

s B

,C,

and

D

BEC D

A

Am

TQ

L

S

Pra

ctic

e (

Ave

rag

e)

Po

stu

late

s an

d P

arag

rap

h P

roo

fs

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-5

2-5



Readin

g t

o L

earn

Math

em

ati

csP

ost

ula

tes

and

Par

agra

ph

Pro

ofs

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-5

2-5

©G

lenc

oe/M

cGra

w-H

ill85

Gle

ncoe

Geo

met

ry

Lesson 2-5

Pre-

Act

ivit

yH

ow a

re p

ostu

late

s u

sed

by

the

fou

nd

ing

fath

ers

of t

he

Un

ited

Sta

tes?

Rea

d th

e in

trod

uct

ion

to

Les

son

2-5

at

the

top

of p

age

89 i

n y

our

text

book

.

Pos

tula

tes

are

ofte

n d

escr

ibed

as

stat

emen

ts t

hat

are

so

basi

c an

d so

cle

arly

corr

ect

that

peo

ple

wil

l be

wil

lin

g to

acc

ept

them

as

tru

e w

ith

out

aski

ng

for

evid

ence

or

proo

f.G

ive

a st

atem

ent

abou

t n

um

bers

th

at y

ou t

hin

k m

ost

peop

le w

ould

acc

ept

as t

rue

wit

hou

t ev

iden

ce.

Sam

ple

an

swer

:E

very

nu

mb

er is

eq

ual

to

itse

lf.

Rea

din

g t

he

Less

on

1.D

eter

min

e w

het

her

eac

h o

f th

e fo

llow

ing

is a

cor

rect

or i

nco

rrec

tst

atem

ent

of a

geom

etri

c po

stu

late

.If

the

stat

emen

t is

in

corr

ect,

repl

ace

the

un

derl

ined

wor

ds t

o m

ake

the

stat

emen

t co

rrec

t.in

corr

ect;

a.A

pla

ne

con

tain

s at

lea

st

that

do

not

lie

on

th

e sa

me

lin

e.th

ree

po

ints

b.

If

inte

rsec

t,th

en t

he

inte

rsec

tion

is

a li

ne.

corr

ect

inco

rrec

t;c.

Thr

ough

any

no

t on

the

sam

e lin

e,th

ere

is e

xact

ly o

ne p

lane

.th

ree

poin

tsd

.A

lin

e co

nta

ins

at l

east

.

inco

rrec

t;tw

o p

oin

tsin

corr

ect;

e.If

tw

o li

nes

,t

hen

th

eir

inte

rsec

tion

is

exac

tly

one

poin

t.in

ters

ect

f.T

hro

ugh

an

y tw

o po

ints

,th

ere

is

one

lin

e.in

corr

ect;

exac

tly

2.D

eter

min

e w

het

her

eac

h s

tate

men

t is

alw

ays,

som

etim

es,o

r n

ever

tru

e.If

th

e st

atem

ent

is n

ot a

lway

s tr

ue,

expl

ain

wh

y.

a.If

tw

o pl

anes

in

ters

ect,

thei

r in

ters

ecti

on i

s a

lin

e.al

way

sb

.T

he

mid

poin

t of

a s

egm

ent

divi

des

the

segm

ent

into

tw

o co

ngr

uen

t se

gmen

ts.a

lway

sc.

Th

ere

is e

xact

ly o

ne

plan

e th

at c

onta

ins

thre

e co

llin

ear

poin

ts.

nev

er;

Sam

ple

answ

er:T

her

e ar

e in

fin

itel

y m

any

pla

nes

if t

he

thre

e p

oin

ts a

reco

llin

ear,

but

on

ly o

ne

pla

ne

if t

he

po

ints

are

no

nco

llin

ear.

d.

If t

wo

lin

es i

nte

rsec

t,th

eir

inte

rsec

tion

is

one

poin

t.al

way

s

3.U

se t

he

wal

ls,f

loor

,an

d ce

ilin

g of

you

r cl

assr

oom

to

desc

ribe

a m

odel

for

eac

h o

f th

efo

llow

ing

geom

etri

c si

tuat

ion

s.

a.tw

o pl

anes

th

at i

nte

rsec

t in

a l

ine

Sam

ple

an

swer

:tw

o a

dja

cen

t w

alls

th

atin

ters

ect

at a

n e

dg

e o

f b

oth

wal

ls in

th

e co

rner

of

the

roo

mb

.tw

o pl

anes

th

at d

o n

ot i

nte

rsec

tS

amp

le a

nsw

er:

the

ceili

ng

an

d t

he

flo

or

(or

two

op

po

site

wal

ls)

c.th

ree

plan

es t

hat

inte

rsec

t in

a p

oint

Sam

ple

an

swer

:th

e fl

oo

r (o

r ce

ilin

g)

and

tw

o a

dja

cen

t w

alls

th

at in

ters

ect

at a

co

rner

of

the

flo

or

(or

ceili

ng

)

Hel

pin

g Y

ou

Rem

emb

er4.

A g

ood

way

to

rem

embe

r a

new

mat

hem

atic

al t

erm

is t

o re

late

it t

o a

wor

d yo

u al

read

ykn

ow.E

xpla

in h

ow t

he id

ea o

f a

mat

hem

atic

al t

heor

emis

rel

ated

to

the

idea

of

a sc

ient

ific

theo

ry.

Sam

ple

an

swer

:Sci

entis

ts d

o e

xper

imen

ts t

o p

rove

th

eori

es;

mat

hem

atic

ian

s u

se d

edu

ctiv

e re

aso

nin

g t

o p

rove

th

eore

ms.

Bo

thp

roce

sses

invo

lve

usi

ng

evi

den

ce t

o s

how

th

at c

erta

in s

tate

men

ts a

re t

rue.

at m

ost

are

para

llel

one

poin

t

fou

r po

ints

two

plan

es

two

poin

ts

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Geo

met

ry

Lo

gic

Pro

blem

sT

he

foll

owin

g pr

oble

ms

can

be

solv

ed b

y el

imin

atin

g po

ssib

ilit

ies.

It m

ay b

e h

elpf

ul

to u

se c

har

ts s

uch

as

the

one

show

n i

n t

he

firs

tpr

oble

m.M

ark

an X

in

th

e ch

art

to e

lim

inat

e a

poss

ible

an

swer

.

Sol

ve e

ach

pro

ble

m.

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

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____

____

__P

ER

IOD

____

_

2-5

2-5

Nan

cyO

livia

Mar

ioK

enji

Pea

chX

XX

Ora

nge

XX

XB

anan

aX

XX

App

leX

XX

1.N

ancy

,Oliv

ia,M

ario

,and

Ken

ji ea

ch h

ave

one

piec

e of

frui

t in

the

ir s

choo

l lun

ch.

The

y ha

ve a

pea

ch,a

n or

ange

,a b

anan

a,an

d an

app

le.M

ario

doe

s no

t ha

ve a

peac

h or

a b

anan

a.O

livia

and

Mar

io ju

stca

me

from

cla

ss w

ith

the

stud

ent

who

has

an a

pple

.Ken

ji an

d N

ancy

are

sit

ting

next

to

the

stud

ent

who

has

a b

anan

a.N

ancy

doe

s no

t ha

ve a

pea

ch.W

hich

stud

ent

has

each

pie

ce o

f fru

it?

Nan

cy—

app

le,

Oliv

ia—

ban

ana,

Mar

io—

ora

ng

e,K

enji—

pea

ch

3.M

r.G

uth

rie,

Mrs

.Hak

oi,M

r.M

irza

,an

dM

rs.R

iva

have

jobs

of

doct

or,a

ccou

ntan

t,te

ach

er,a

nd

offi

ce m

anag

er.M

r.M

irza

live

s n

ear

the

doct

or a

nd

the

teac

her

.M

rs.R

iva

is n

ot t

he

doct

or o

r th

e of

fice

man

ager

.Mrs

.Hak

oi i

s n

ot t

he

acco

un

tan

t or

th

e of

fice

man

ager

.Mr.

Gu

thri

e w

ent

to l

un

ch w

ith

th

e do

ctor

.M

rs.R

iva’

s so

n i

s a

hig

h s

choo

l st

ude

nt

and

is o

nly

sev

en y

ears

you

nge

r th

anh

is a

lgeb

ra t

each

er.W

hic

h p

erso

n h

asea

ch o

ccu

pati

on?

Mr.

Gu

thri

e—te

ach

er,

Mrs

.Hak

oi—

do

cto

r,M

r.M

irza

—o

ffic

e m

anag

er,

Mrs

.Riv

a—ac

cou

nta

nt

2.V

icto

r,L

eon

,Kas

ha,

and

Sh

eri

each

pla

yon

e in

stru

men

t.T

hey

pla

y th

e vi

ola,

clar

inet

,tru

mpe

t,an

d fl

ute

.Sh

eri

does

not

pla

y th

e fl

ute

.Kas

ha

live

s n

ear

the

stu

den

t w

ho

play

s fl

ute

an

d th

e on

ew

ho

play

s tr

um

pet.

Leo

n d

oes

not

pla

ya

bras

s or

win

d in

stru

men

t.W

hic

hst

ude

nt

play

s ea

ch i

nst

rum

ent?

Vic

tor—

flu

te,

Leo

n—

vio

la,

Kas

ha—

clar

inet

,S

her

i—tr

um

pet

4.Y

vett

e,L

ana,

Bor

is,a

nd

Sco

tt e

ach

hav

ea

dog.

Th

e br

eeds

are

col

lie,

beag

le,

pood

le,a

nd

terr

ier.

Yve

tte

and

Bor

isw

alke

d to

th

e li

brar

y w

ith

th

e st

ude

nt

wh

o h

as a

col

lie.

Bor

is d

oes

not

hav

e a

pood

le o

r te

rrie

r.S

cott

doe

s n

ot h

ave

aco

llie

.Yve

tte

is i

n m

ath

cla

ss w

ith

th

est

ude

nt

wh

o h

as a

ter

rier

.Wh

ich

stu

den

t h

as e

ach

bre

ed o

f do

g?

Yve

tte—

po

od

leL

ana—

colli

eB

ori

s—b

eag

leS

cott

—te

rrie

r



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swer

s

Stu

dy G

uid

e a

nd I

nte

rven

tion

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ebra

ic P

roo

f

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-6

2-6

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met

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ebra

ic P

roo

fT

he

foll

owin

g pr

oper

ties

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alge

bra

can

be

use

d to

just

ify

the

step

sw

hen

sol

vin

g an

alg

ebra

ic e

quat

ion

.

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per

tyS

tate

men

t

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lexi

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or e

very

num

ber

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�a.

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met

ric

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bers

aan

d b,

if a

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then

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a.

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siti

veF

or a

ll nu

mbe

rs a

, b,

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en a

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dit

ion

an

d S

ub

trac

tio

nF

or a

ll nu

mbe

rs a

, b,

and

c,

if a

�b

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a�

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b�

can

d a

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ltip

licat

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d D

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num

bers

a,

b, a

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, if

a�

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�0

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�a c��

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bst

itu

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or a

ll nu

mbe

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atio

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ere

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som

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s

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veA

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m�

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, th

en C

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m�

A�

m�

B,

then

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m�

A.

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siti

veIf

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d C

D�

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, th

en A

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m�

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and

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o-co

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te t

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pro

per

ty t

hat

ju

stif

ies

each

sta

tem

ent.

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1 �

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�m

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Sym

met

ric

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per

ty

2.If

m�

1�90

an

d m

�2

�m

�1,

then

m�

2�90

.S

ub

stit

uti

on

3.If

AB

�R

San

d R

S�

WY

,th

en A

B�

WY

.Tr

ansi

tive

Pro

per

ty

4.If

AB

�C

D,t

hen

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��1 2� C

D.

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ltip

licat

ion

Pro

per

ty

5.If

m�

1 �

m�

2 �

110

and

m�

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m�

3,th

en m

�1

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0.S

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6.R

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AB

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San

d T

U�

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dd

itio

n P

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erty

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m�

1 �

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d m

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�m

�3,

then

m�

1 �

m�

3.Tr

ansi

tive

Pro

per

ty

9.A

for

mu

la f

or t

he

area

of

a tr

ian

gle

is

A�

�1 2� bh

.Pro

ve t

hat

bh

is e

qual

to

2 t

imes

th

e ar

ea o

f th

e tr

ian

gle.

DA

R

TS

CB

12 3

Stu

dy G

uid

e a

nd I

nte

rven

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(con

tinued

)

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ebra

ic P

roo

f

NA

ME

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____

____

____

____

____

____

____

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AT

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ER

IOD

____

_

2-6

2-6

Exam

ple

Exam

ple

Exer

cises

Exer

cises

Giv

en:

A�

�1 2� bh

Pro

ve:

2A�

bh

Sta

tem

ents

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son

s

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h1.

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en

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ult

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bst

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AT

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ER

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____

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2-6

2-6

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met

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Sta

te t

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pro

per

ty t

hat

ju

stif

ies

each

sta

tem

ent.

1.If

80

�m

�A

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en m

�A

�80

.S

ymm

etri

c P

rop

erty

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ansi

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per

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erty

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N.

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btr

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on

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per

ty

5.If

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nd

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en m

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stit

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ple

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90,m

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olve

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____

____

____

____

____

____

____

____

____

____

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____

____

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ER

IOD

____

_

2-6

2-6



An

swer

s

Readin

g t

o L

earn

Math

em

ati

csA

lgeb

raic

Pro

of

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-6

2-6

©G

lenc

oe/M

cGra

w-H

ill91

Gle

ncoe

Geo

met

ry

Lesson 2-6

Pre-

Act

ivit

yH

ow i

s m

ath

emat

ical

evi

den

ce s

imil

ar t

o ev

iden

ce i

n l

aw?

Rea

d th

e in

trod

uct

ion

to

Les

son

2-6

at

the

top

of p

age

94 i

n y

our

text

book

.

Wh

at a

re s

ome

of t

he

thin

gs t

hat

law

yers

mig

ht

use

in

pre

sen

tin

g th

eir

clos

ing

argu

men

ts t

o a

tria

l ju

ry i

n a

ddit

ion

to

evid

ence

gat

her

ed p

rior

to

the

tria

l an

d te

stim

ony

hea

rd d

uri

ng

the

tria

l?S

amp

le a

nsw

er:T

hey

mig

ht

tell

the

jury

ab

ou

t la

ws

rela

ted

to

th

e ca

se,c

ou

rtru

ling

s,an

d p

rece

den

ts s

et b

y ea

rlie

r tr

ials

.

Rea

din

g t

he

Less

on

1.N

ame

the

prop

erty

ill

ust

rate

d by

eac

h s

tate

men

t.a.

If a

�4.

75 a

nd

4.75

�b,

then

a�

b.Tr

ansi

tive

Pro

per

ty o

f E

qu

alit

yb

.If

x�

y,th

en x

�8

�y

�8.

Ad

dit

ion

Pro

per

ty o

f E

qu

alit

yc.

5(12

�19

) �

5 �

12 �

5 �

19D

istr

ibu

tive

Pro

per

tyS

ub

stit

uti

on

Pro

per

tyd

.If

x�

5,th

en x

may

be

repl

aced

wit

h 5

in

an

y eq

uat

ion

or

expr

essi

on.

of

Eq

ual

ity

e.If

x�

y,th

en 8

x�

8y.

Mu

ltip

licat

ion

Pro

per

ty o

f E

qu

alit

yf.

If x

�23

.45,

then

23.

45 �

x.S

ymm

etri

c P

rop

erty

of

Eq

ual

ity

g.If

5x

�7,

then

x�

�7 5� .D

ivis

ion

Pro

per

ty o

f E

qu

alit

yh

.If

x�

12,t

hen

x�

3 �

9.S

ub

trac

tio

n P

rop

erty

of

Eq

ual

ity

2.G

ive

the

reas

on f

or e

ach

sta

tem

ent

in t

he

foll

owin

g tw

o-co

lum

n p

roof

.G

iven

:5(n

�3)

�4(

2n�

7) �

14P

rove

:n�

9S

tate

men

tsR

easo

ns

1.5(

n�

3) �

4(2n

�7)

�14

1.G

iven

2.5n

�15

�8n

�28

�14

2.D

istr

ibu

tive

Pro

per

ty3.

5n�

15 �

8n�

423.

Su

bst

itu

tio

n P

rop

erty

4.5n

�15

�15

�8n

�42

�15

4.A

dd

itio

n P

rop

erty

5.5n

�8n

�27

5.S

ub

stit

uti

on

Pro

per

ty6.

5n�

8n�

8n�

27 �

8n6.

Su

btr

acti

on

Pro

per

ty7.

�3n

��

277.

Su

bst

itu

tio

n P

rop

erty

8.��

3 3n ��

�� 2 37 �

8.D

ivis

ion

Pro

per

ty

9.n

�9

9.S

ub

stit

uti

on

Pro

per

ty

Hel

pin

g Y

ou

Rem

emb

er3.

A g

ood

way

to

rem

embe

r m

athe

mat

ical

ter

ms

is t

o re

late

the

m t

o w

ords

you

alr

eady

kno

w.

Giv

e an

eve

ryda

y w

ord

that

is r

elat

ed in

mea

ning

to

the

mat

hem

atic

al t

erm

ref

lexi

vean

dex

plai

n ho

w t

his

wor

d ca

n he

lp y

ou t

o re

mem

ber

the

Ref

lexi

ve P

rope

rty

and

to d

isti

ngui

shit

fro

m t

he S

ymm

etri

c an

d T

rans

itiv

e P

rope

rtie

s.S

amp

le a

nsw

er:R

efle

ctio

n:I

f yo

ulo

ok

at y

ou

r re

flec

tio

n,y

ou

see

yo

urs

elf.

Th

e R

efle

xive

Pro

per

ty s

ays

that

ever

y n

um

ber

is e

qu

al t

o it

self

.Th

e R

efle

xive

Pro

per

ty in

volv

es o

nly

on

en

um

ber

,wh

ile t

he

Sym

met

ric

and

Tra

nsi

tive

Pro

per

ties

eac

h in

volv

e tw

oo

r th

ree

nu

mb

ers.

©G

lenc

oe/M

cGra

w-H

ill92

Gle

ncoe

Geo

met

ry

Sym

met

ric,

Ref

lexi

ve,a

nd

Tra

nsi

tive

Pro

per

ties

Equ

alit

y h

as t

hre

e im

port

ant

prop

erti

es.

Ref

lexi

veS

ymm

etri

cT

ran

siti

ve

Oth

er r

elat

ion

s h

ave

som

e of

th

e sa

me

prop

erti

es.C

onsi

der

the

rela

tion

“is

nex

t to

”fo

r ob

ject

s la

bele

d X

,Y,a

nd

Z.W

hic

h o

f th

epr

oper

ties

lis

ted

abov

e ar

e tr

ue

for

this

rel

atio

n?

Xis

nex

t to

X.

Fal

seIf

Xis

nex

t to

Y,t

hen

Yis

nex

t to

X.

Tru

eIf

Xis

nex

t to

Yan

d Y

is n

ext

to Z

,th

en X

is n

ext

to Z

.F

alse

On

ly t

he

sym

met

ric

prop

erty

is

tru

e fo

r th

e re

lati

on “

is n

ext

to.”

For

eac

h r

elat

ion

,sta

te w

hic

h p

rop

erti

es (

sym

met

ric,

refl

exiv

e,tr

an

siti

ve)

are

tru

e.

1.is

th

e sa

me

size

as

2.is

a f

amil

y de

scen

dan

t of

sym

met

ric,

refl

exiv

e,tr

ansi

tive

tran

siti

ve

3.is

in

th

e sa

me

room

as

4.is

th

e id

enti

cal

twin

of

sym

met

ric,

refl

exiv

e,sy

mm

etri

c,tr

ansi

tive

tran

siti

ve

5.is

war

mer

th

an6.

is o

n t

he

sam

e li

ne

as

tran

siti

vesy

mm

etri

c,re

flex

ive

7.is

a s

iste

r of

8.is

th

e sa

me

wei

ght

as

tran

siti

vesy

mm

etri

c,re

flex

ive

tran

siti

ve

9.F

ind

two

oth

er e

xam

ples

of

rela

tion

s,an

d te

ll w

hic

h p

rope

rtie

s ar

e tr

ue

for

each

rel

atio

n.

See

stu

den

ts’w

ork

.

a�

aIf

a�

b,th

en b

�a.

If a

�b

and

b�

c,th

en a

�c.

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-6

2-6



Stu

dy G

uid

e a

nd I

nte

rven

tion

Pro

vin

g S

egm

ent

Rel

atio

nsh

ips

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-7

2-7

©G

lenc

oe/M

cGra

w-H

ill93

Gle

ncoe

Geo

met

ry

Lesson 2-7

Seg

men

t A

dd

itio

nT

wo

basi

c po

stu

late

s fo

r w

orki

ng

wit

h s

egm

ents

an

d le

ngt

hs

are

the

Ru

ler

Pos

tula

te,w

hic

h e

stab

lish

es n

um

ber

lin

es,a

nd

the

Seg

men

t A

ddit

ion

Pos

tula

te,

wh

ich

des

crib

es w

hat

it

mea

ns

for

one

poin

t to

be

betw

een

tw

o ot

her

poi

nts

.

Ru

ler

Po

stu

late

The

poi

nts

on a

ny li

ne o

r lin

e se

gmen

t can

be

paire

d w

ith r

eal n

umbe

rs s

o th

at, g

iven

any

two

poin

ts A

and

Bon

a li

ne,

Aco

rres

pond

s to

zer

o an

d B

corr

espo

nds

to a

pos

itive

rea

l num

ber.

Seg

men

t A

dd

itio

n

Po

stu

late

Bis

bet

wee

n A

and

Cif

and

only

if A

B�

BC

�A

C.

Wri

te a

tw

o-co

lum

n p

roof

.G

iven

:Qis

th

e m

idpo

int

of P �

R�.

Ris

th

e m

idpo

int

of Q �

S�.

Pro

ve:P

R�

QS

Sta

tem

ents

Rea

son

s

1.Q

is t

he

mid

poin

t of

P�R�

.1.

Giv

en2.

PQ

�Q

R2.

Def

init

ion

of

mid

poin

t3.

Ris

th

e m

idpo

int

of Q �

S�.

3.G

iven

4.Q

R�

RS

4.D

efin

itio

n o

f m

idpo

int

5.P

Q�

QR

�Q

R�

RS

5.A

ddit

ion

Pro

pert

y6.

PQ

�Q

R�

PR

,QR

�R

S�

QS

6.S

egm

ent

Add

itio

n P

ostu

late

7.P

R�

QS

7.S

ubs

titu

tion

Com

ple

te e

ach

pro

of.

PQ

RS

Exam

ple

Exam

ple

Exer

cises

Exer

cises

1.G

iven

:BC

�D

EP

rove

:AB

�D

E�

AC

Sta

tem

ents

Rea

son

sa.

BC

�D

Ea.

Giv

enb

.AB

�B

C�

AC

b.S

eg.A

dd.P

ost.

c.A

B�

DE

�A

Cc.

Su

bst

itu

tio

n

AB

C

DE

2.G

iven

:Qis

bet

wee

nP

and

R,R

is b

etw

een

Q

and

S,P

R�

QS

.P

rove

:PQ

�R

S

Sta

tem

ents

Rea

son

s

a.Q

is b

etw

een

a.G

iven

Pan

d R

.b

.PQ

�Q

R�

PR

b. S

eg.A

dd.P

ost

.c.

Ris

bet

wee

nc.

Giv

enQ

and

S.

d. Q

R�

RS

�Q

Sd

.Seg

.Add

.Pos

t.e.

PR

�Q

Se.

Giv

enf.

PQ

�Q

R�

f.S

ub

stit

uti

on

QR

�R

Sg.

PQ

�Q

R�

QR

�g.

Su

btr

acti

on

QR

�R

S�

QR

Pro

p.

h.P

Q�

RS

h.S

ubs

titu

tion

PQ

RS

©G

lenc

oe/M

cGra

w-H

ill94

Gle

ncoe

Geo

met

ry

Seg

men

t C

on

gru

ence

Th

ree

prop

erti

es o

f al

gebr

a—th

e R

efle

xive

,Sym

met

ric,

and

Tra

nsi

tive

Pro

pert

ies

of E

qual

ity—

hav

e co

un

terp

arts

as

prop

erti

es o

f ge

omet

ry.T

hes

epr

oper

ties

can

be

prov

ed a

s a

theo

rem

.As

wit

h o

ther

th

eore

ms,

the

prop

erti

es c

an t

hen

be

use

d to

pro

ve r

elat

ion

ship

s am

ong

segm

ents

.

Seg

men

t C

on

gru

ence

Th

eore

mC

ongr

uenc

e of

seg

men

ts is

ref

lexi

ve,

sym

met

ric,

and

tran

sitiv

e.

Ref

lexi

ve P

rop

erty

A �B�

�A�

B�

Sym

met

ric

Pro

per

tyIf

A�B�

�C�

D�,

then

C�D�

�A�

B�.

Tran

siti

ve P

rop

erty

If A�

B��

C�D�

and

C�D�

�E�

F�, t

hen

A�B�

�E�

F�.

Wri

te a

tw

o-co

lum

n p

roof

.G

iven

:A �B�

�D�

E�;B�

C��

E�F�

Pro

ve:A�

C��

D�F�

Sta

tem

ents

Rea

son

s1.

A�B�

�D�

E�1.

Giv

en2.

AB

�D

E2.

Def

init

ion

of

con

gru

ence

of

segm

ents

3.B �

C��

E�F�

3.G

iven

4.B

C�

EF

4.D

efin

itio

n o

f co

ngr

uen

ce o

f se

gmen

ts5.

AB

�B

C�

DE

�E

F5.

Add

itio

n P

rope

rty

6.A

B�

BC

�A

C,D

E�

EF

�D

F6.

Seg

men

t A

ddit

ion

Pos

tula

te7.

AC

�D

F7.

Su

bsti

tuti

on8.

A �C�

�D�

F�8.

Def

init

ion

of

con

gru

ence

of

segm

ents

Ju

stif

y ea

ch s

tate

men

t w

ith

a p

rop

erty

of

con

gru

ence

.

1.If

D�E�

�G �

H�,t

hen

G�H�

�D �

E�.

Sym

met

ric

Pro

per

ty

2.If

A�B�

�R�

S�an

d R�

S��

W�Y�

,th

en A�

B��

W�Y�

.Tr

ansi

tive

Pro

p.

3.R�

S��

R �S�

Ref

lexi

ve P

rop

.

4.C

ompl

ete

the

proo

f.G

iven

:P �R�

�Q �

S�P

rove

:P�Q�

�R�

S�

Sta

tem

ents

Rea

son

s

a.P�

R��

Q�S�

a.G

iven

b.P

R�

QS

b.D

efin

itio

n o

f co

ng

ruen

ce o

f se

gm

ents

c.P

Q�

QR

�P

Rc.

Seg

men

t A

dd

itio

n P

ost

ula

ted

.QR

�R

S�

QS

d.S

egm

ent

Add

itio

n P

ostu

late

e.P

Q�

QR

�Q

R�

RS

e.S

ub

stit

uti

on

f.P

Q�

RS

f.S

ubt

ract

ion

Pro

pert

yg.

P�Q�

�R�

S�g.

Def

init

ion

of

con

gru

ence

of

segm

ents

PQ

RS

DE

FA

BC

Stu

dy G

uid

e a

nd I

nte

rven

tion

(con

tinued

)

Pro

vin

g S

egm

ent

Rel

atio

nsh

ips

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-7

2-7

Exam

ple

Exam

ple

Exer

cises

Exer

cises



An

swer

s

Skil

ls P

ract

ice

Pro

vin

g S

egm

ent

Rel

atio

nsh

ips

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-7

2-7

©G

lenc

oe/M

cGra

w-H

ill95

Gle

ncoe

Geo

met

ry

Lesson 2-7

Ju

stif

y ea

ch s

tate

men

t w

ith

a p

rop

erty

of

equ

alit

y,a

pro

per

ty o

f co

ngr

uen

ce,o

r a

pos

tula

te.

1.Q

A�

QA

Ref

lexi

ve P

rop

erty

of

Eq

ual

ity

2.If

A�B�

�B�

C�an

d B�

C��

C�E�

,th

en A�

B��

C�E�

.

Tran

siti

ve P

rop

erty

of

Co

ng

ruen

ce

3.If

Qis

bet

wee

n P

and

R,t

hen

PR

�P

Q�

QR

.

Seg

men

t A

dd

itio

n P

ost

ula

te

4.If

AB

�B

C�

EF

�F

Gan

d A

B�

BC

�A

C,t

hen

EF

�F

G�

AC

.

Su

bst

itu

tio

n P

rop

erty

Com

ple

te e

ach

pro

of.

5.G

iven

:S�U�

�L�

R�T�

U��

L�N�

Pro

ve:S�

T��

N�R�

Pro

of:

Sta

tem

ents

Rea

son

s

a.S�

U��

L�R�

,T�U�

�L�

N�a.

Giv

enb

.SU

�L

R,T

U�

LN

b.

Def

init

ion

of

�se

gmen

ts

c.S

U�

ST

�T

Uc.

Seg

men

t A

dd

itio

n P

ost

ula

teL

R�

LN

�N

R

d.S

T�

TU

�L

N�

NR

d.

Su

bst

itu

tio

n P

rop

erty

e.S

T�

LN

�L

N�

NR

e.S

ub

stit

uti

on

Pro

per

tyf.

ST

�L

N�

LN

�L

N�

NR

�L

Nf.

Su

btr

acti

on

Pro

per

tyg.

ST

�N

Rg.

Su

bsti

tuti

on P

rope

rty

h.S�

T��

N�R�

h.

Def

init

ion

of

�se

gm

ents

6.G

iven

:A�B�

�C�

D�P

rove

:C�D�

�A �

B�P

roof

:

Sta

tem

ents

Rea

son

s

a.A�

B��

C�D�

a.G

iven

b.A

B�

CD

b.

Def

init

ion

of

�se

gm

ents

c.C

D�

AB

c.S

ymm

etri

c P

rop

erty

of

�

d.C�

D��

A�B�

d.

Def

init

ion

of

�se

gmen

ts

US

T

RL

N

©G

lenc

oe/M

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w-H

ill96

Gle

ncoe

Geo

met

ry

Com

ple

te t

he

foll

owin

g p

roof

.

1.G

iven

:A�B�

�D�

E�B

is t

he

mid

poin

t of

A�C�

.E

is t

he

mid

poin

t of

D �F�

.P

rove

:B �C�

�E�

F�P

roof

:

Sta

tem

ents

Rea

son

s

a.A�

B��

D�E�

a.G

iven

Bis

th

e m

idp

oin

t o

f A�

C�.

Eis

th

e m

idp

oin

t o

f D�

F�.b

.AB

�D

Eb

.D

efin

itio

n o

f �

seg

men

tsc.

AB

�B

Cc.

Def

init

ion

of

Mid

poin

t

DE

�E

Fd

.AC

�A

B�

BC

d.

Seg

men

t A

dd

itio

n P

ost

ula

teD

F�

DE

�E

F

e.A

B�

BC

�D

E�

EF

e.S

ub

stit

uti

on

Pro

per

tyf.

AB

�B

C�

AB

�E

Ff.

Su

bst

itu

tio

n P

rop

erty

g.A

B�

BC

�A

B�

AB

�E

F�

AB

g.S

ubt

ract

ion

Pro

pert

y

h.B

C�

EF

h.

Su

bst

itu

tio

n P

rop

erty

i.B�

C��

E�F�

i.D

efin

itio

n o

f �

seg

men

ts

2.TR

AV

ELR

efer

to

the

figu

re.D

eAn

ne

know

s th

at t

he

dist

ance

fro

m G

rays

on t

o A

pex

is t

he

sam

e as

th

e di

stan

cefr

om R

eddi

ng

to P

ine

Blu

ff.P

rove

th

at t

he

dist

ance

fro

mG

rays

on t

o R

eddi

ng

is e

qual

to

the

dist

ance

fro

m A

pex

to P

ine

Blu

ff.

Giv

en:

G�A�

�R�

P�P

rove

:G�

R��

A�P�

Pro

of:

Sta

tem

ents

Rea

son

s

1.G�

A��

R�P�

1.G

iven

2.G

A�

RP

2.D

efin

itio

n o

f �

seg

men

ts3.

GA

�A

R�

AR

�R

P3.

Ad

dit

ion

Pro

per

ty4.

GR

�G

A�

AR

,AP

�A

R�

RP

4.S

egm

ent

Ad

dit

ion

Po

stu

late

5.G

R�

AP

5.S

ub

stit

uti

on

Pro

per

ty6.

G�R�

�A�

P�6.

Def

init

ion

of

�se

gm

ents

Gray

son

Apex

Redd

ing

Pine

Blu

ff

GA

RP

CA

B

FD

E

Pra

ctic

e (

Ave

rag

e)

Pro

vin

g S

egm

ent

Rel

atio

nsh

ips

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-7

2-7



Readin

g t

o L

earn

Math

em

ati

csP

rovi

ng

Seg

men

t R

elat

ion

ship

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-7

2-7

©G

lenc

oe/M

cGra

w-H

ill97

Gle

ncoe

Geo

met

ry

Lesson 2-7

Pre-

Act

ivit

yH

ow c

an s

egm

ent

rela

tion

ship

s b

e u

sed

for

tra

vel?

Rea

d th

e in

trod

uct

ion

to

Les

son

2-7

at

the

top

of p

age

101

in y

our

text

book

.

•W

hat

is

the

tota

l di

stan

ce t

hat

th

e pl

ane

wil

l fl

y to

get

fro

m S

an D

iego

to

Dal

las?

1430

mi

•B

efor

e le

avin

g h

ome,

a pa

ssen

ger

use

d a

road

atl

as t

o de

term

ine

that

th

edi

stan

ce b

etw

een

San

Die

go a

nd

Dal

las

is a

bou

t 13

50 m

iles

.Wh

y is

th

efl

yin

g di

stan

ce g

reat

er t

han

th

at?

Sam

ple

an

swer

:P

ho

enix

is n

ot

on

a s

trai

gh

t lin

e b

etw

een

San

Die

go

an

d D

alla

s,so

th

e st

op

add

ed t

o t

he

dis

tan

ce t

rave

led

.A n

on

sto

p f

ligh

t w

ou

ld h

ave

bee

n s

ho

rter

.

Rea

din

g t

he

Less

on

1.If

Eis

bet

wee

n Y

and

S,w

hic

h o

f th

e fo

llow

ing

stat

emen

ts a

re a

lway

str

ue?

B,E

A.Y

S�

ES

�Y

EB

.YS

�E

S�

YE

C.Y

E�

ES

D.Y

E�

ES

�Y

SE

.SE

�E

Y�

SY

F.E

is t

he

mid

poin

t of

Y �S�

.

2.G

ive

the

reas

on f

or e

ach

sta

tem

ent

in t

he

foll

owin

g tw

o-co

lum

n p

roof

.G

iven

:Cis

th

e m

idpo

int

of B �

D�.

Dis

th

e m

idpo

int

of C �

E�.

Pro

ve:B �

D��

C�E�

Sta

tem

ents

Rea

son

s

1.C

is t

he

mid

poin

t of

B�D�

.1.

Giv

en2.

BC

�C

D2.

Def

init

ion

of

mid

po

int

3.D

is t

he

mid

poin

t of

C�E�

.3.

Giv

en4.

CD

�D

E4.

Def

init

ion

of

mid

po

int

5.B

C�

DE

5.Tr

ansi

tive

Pro

per

ty o

f E

qu

alit

y6.

BC

�C

D�

CD

�D

E6.

Ad

dit

ion

Pro

per

ty o

f E

qu

alit

y7.

BC

�C

D�

BD

7.S

egm

ent

Ad

dit

ion

Po

stu

late

CD

�D

E�

CE

8.B

D�

CE

8.S

ub

stit

uti

on

Pro

per

ty9.

B�D�

�C�

E�9.

Def

.of

�se

gm

ents

Hel

pin

g Y

ou

Rem

emb

er3.

On

e w

ay t

o ke

ep t

he

nam

es o

f re

late

d po

stu

late

s st

raig

ht

in y

our

min

d is

to

asso

ciat

eso

met

hing

in t

he n

ame

of t

he p

ostu

late

wit

h th

e co

nten

t of

the

pos

tula

te.H

ow c

an y

ou u

seth

is i

dea

to d

isti

ngui

sh b

etw

een

the

Rul

er P

ostu

late

and

the

Seg

men

t A

ddit

ion

Pos

tula

te?

Sam

ple

an

swer

:Th

ere

are

two

wo

rds

in “

Ru

ler

Po

stu

late

”an

d t

hre

e w

ord

sin

“S

egm

ent

Ad

dit

ion

Po

stu

late

.”T

he

stat

emen

t o

f th

e R

ule

r P

ost

ula

tem

enti

on

s tw

o p

oin

ts,a

nd

th

e st

atem

ent

of

the

Seg

men

t A

dd

itio

nP

ost

ula

te m

enti

on

s th

ree

po

ints

.

A

BE

DC

©G

lenc

oe/M

cGra

w-H

ill98

Gle

ncoe

Geo

met

ry

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-7

2-7

Geo

met

ry C

ross

wo

rd P

uzz

le

AC

RO

SS

3.P

oin

ts o

n t

he

sam

e li

ne

are

��

��

.

4.A

poi

nt

on a

lin

e an

d al

lpo

ints

of

the

lin

e to

on

e si

de

of i

t.

9.A

n a

ngl

e w

hos

e m

easu

re i

sgr

eate

r th

an 9

0.

10.T

wo

endp

oin

ts a

nd

all

poin

tsbe

twee

n t

hem

.

16.A

fla

t fi

gure

wit

h n

o th

ickn

ess

that

ext

ends

in

defi

nit

ely

in a

lldi

rect

ion

s.

17.S

egm

ents

of

equ

al l

engt

h a

re��

��

��se

gmen

ts.

18.T

wo

non

coll

inea

r ra

ys w

ith

aco

mm

on e

ndp

oin

t.

19.I

f m

�A

�m

�B

�18

0,th

en�

Aan

d �

Bar

e ��

��

��an

gles

.

DO

WN

1.T

he

set

of a

ll p

oin

ts c

olli

nea

r to

tw

o po

ints

is a

��

��

�.

2.T

he

poin

t w

her

e th

e x-

an

d y-

axis

mee

t.

5.A

n a

ngl

e w

hos

e m

easu

re i

s le

ss t

han

90.

6.If

m�

A�

m�

D�

90,t

hen

�A

and

�D

are

��

��

angl

es.

7.L

ines

th

at m

eet

at a

90°

angl

e ar

e ��

��

��.

8.T

wo

angl

es w

ith

a c

omm

on s

ide

but

no

com

mon

inte

rior

poi

nts

are

��

��

�.

10.A

n “

angl

e”fo

rmed

by

oppo

site

ray

s is

a

��

��

angl

e.

11.T

he

mid

dle

poin

t of

a l

ine

segm

ent.

12.P

oin

ts t

hat

lie

in

th

e sa

me

plan

e ar

e ��

��

��.

13.T

he

fou

r pa

rts

of a

coo

rdin

ate

plan

e.

14.T

wo

non

adja

cen

t an

gles

for

med

by

two

inte

rsec

tin

g li

nes

are

��

��

�an

gles

.

15.I

n a

ngl

e A

BC

,poi

nt

Bis

th

e ��

��

��.

CO

LL

I N EL

NE

AR I G I N

EM I D P O

NG

R T E XEV

UEMELPMO

BT

U T ECAY

R

SE R P E N D I C U L

EG

NA LCI

A RPC N T A R Y

RA

TN

EM

EL

PP

USTNARDA

LP L A N A ROC

NECAJDA N T

UQ

NTREV

C

I N T

GE

S T R A I G H T

T

O1

2

34

5

67

9

14 18

15

17

1110

8

1312 16

19



An

swer

s

Stu

dy G

uid

e a

nd I

nte

rven

tion

Pro

vin

g A

ng

le R

elat

ion

ship

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-8

2-8

©G

lenc

oe/M

cGra

w-H

ill99

Gle

ncoe

Geo

met

ry

Lesson 2-8

Sup

ple

men

tary

an

d C

om

ple

men

tary

An

gle

sT

her

e ar

e tw

o ba

sic

post

ula

tes

for

wor

kin

g w

ith

an

gles

.Th

e P

rotr

acto

r P

ostu

late

ass

ign

s n

um

bers

to

angl

e m

easu

res,

and

the

An

gle

Add

itio

n P

ostu

late

rel

ates

par

ts o

f an

an

gle

to t

he

wh

ole

angl

e.

Pro

trac

tor

Giv

en A

B��

�an

d a

num

ber

rbe

twee

n 0

and

180,

the

re is

exa

ctly

one

ray

P

ost

ula

tew

ith e

ndpo

int

A,

exte

ndin

g on

eith

er s

ide

of A

B��

� , s

uch

that

the

mea

sure

of

the

ang

le f

orm

ed is

r.

An

gle

Ad

dit

ion

Ris

in t

he in

terio

r of

�P

QS

if an

d on

ly if

P

ost

ula

tem

�P

QR

�m

�R

QS

�m

�P

QS

.

Th

e tw

o po

stu

late

s ca

n b

e u

sed

to p

rove

th

e fo

llow

ing

two

theo

rem

s.

Su

pp

lem

ent

If tw

o an

gles

for

m a

line

ar p

air,

then

the

y ar

e su

pple

men

tary

ang

les.

Th

eore

mIf

�1

and

�2

form

a li

near

pai

r, th

en m

�1

�m

�2

�18

0.

Co

mp

lem

ent

If th

e no

ncom

mon

sid

es o

f tw

o ad

jace

nt a

ngle

s fo

rm a

rig

ht a

ngle

,

Th

eore

mth

en t

he a

ngle

s ar

e co

mpl

emen

tary

ang

les.

IfG

F��

� ⊥G

H��

� , t

hen

m�

3 �

m�

4 �

90.

HG

43

FJ

CB

12

AD

SR

P

Q

If �

1 an

d �

2 fo

rm a

lin

ear

pai

r an

d m

�2

�11

5,fi

nd

m�

1.

m�

1 �

m�

2�

180

Sup

pl. T

heor

em

m�

1 �

115

�18

0S

ubst

itutio

n

m�

1�

65S

ubtr

actio

n P

rop.

P

Q

NM

12

If �

1 an

d �

2 fo

rm a

righ

t an

gle

and

m�

2 �

20,f

ind

m�

1.

m�

1 �

m�

2�

90C

ompl

. The

orem

m�

1 �

20�

90S

ubst

itutio

n

m�

1�

70S

ubtr

actio

n P

rop.

T

WR S

12

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Fin

d t

he

mea

sure

of

each

nu

mb

ered

an

gle.

1.2.

3.

m�

7 �

5x�

5,m

�5

�5x

,m�

6 �

4x�

6,m

�11

�11

x,m

�8

�x

�5

m�

7 �

10x,

m�

12 �

10x

�10

m�

7 �

155,

m�

8 �

12x

�12

m�

11 �

110,

m�

8 �

25m

�5

�30

,m�

6 �

30,

m�

12 �

110,

m�

7 �

60,m

�8

�60

m�

13 �

70

FC

JH

A11 12

13

WU

XY

Z

V5

86

7R

TP

Q

S

87

©G

lenc

oe/M

cGra

w-H

ill10

0G

lenc

oe G

eom

etry

Co

ng

ruen

t an

d R

igh

t A

ng

les

Th

ree

prop

erti

es o

f an

gles

can

be

prov

ed a

s th

eore

ms.

Con

grue

nce

of a

ngle

s is

ref

lexi

ve,

sym

met

ric,

and

tran

sitiv

e.

Ang

les

supp

lem

enta

ry t

o th

e sa

me

angl

e A

ngle

s co

mpl

emen

tary

to

the

sam

e an

gle

or t

o or

to

cong

ruen

t an

gles

are

con

grue

nt.

cong

ruen

t an

gles

are

con

grue

nt.

If �

1 an

d �

2 ar

e su

pple

men

tary

to

�3,

If

�4

and

�5

are

com

plem

enta

ry t

o �

6,

then

�1

��

2.th

en �

4 �

�5.

Wri

te a

tw

o-co

lum

n p

roof

.G

iven

:�A

BC

and

�C

BD

are

com

plem

enta

ry.

�D

BE

and

�C

BD

form

a r

igh

t an

gle.

Pro

ve:�

AB

C�

�D

BE

Sta

tem

ents

Rea

son

s1.

�A

BC

and

�C

BD

are

com

plem

enta

ry.

1.G

iven

�D

BE

and

�C

BD

form

a r

igh

t an

gle.

2.�

DB

Ean

d �

CB

Dar

e co

mpl

emen

tary

.2.

Com

plem

ent

Th

eore

m3.

�A

BC

��

DB

E3.

An

gles

com

plem

enta

ry t

o th

e sa

me

�ar

e �

.

Com

ple

te e

ach

pro

of.

BE

D

CA

ML

NK

STR

PQ

45

6A

B

C

F HJ

G

ED

1

23

Stu

dy G

uid

e a

nd I

nte

rven

tion

(con

tinued

)

Pro

vin

g A

ng

le R

elat

ion

ship

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-8

2-8

Exam

ple

Exam

ple

Exer

cises

Exer

cises

1.G

iven

:A�B�

⊥B�

C�;

�1

and

�3

are

com

plem

enta

ry.

Pro

ve:�

2 �

�3

Sta

tem

ents

Rea

son

s

a.A�

B�⊥

B�C�

a.G

iven

b.�

AB

Cis

a

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Pre-

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2-8

at

the

top

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age

107

in y

our

text

book

.

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pai

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sci

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th

at t

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angl

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expl

ain

how

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ould

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des

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ang

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on

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each

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ce t

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d .

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If t

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re c

ompl

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to

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re

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a(n

)

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m c

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t ad

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very

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its

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it

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at a

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at t

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ER

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_

2-8

2-8



Chapter 2 Assessment Answer Key Form 1 Form 2APage 105 Page 106 Page 107


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

C

B

C

A

B

D

C

D

A

D

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

B

A

C

D

C

C

B

A

D

A

120

1.

2.

3.

4.

5.

6.

7.

8.

9.

B

D

B

C

A

A

C

B

A


Chapter 2 Assessment Answer KeyForm 2A (continued) Form 2BPage 108 Page 109 Page 110

An

swer

s

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

B

C

D

D

A

B

C

B

C

A

D

5

1.

2.

3.

4.

5.

6.

7.

8.

9.

D

A

A

C

D

C

D

C

A

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

D

A

C

B

C

C

A

B

B

B

B

5


Chapter 2 Assessment Answer KeyForm 2CPage 111 Page 112

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

9

Sample answer:X, Y, and Z arenot collinear.

false

true

If an animal is a dog,then it has four feet.

you live in Chicago

If two lines are ||,then they are ⊥ to

the same line.

m�X � m�Y � 180

If today is January 1,then school is closed.

subtracting 6 fromeach side, x � 7

Division Property

85

13.

14.

15.

16.

17.

18.

19.

20.

B:

Def. of � bisector

Substitution

MidpointTheorem

TransitiveProperty

Substitution

Anglescomplementary

to same � or � � are �.

Def. of �segments

Segment AdditionPostulate

If a figure is nota �, then it is

not a rhombus.


Chapter 2 Assessment Answer KeyForm 2DPage 113 Page 114

An

swer

s

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

80

Sample answer:M, N, and P arenot collinear.

true

false

If a bird is a chicken,then it has two wings.

you live in Illinois

If two lines are not ||to a third line, then

they are not || to each other.

Ohio State will playin the Rose Bowl.

If today is Thursday,then Amy will stay

home.

adding 2 to eachside, x � �3

MultiplicationProperty

170

13.

14.

15.

16.

17.

18.

19.

20.

B:

Def. of � bisector

Complements of � � are �.

Midpoint Theorem

SymmetricProperty

Segment AdditionProperty

TransitiveProperty

SupplementaryAngles Theorem

Def. of � �

40


Chapter 2 Assessment Answer KeyForm 3Page 115 Page 116

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

m�A � m�C

ABDC is aparallelogram or

a rhombus.

false

true

3

If an animal is anelephant, then it

is a mammal.

If two � are not�, then they arenot supplementsof the same �.

Matt has practiceon Saturday.

If x � 6 � 10,then x2 � 16.

MidpointTheorem

Basketball5

Football10

12

3

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

Distributive Property

Addition Property

� Addition Postulate

Complements of thesame angle are �.

Three points lie inthe same plane.

Seg. Addition Post.

Symmetric Prop.

Distributive Prop.

Vertical � Theorem

All right � are �.

x � 4, y � 9

Chapter 2 Assessment Answer KeyPage 117, Open-Ended Assessment

Scoring Rubric


Score General Description Specific Criteria

• Shows thorough understanding of the concepts of truthtables, logic, algebraic properties, postulates, theorems,conditional statements, and segment and anglerelationships.

• Uses appropriate strategies to solve problems.• Computations are correct.• Written explanations are exemplary.• Diagrams and truth tables are accurate and appropriate.• Goes beyond requirements of some or all problems.

• Shows an understanding of the concepts of truth tables,logic, algebraic properties, postulates, theorems, conditionalstatements, and segment and angle relationships.

• Uses appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are effective.• Diagrams and truth tables are mostly accurate and

appropriate.• Satisfies all requirements of problems.

• Shows an understanding of most of the concepts of truthtables, logic, algebraic properties, postulates, theorems,conditional statements, and segment and anglerelationships.

• May not use appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are satisfactory.• Diagrams and truth tables are mostly accurate.• Satisfies the requirements of most of the problems.

• Final computation is correct.• No written explanations or work shown to substantiate the

final computation.• Diagrams and truth tables may be accurate but lack detail

or explanation.• Satisfies minimal requirements of some of the problems.

• Shows little or no understanding of most of the concepts oftruth tables, logic, algebraic properties, postulates, theorems,conditional statements, and segment and angle relationships.

• Does not use appropriate strategies to solve problems.• Computations are incorrect.• Written explanations are unsatisfactory.• Diagrams and truth tables are inaccurate or inappropriate.• Does not satisfy requirements of problems.• No answer given.

0 UnsatisfactoryAn incorrect solutionindicating no mathematicalunderstanding of theconcept or task, or nosolution is given

1 Nearly Unsatisfactory A correct solution with nosupporting evidence orexplanation

2 Nearly SatisfactoryA partially correctinterpretation and/orsolution to the problem

3 SatisfactoryA generally correct solution,but may contain minor flawsin reasoning or computation

4 SuperiorA correct solution that is supported by well-developed, accurateexplanations

An

swer

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Chapter 2 Assessment Answer KeyPage 117, Open-Ended Assessment

Sample Answers

1. Truth table with the following columns:

Since column 5 is the same as column 8, the conditional is equivalent to itscontrapositive and since column 6 is the same as column 7, the converse and the inverse are equivalent.

p q �p �q p → q q → p �p → �q �q → �p

T T F F T T T T

T F F T F T T F

F T T F T F F T

F F T T T T T T

In addition to the scoring rubric found on page A31, the following sample answers may be used as guidance in evaluating open-ended assessment items.

2. (1) If you are at the Sears Tower, thenyou are in Chicago.

(2) If you are in Chicago, then you are inIllinois.

(3) Therefore, if you are at the SearsTower, then you are in Illinois.

3. (1) If two numbers are even, then theirsum is even.

(2) The numbers are 4 and 6.(3) The sum of 4 and 6 is even.

4. An example of the Transitive Propertycould be If AB � BC and BC � EF, thenAB � EF. A similar example thatillustrates the Substitution Propertywould be If AB � BC and AB � EF � GH,then BC � EF � GH.

5.

For m�1 � 7x � 9 and m�2 � 5x � 5,find x and m�1.7x � 9 � 5x � 5

2x � 14x � 7

Therefore, m�1 � 49 � 9 � 40 and m�2 � 35 � 5 � 40.

6. No, two lines do not have to intersect. Ifthey do not intersect they could beparallel or skew. If they do not have to bedistinct lines they could also be the sameline. Students should draw a diagramshowing two parallel lines.

7. Any three distinct points lie in one plane.Therefore, if the stool has 3 legs thebottoms of all 3 legs would always lie inthe same plane, i.e. the plane of the floor,so it would not rock. If the stool had 4 legsand they were not all exactly the samelength, then only 3 of them could be on thefloor at one time and the stool would rock.

8a. Students should write a true statementwith a false converse such as If you arein Houston, then you are in Texas.

b. Converse: If you are in Texas, then youare in Houston.Inverse: If you are not in Houston, thenyou are not in Texas.Contrapositive: If you are not in Texas,then you are not in Houston.

c. Converse: falseInverse: falseContrapositive: true

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Chapter 2 Assessment Answer KeyVocabulary Test/Review Quiz 1 Quiz 3Page 118 Page 119 Page 120

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Quiz 2Page 119 Quiz 4

Page 120

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13.

false, theoremfalse, postulate

true

true

false, conclusion

false, hypothesis

true

true

true

false, deductivereasoning

a statement writtenin if-then form

a convenient method fororganizing the truthvalues of statements

a compound statementformed by joining two or

more statements withthe word and

1.

2.

3.

4.

5.

AB � BC � AC

Sample answer:m�A � 40 when

m�B � 50.true

true

B

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3.

4.

5.

x � 2 or x � �2

two � are right �

supplementaryand �

not valid

All isoscelestrapezoids arequadrilaterals.

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2.

3.

4.

5.

10

midpoint

SymmetricProperty

SubtractionProperty

TransitiveProperty

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2.

3.

4.

5.

Symmetric Prop.

SegmentAddition Prop.

� supplementary tothe same � are �.

Vertical � are �.

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Chapter 2 Assessment Answer KeyMid-Chapter Test Cumulative ReviewPage 121 Page 122

Part I

Part II

6.

7.

8.

9.

10.

a � �6

If two � are right �,then they are �.

Charlie can swim.

If the Giants score atouchdown, then

they will play in theSuper Bowl.

true

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5.

A

B

A

C

D

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8.06 units

13.45 units

point J

JE�� and JF��

�EJG or �DJH

�3 and �4

25.5 in.

AB�� and CD�� do notintersect.

true

If a ray extends fromthe vertex of an �, it

bisects the �.

�A and �B have the same measure.

3 � 14x � 53

Addition Property

�5164� � �

1144x

�

A BC D

AB R

T


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Chapter 2 Assessment Answer KeyStandardized Test Practice

Page 123 Page 124

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10. E F G H

A B C D

E F G H

A B C D

E F G H

A B C D

E F G H

A B C D

E F G H

A B C D11. 12.

13. 14.

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

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99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

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99 9 987654321

87654321

87654321

87654321

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155

50

150

65.3

1 4 1 3

3 6 4

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