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Page 1: Proof, Parallel and 1 Perpendicular Lines 1... · Proof, Parallel and Perpendicular Lines ... parallel perpendicular postulate ... 2 SpringBoard® Mathematics with Meaning™ Geometry

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Unit

1

1

??

Proof, Parallel and Perpendicular Lines

Essential Questions

Why are properties, postulates, and theorems important in mathematics?

How are angles and parallel and perpendicular lines used in real-world settings?

Unit OverviewIn this unit you will begin the study of an axiomatic system, Geometry. You will investigate the concept of proof and discover the importance of proof in mathematics. You will extend your knowledge of the characteristics of angles and parallel and perpendicular lines and explore practical applications involving angles and lines.

Academic VocabularyAdd these words and others you encounter in this unit to your Math Notebook.

angle bisector complementary angles conditional statement congruent conjecture counterexample deductive reasoning inductive reasoning

midpoint of a segment parallel perpendicular postulate proof supplementary angles theorem

These assessments, following Activities 1.4, 1.7, and 1.9, will give you an opportunity to demonstrate what you have learned about reasoning, proof, and some basic geometric fi gures.

Embedded Assessment 1

Conditional Statements and Logicp. 37

Embedded Assessment 2

Angles and Parallel Lines p. 65

Embedded Assessment 3

Slope, Distance, and Midpoint p. 81

EMBEDDED ASSESSMENTS

??

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2 SpringBoard® Mathematics with Meaning™ Geometry

1. Solve each equation.a. 5x - 2 = 8b. 4x - 3 = 2x + 9c. 6x + 3 = 2x + 8

2. Graph y = 3x - 3 and label the x- and y-intercepts.

3. Tell the slope of a line that contains the points (5, -3) and (7, 3).

4. Write the equation of a line that has slope 1 __ 3 and y-intercept 4.

5. Write the equation of the line graphed below.

x

y

8

10

6

4

2

–8–10 –6 –4 –2 2 4 6 8 10–2

–4

–6

–8

–10

6. Give the characteristics of a rectangular prism.

7. Draw a right triangle and label the hypotenuse and legs.

8. Use a protractor to fi nd the measure of each angle.a.

b.

c.

UNIT 1

Getting Ready

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Unit 1 • Proof, Parallel and Perpendicular Lines 3

ACTIVITY

1.1Geometric FiguresWhat’s My Name?SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Interactive Word Wall, Activating Prior Knowledge, Group Presentation

Below are some types of fi gures you have seen in earlier mathematics courses. Describe each fi gure. Using geometric terms and symbols, list as many names as possible for each fi gure.

1. Q 2.

3. X Y Z 4. D E

5. � m 6.

N

χ

P

KT

J

7.

D B

A

C

TRY THESE A

Identify each geometric fi gure. Th en use symbols to write two diff erent names for each.

a. Q P b. N C

F G HF G H

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4 SpringBoard® Mathematics with Meaning™ Geometry

My Notes

ACTIVITY 1.1continued

Geometric FiguresWhat’s My Name?What’s My Name?

8. Draw four angles with diff erent characteristics. Describe each angle. Name the angles using numbers and letters.

9. Compare and contrast each pair of angles.

a. b.

1 21 2

CD

BA

CD

BA

SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Interactive Word Wall, Activating Prior Knowledge, Discussion Group, Self/Peer Revision

When you compare and contrast two fi gures, you describe how they are alike or different.

MATH TERMS

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Unit 1 • Proof, Parallel and Perpendicular Lines 5

My Notes c.

d.

10. a. Th e fi gure below shows two intersecting lines. Name two angles that are supplementary to ∠4.

b. Explain why the angles you named in part a must have the same measure.

F

D

E30°

T

R

S

60°

F

D

E30°

T

R

S

60°

F

D

E30°

Z

X

Y

150°

F

D

E30°

Z

X

Y

150°

SUGGESTED LEARNING STRATEGIES: Discussion Group, Peer/Self Revision

Geometric FiguresWhat’s My Name?What’s My Name?

ACTIVITY 1.1continued

436 5

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6 SpringBoard® Mathematics with Meaning™ Geometry

My NotesMy Notes

Geometric FiguresWhat’s My Name?What’s My Name?

ACTIVITY 1.1continued

TRY THESE B

Complete the chart by naming all the listed angle types in each fi gure.

A B

F

E

C D

Acute angles

Obtuse angles

Angles with the same measure

Supplementary angles

Complementary angles

11. In the circle below, draw and label each geometric term.

a. Radius OA

b. Chord BA

c. Diameter CA

12. Refer to your drawings in circle above. What is the geometric term for point O?

SUGGESTED LEARNING STRATEGIES: Discussion Group, Create Representations, Activating Prior Knowledge

78

10 9

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Unit 1 • Proof, Parallel and Perpendicular Lines 7

My Notes 13. In the space below, draw a circle with center P and radius PQ = 1 in. Locate a point A so that PA = 1 1 __ 2 in. Locate a point B so that PB = 3 __ 4 in.

14. Use your diagram to complete these statements.

a. A lies _________________ the circle because ___________________________________.

b. B lies _________________ the circle because ___________________________________.

15. In your diagram above, draw circles with radii PA and PB.Th ese three circles are called _______________________.

SUGGESTED LEARNING STRATEGIES: Discussion Group, Create Representations, Activating Prior Knowledge

ACTIVITY 1.1continued

Geometric FiguresWhat’s My Name?What’s My Name?

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8 SpringBoard® Mathematics with Meaning™ Geometry

My Notes

Geometric FiguresWhat’s My Name?What’s My Name?

ACTIVITY 1.1continued

Write your answers on notebook paper. Show your work.

CHECK YOUR UNDERSTANDING

Write your answers on notebook paper. Show your work.

1. Describe all acceptable ways to name a line.

2. Describe all acceptable ways to name a plane.

3. Compare and contrast collinear and coplanar points.

4. What are some acceptable ways to name an angle?

5. a. Why is there a problem with using ∠B to name the angle below?

b. How many angles are in the fi gure?

6. Compare and contrast the terms acute angle, obtuse angle, right angle, and straight angle.

7. Th e measure of ∠A is 42°.

a. What is the measure of an angle that is complementary to ∠A?

b. What is the measure of an angle that is supplementary to ∠A?

8. Draw a circle P.

a. Draw a segment that has one endpoint on the circle but is not a chord.

b. Draw a segment that intersects the circle in two points, contains the center, but is not a radius, diameter, or chord.

9. MATHEMATICAL R E F L E C T I O N

How can you use the fi gures in this activity to

describe real-world objects and situations? Give examples.

TRY THESE C

Classify each segment in circle O. Use all terms that apply.___

AF : ___________________________

BO : ___________________________

CO : ___________________________

DO : ___________________________

EO : ___________________________

CE : ________________________

B

AC

D

O

CB

A

FE D

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Unit 1 • Proof, Parallel and Perpendicular Lines 9

My Notes

ACTIVITY

1.2Logical ReasoningRiddle Me ThisSUGGESTED LEARNING STRATEGIES: Interactive Word Wall, Vocabulary Organizer, Think/Pair/Share, Look for a Pattern

Th e ability to recognize patterns is an important aspect of mathematics. But when is an observed pattern actually a real pattern that continues beyond just the observed cases? In this activity, you will explore patterns and check to see if the patterns hold true beyond the observed cases.

Inductive reasoning is the process of observing data, recognizing patterns, and making a generalization. Th is generalization is a conjecture.

1. Five students attended a party and ate a variety of foods. Something caused some of them to become ill. JT ate a hamburger, pasta salad, and coleslaw. She became ill. Guy ate coleslaw and pasta salad but not a hamburger. He became ill. Dean ate only a hamburger and felt fi ne. Judy didn’t eat anything and also felt fi ne. Cheryl ate a hamburger and pasta salad but no coleslaw, and she became ill. Use inductive reasoning to make a conjecture about which food probably caused the illness.

2. Use inductive reasoning to make a conjecture about the next two terms in each sequence. Explain the pattern you used to determine the terms.

a. A, 4, C, 8, E, 12, G, 16, ,

b. 3, 9, 27, 81, 243, ,

c. 3, 8, 15, 24, 35, 48, ,

d. 1, 1, 2, 3, 5, 8, 13, ,

ACADEMIC VOCABULARY

conjectureinductive reasoning

CONNECT TO HISTORYHISTORY

The sequence of numbers in Item 2d is named after Leonardo of Pisa, who was known as Fibonacci. Fibonacci’s 1202 book Liber Abaci introduced the sequence to Western European mathematics.

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My Notes

Logical Reasoning ACTIVITY 1.2continued Riddle Me ThisRiddle Me This

3. Use the four circles on the next page. From each of the given points, draw all possible chords. Th ese chords will form a number of non-overlapping regions in the interior of each circle. For each circle, count the number of these regions. Th en enter this number in the appropriate place in the table below.

Number of Points on the Circle

Number of Non-Overlapping Regions Formed

2345

4. Look for a pattern in the table above.

a. Describe, in words, any patterns you see for the numbers in the column labeled Number of Non-Overlapping Regions Formed.

b. Use the pattern that you described to predict the number of non-overlapping regions that will be formed if you draw all possible line segments that connect six points on a circle.

SUGGESTED LEARNING STRATEGIES: Look for a Pattern, Group Presentation

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Unit 1 • Proof, Parallel and Perpendicular Lines 11

My Notes

ACTIVITY 1.2continued

Logical ReasoningRiddle Me ThisRiddle Me This

SUGGESTED LEARNING STRATEGIES: Visualization, Look for a Pattern

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My NotesMy Notes

Logical Reasoning ACTIVITY 1.2continued Riddle Me ThisRiddle Me This

5. Use the circle on the next page. Draw all possible chords connecting any two of the six points.

a. What is the number of non-overlapping regions formed by chords connecting the points on this circle?

b. Is the number you obtained above the same number you predicted from the pattern in the table in Item 4b?

c. Describe what you would do to further investigate the pattern in the number of regions formed by chords joining n points on a circle, where n represents any number of points placed on a circle.

d. Try to fi nd a new pattern for predicting the number of regions formed by chords joining points on a circle when two, three, four, fi ve, and six points are placed on a circle. Describe the pattern algebraically or in words.

SUGGESTED LEARNING STRATEGIES: Discussion Group

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Unit 1 • Proof, Parallel and Perpendicular Lines 13

My Notes

ACTIVITY 1.2continued

Logical ReasoningRiddle Me ThisRiddle Me This

SUGGESTED LEARNING STRATEGIES: Visualization, Look for a Pattern

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My Notes

Logical Reasoning ACTIVITY 1.2continued Riddle Me ThisRiddle Me This

ACTIVITY 1.2continued

6. Write each sum.

1 + 3 =

1 + 3 + 5 =

1 + 3 + 5 + 7 =

1 + 3 + 5 + 7 + 9 =

1 + 3 + 5 + 7 + 9 + 11 =

7. Look for a pattern in the sums in Item 6.

a. Describe algebraically or in words the pattern you see for the sums in Item 6.

b. Compare your description above with the descriptions that others in your group or class have written. Below, record any descriptions that were diff erent from yours.

c. Check the patterns in Items 7a and 7b to see if they predict the next few sums beyond the list of sums given in Item 6. Record or explain your fi ndings below.

SUGGESTED LEARNING STRATEGIES: Think/Pair/Share

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Unit 1 • Proof, Parallel and Perpendicular Lines 15

My Notes

ACTIVITY 1.2continued

Logical ReasoningRiddle Me ThisRiddle Me This

8. Consider this array of squares.

A B C D E F

B B C D E F

C C C D E F

D D D D E F

E E E E E F

F F F F F F

a. Round 1: Shade all the squares labeled A. How many A squares did you shade? How many total squares are shaded?

b. Round 2: Shade all the squares labeled B. How many B squares did you shade? How many total squares are shaded?

c. Complete the table for letters C, D, E, and F.

Letter Squares Shaded in this Round

Total Number of Shaded Squares

Round 1: A 1 1Round 2: B 3 4Round 3: CRound 4: DRound 5: ERound 6: F

SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Interactive Word Wall, Marking the Text, Predict and Confirm, Look for a Pattern, Group Presentation, Discussion Group

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My Notes

Logical Reasoning ACTIVITY 1.2continued Riddle Me ThisRiddle Me This

SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Interactive Word Wall, Marking the Text, Predict and Confirm, Look for a Pattern, Group Presentation, Discussion Group

9. Th ink about the process you used to complete the table for Item 8c. Use your fi ndings in Item 8 to answer the following questions.

a. One method of describing the pattern in Item 6 would be to say, “Th e new sum is always the next larger perfect square.” Analyze the process you used in Item 8 and then write a convincing argument that this pattern description will continue to hold true as the array is expanded and more squares are shaded.

b. Another method of describing the pattern in Item 6 would be to say, “Th e new sum is always the result of adding the next odd integer to the preceding sum.” Analyze the process you used in Item 8 and write a convincing argument that this pattern description will continue to hold true as the array is expanded and more squares are shaded.

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Unit 1 • Proof, Parallel and Perpendicular Lines 17

My Notes

ACTIVITY 1.2continued

Logical ReasoningRiddle Me ThisRiddle Me This

SUGGESTED LEARNING STRATEGIES: Interactive Word Wall, Vocabulary Organizer, Marking the Text

In this activity, you have described patterns and made conjectures about how these patterns would extend beyond your observed cases, based on collected data. Some of your conjectures were probably shown to be untrue when additional data were collected. Other conjectures took on a greater sense of certainty as more confi rming data were collected.

In mathematics, there are certain methods and rules of argument that mathematicians use to convince someone that a conjecture is true, even for cases that extend beyond the observed data set. Th ese rules are called rules of logical reasoning or rules of deductive reasoning. An argument that follows such rules is called a proof. A statement or conjecture that has been proven, that is, established as true without a doubt, is called a theorem. A proof transforms a conjecture into a theorem.

Below are some defi nitions from arithmetic.Even integer: An integer that has a remainder of 0 when it is divided

by 2.Odd integer: An integer that has a remainder of 1 when it is divided

by 2.In the following items, you will make some conjectures about the

sums of even and odd integers.

10. Calculate the sum of some pairs of even integers. Show the examples you use and make a conjecture about the sum of two even integers.

11. Calculate the sum of some pairs of odd integers. Show the examples you use and make a conjecture about the sum of two odd integers.

12. Calculate the sum of pairs of integers consisting of one even inte-ger and one odd integer. Show the examples you use and make a conjecture about the sum of an even integer and an odd integer.

ACADEMIC VOCABULARY

deductive reasoningprooftheorem

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My Notes

Logical Reasoning ACTIVITY 1.2continued Riddle Me ThisRiddle Me This

SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Look for a Pattern, Use Manipulatives

Th e following items will help you write a convincing argument (a proof) that supports each of the conjectures you made in Items 10–12.

13. Figures A, B, C, and D are puzzle pieces. Each fi gure represents an integer determined by counting the square pieces in the fi gure. Use these fi gures to answer the following questions.

a. Which of the fi gures can be used to model an even integer?

b. Which of the fi gures can be used to model an odd integer?

c. Compare and contrast the models of even and odd integers.

Figure A Figure B

Figure C Figure D

Figure A Figure B

Figure C Figure D

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My Notes

ACTIVITY 1.2continued

Logical ReasoningRiddle Me ThisRiddle Me This

SUGGESTED LEARNING STRATEGIES: Use Manipulatives, Discussion Group

14. Which pairs of puzzle pieces can fi t together to form rectangles? Make sketches to show how they fi t.

15. Explain how the fi gures (when used as puzzle pieces) can be used to show that each of the conjectures in Items 10 through 12 is true.

In Items 14 and 15, you proved the conjectures in Items 10 through 12 geometrically. In Items 16 through 18, you will prove the same conjectures algebraically.

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My Notes

Logical Reasoning ACTIVITY 1.2continued Riddle Me ThisRiddle Me This

SUGGESTED LEARNING STRATEGIES: Discussion Group

Th is is an algebraic defi nition of even integer: An integer is even if and only if it can be written in the form 2p, where p is an integer. (You can use other variables, such as 2m, to represent an even integer, where m is an integer.)

16. Use the expressions 2p and 2m, where p and m are integers, to confi rm the conjecture that the sum of two even integers is an even integer.

Th is is an algebraic defi nition of odd integer: An integer is odd if and only if it can be written in the form 2t + 1, where t is an integer. (Again, you do not have to use t as the variable.)

17. Use expressions for odd integers to confi rm the conjecture that the sum of two odd integers is an even integer.

18. Use expressions for even and odd integers to confi rm the conjecture that the sum of an even integer and an odd integer is an odd integer.

CONNECT TO PROPERTIESPROPERTIES

• A set has closure under an operation if the result of the operation on members of the set is also in the same set.

• In symbols, distributive property of multiplication over addition can be written as a(b + c) = a(b) + a(c).

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Unit 1 • Proof, Parallel and Perpendicular Lines 21

My Notes

ACTIVITY 1.2continued

Logical ReasoningRiddle Me ThisRiddle Me This

SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Summarize/Paraphrase/Retell, Create Representations, Group Presentation

In Items 14 and 15, you developed a geometric puzzle-piece argument to confi rm conjectures about the sums of even and odd integers. In Items 16–18, you developed an algebraic argument to confi rm these same conjectures. Even though one method is considered geometric and the other algebraic, they are oft en seen as the same basic argument.

19. Explain the link between the geometric and the algebraic methods of the proof.

20. State three theorems that you have proved about the sums of even and odd integers.

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Logical Reasoning ACTIVITY 1.2continued Riddle Me ThisRiddle Me This

Write your answers on notebook paper. Show your work.

CHECK YOUR UNDERSTANDING

Write your answers on notebook paper. Show your work.

1. Use inductive reasoning to determine the next two terms in the sequence.

a. 1 __ 2 , 1 __ 4 , 1 __ 8 , 1 ___ 16 , …

b. A, B, D, G, K, …

2. Write the fi rst fi ve terms of two diff erent sequences that have 10 as the second term.

3. Generate a sequence using the description: the fi rst term in the sequence is 4 and each term is three more than twice the previous term.

Use this picture pattern for Item 4 at the top of the next column.

4. a. Draw the next shape in the pattern,

b. Write a sequence of numbers that could be used to express the pattern,

c. Verbally describe the pattern of the sequence.

5. Use expressions for even and odd integers to confi rm the conjecture that the product of an even integer and an odd integer is an even integer.

6. MATHEMATICAL R E F L E C T I O N

Suppose you are given the fi rst fi ve terms of a

numerical sequence. What are some approaches you could use to determine the rule for the sequence and then write the next two terms of the sequence? You may use an example to illustrate your answer.

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Unit 1 • Proof, Parallel and Perpendicular Lines 23

My Notes

ACTIVITY

1.3The Axiomatic System of GeometryBack to the BeginningSUGGESTED LEARNING STRATEGIES: Close Reading, Quickwrite, Think/Pair/Share, Vocabulary Organizer, Interactive Word Wall

Geometry is an axiomatic system. Th at means that from a small, basic set of agreed-upon assumptions and premises, an entire structure of logic is devised. Many interactive computer games are designed with this kind of structure. A game may begin with basic set of scenarios. From these scenarios, a gamer can devise tools and strategies to win the game.

In geometry, it is necessary to agree on clear-cut meanings, or defi nitions, for words used in a technical manner. For a defi nition to be helpful, it must be expressed in words whose meanings are already known and understood.

Compare the following defi nitions.

Fountain: a roundel that is barry wavy of six argent and azure.

Guige: a belt that is worn over the right shoulder and used to support a shield.

1. Which of the two defi nitions above is easier to understand? Why?

For a new vocabulary term to be helpful, it should be defi ned using words that have already been defi ned. Th e fi rst defi nitions used in building a system, however, cannot be defi ned in terms of other vocabulary words because no other vocabulary words have been defi ned yet. In geometry, it is traditional to start with the simplest and most fundamental terms—without trying to defi ne them—and use these terms to defi ne other terms and develop the system of geometry. Th ese fundamental undefi ned terms are point, line, and plane.

2. Defi ne each term using the undefi ned terms.

a. Ray

b. Collinear points

c. Coplanar points

The term, line segment, can be defi ned in terms of undefi ned terms: A line segment is part of a line bounded by two points on the line called endpoints.

MATH TERMS

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My Notes

The Axiomatic System of Geometry ACTIVITY 1.3continued Back to the BeginningBack to the Beginning

Aft er a term has been defi ned, it can be used to defi ne other terms. For example, an angle is defi ned as a fi gure formed by two rays with a common endpoint.

3. Defi ne each term using the already defi ned terms.

a. Complementary angles

b. Supplementary angles

Th e process of deductive reasoning, or deduction, must have a starting point. A conclusion based on deduction cannot be made unless there is an established assertion to work from. To provide a starting point for the process of deduction, a number of assertions are accepted as true without proof. Th ese assertions are called axioms, or postulates.

When you solve algebraic equations, you are using deduction. Th e properties you use are like postulates in geometry.

4. Using one operation or property per step, show how to solve the equation 4x + 9 = 18 - 1 _ 2 x. Name each operation or property used to justify each step.

ACADEMIC VOCABULARY

complementary anglessupplementary anglespostulate

SUGGESTED LEARNING STRATEGIES: Vocabulary Organizer, Interactive Word Wall, Activating Prior Knowledge

CONNECT TO ALGEBRAALGEBRA

Addition Property of Equality

If a = b and c = d, then a + c = b + d.

Subtraction Property of Equality

If a = b and c = d, then a - c = b - d.

Multiplication Property of Equality

If a = b, then ca = cb.

Division Property of Equality

If a = b and c ≠ 0, then a __ c = b __ c .

Distributive Property

a(b + c) = ab + ac

Refl exive Property

a = a

Symmetric Property

If a = b, then b = a.

Transitive Property

If a = b and b = c, then a = c.

Substitution Property

If a = b, then a can be substituted for b in any equation or inequality.

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My Notes

ACTIVITY 1.3continued

The Axiomatic System of Geometry Back to the BeginningBack to the Beginning

You can organize the steps and the reasons used to justify the steps in two columns with statements (steps) on the left and reasons (properties) on the right. Th is format is called a two-column proof.

EXAMPLE 1

Given: 3(x + 2) - 1 = 5x + 11 Prove: x = -3

Statements Reasons1. 3(x + 2) - 1 = 5x + 11 1. Given equation2. 3(x + 2) = 5x + 12 2. Addition Property of Equality3. 3x + 6 = 5x + 12 3. Distributive Property4. 6 = 2x + 12 4. Subtraction Property of Equality5. -6 = 2x 5. Subtraction Property of Equality6. -3 = x 6. Division Property of Equality7. x = -3 7. Symmetric Property of Equality

TRY THESE A

a. Supply the reasons to justify each statement in the proof below.

Given: x - 3 _____ 2 = 6 + x _____ 5 Prove: x = 9

Statements Reasons

1. x - 3 _____ 2 = 6 + x _____ 5 1.

2. 10 ( x - 3 _____ 2 ) = 10 ( 6 + x _____ 5 ) 2.

3. 5(x - 3) = 2(6 + x) 3.

4. 5x - 15 = 12 + 2x 4.

5. 3x - 15 = 12 5.

6. 3x = 27 6.

7. x = 9 7.

SUGGESTED LEARNING STRATEGIES: Think/Pair/Share

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My NotesMy Notes

The Axiomatic System of Geometry ACTIVITY 1.3continued Back to the BeginningBack to the Beginning

TRY THESE A (continued)

b. Complete the Prove statement and write a two-column proof for the equation given in Item 4. Number each statement and corresponding reason.

Given: 4x + 9 = 18 - 1 __ 2 x Prove:

Rules of logical reasoning involve using a set of given statements along with a valid argument to reach a conclusion. Statements to be proved are oft en written in if-then form. An if-then statement is called a conditional statement. In such statements, the if clause is the hypothesis, and the then clause is the conclusion.

EXAMPLE 2

Conditional statement: If 3(x + 2) - 1 = 5x + 11, then x = -3.Hypothesis Conclusion

3(x + 2) - 1 = 5x + 11 x = -3

TRY THESE B

Use the conditional statement: If x + 7 = 10, then x = 3.

a. What is the hypothesis?

b. What is the conclusion?

c. State the property of equality that justifi es the conclusion of the statement.

ACADEMIC VOCABULARY

conditional statement

WRITING MATH

The letters p and q are often used to represent the hypo thesis and conclusion, respectively, in a conditional statement. The basic form of an if-then statement would then be, “If p, then q.”

SUGGESTED LEARNING STRATEGIES: Vocabulary Organizer, Think/Pair/Share, Interactive Word Wall, Marking the Text, Group Presentation

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My Notes

ACTIVITY 1.3continued

The Axiomatic System of Geometry Back to the BeginningBack to the Beginning

SUGGESTED LEARNING STRATEGIES: Vocabulary Organizer, Interactive Word Wall

Conditional statements may not always be written in if-then form. You can restate such conditional statements in if-then form.

5. Restate each conditional statement in if-then form.

a. I’ll go if you go.

b. Th ere is smoke only if there is fi re.

c. x = 4 implies x2 = 16.

An if-then statement is false if an example can be found for which the hypothesis is true and the conclusion is false. Th is type of example is a counterexample.

6. Th is is a false conditional statement.

If two numbers are odd, then their sum is odd.

a. Identify the hypothesis of the statement.

b. Identify the conclusion of the statement.

c. Give a counterexample for the conditional statement and justify your choice for this example.

READING MATH

Forms of conditional statements include:

• If p, then q.

• p only if q

• p implies q.

• q if p

ACADEMIC VOCABULARY

counterexample

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My Notes

The Axiomatic System of Geometry ACTIVITY 1.3continued Back to the BeginningBack to the Beginning

SUGGESTED LEARNING STRATEGIES: Vocabulary Organizer, Interactive Word Wall, Think/Pair/Share

Every conditional statement has three related conditionals. Th ese are the converse, the inverse, and the contrapositive of the conditional statement. Th e converse of a conditional is formed by interchanging the hypothesis and conclusion of the statement. Th e inverse is formed by negating both the hypothesis and the conclusion. Finally, the contrapositive is formed by interchanging and negating both the hypothesis and the conclusion.

Conditional: If p, then q.

Converse: If q, then p.

Inverse: If not p, then not q.

Contrapositive: If not q, then not p.

7. Given the conditional statement:

If a fi gure is a triangle, then it is a polygon.

Complete the table.

Form of the statement

Write the statement

True or False?

If the statement is false, give a

counterexample.

Conditional statement

If a fi gure is a triangle, then it is a polygon.

Converse of the conditional statementInverse of the conditional statementContrapositive of the conditional statement

converseinversecontrapositive

MATH TERMS

CONNECT TO APAP

When both a statement and its converse are true, you can connect the hypothesis and conclusion with the words “if and only if.”

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Unit 1 • Proof, Parallel and Perpendicular Lines 29

My Notes

ACTIVITY 1.3continued

The Axiomatic System of Geometry Back to the BeginningBack to the Beginning

SUGGESTED LEARNING STRATEGIES: Group Presentation

If a given conditional statement is true, the converse and inverse are not necessarily true. However, the contrapositive of a true conditional is always true, and the contrapositive of a false conditional is always false. Likewise, the converse and inverse of a conditional are either both true or both false. Statements with the same truth values are logically equivalent.

8. Write a true conditional statement whose inverse is false.

9. Write a true conditional statement that is logically equivalent to its converse.

When a statement and its converse are both true, they can be combined into one statement using the words “if and only if ”. All defi nitions you have learned can be written as “if and only if ” statements.

10. Write the defi nition of perpendicular lines in if and only if form.

TRY THESE C

Use this statement: Numbers that do not end in 2 are not even.

a. Rewrite the statement in if-then form and state whether it is true or false.

b. Write the converse and state whether it is true or false. If false, give a counterexample.

c. Write the inverse and state whether it is true or false.

d. Write the contrapositive and state whether it is true or false. If false, give a counterexample.

The truth value of a statement is the truth or falsity of that statement

MATH TERMS

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The Axiomatic System of GeometryBack to the BeginningBack to the Beginning

ACTIVITY 1.3continued

Write your answers on notebook paper. Show your work.

CHECK YOUR UNDERSTANDING

Write your answers on notebook paper. Show your work.

1. Identify the property that justifi es the statement: If 4x - 3 = 7, then 4x = 10.

2. Complete the prove statement and write a two column proof for the equation:

Given: x - 2 = 3(x - 4)

Prove:

3. Write the statement in if-then form.

Two angles have measures that add up to 90° only if they are complements of each other.

Use the following statement for Items 4–7.

If a vehicle has four wheels, then it is a car.

4. State the hypothesis.

5. Write the converse.

6. Write the inverse.

7. Write the contrapositive.

8. Given the false conditional statement:

If a vehicle has four wheels, then it is a car.

Write a counterexample.

9. Which of the following is a counterexample of this statement?

If an angle is acute, then it measures 80°.

a. a 100° angle b. a 70° angle

c. an 80° angle d. a 90° angle

10. Give an example of a false statement that has a true converse.

11. A certain conditional statement is true. Which of the following must also be true?

a. converse b. inverse

c. contrapositive d. all of the above

12. Given: (1) If X is blue, then Y is gold.(2) Y is not gold.

Which of the following must be true?

a. Y is blue. b. Y is not blue.

c. X is not blue. d. X is gold.

13. MATHEMATICAL R E F L E C T I O N

Some test questions ask you to tell if a statement is

sometimes true, always true, or never true. How can counterexamples help you to answer these types of questions?

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My Notes

ACTIVITY

1.4Truth Tables The Truth of the MatterSUGGESTED LEARNING STRATEGIES: Close Reading, Activating Prior Knowledge, Group Discussion, Interactive Word Wall, Vocabulary Organizer, Think/Pair/Share

Symbolic logic allows you to determine the validity of statements without being distracted by a lot of text. You can use symbols, such as p and q, to represent simple statements. A compound statement is formed when two or more simple statements are connected as a conditional (if-then) a biconditional (if and only if), a conjunction (and), or a disjunction (or).

1. Th e symbol → represents a conditional. You read p → q as, “if p, then q,” or “p implies q.” Let p represent the statement “you arrive before 7 PM,” and let q represent the statement “you will get a good seat.” Use the information in the play poster to the right to write the statement that is represented by p → q.

Truth tables are used to determine the conditions under which a statement is true or false. Th is truth table displays the truth values for p → q, which are dependent on the truth values for p and q.

2. Row 1 addresses the case when both p and q are true. Refer to the play announcement that you completed in Item 1. Explain why p → q would be true if both p and q are true.

3. Row 2 addresses the case when p is true and q is false. In terms of the play announcement, explain why p → q would be false if p is true and q is false.

4. Refer to Rows 3 and 4. In terms of the play announcement, explain why p → q is true if p is false.

p q p → qRow 1 T T TRow 2 T F FRow 3 F T TRow 4 F F T

p q p → qRow 1 T T TRow 2 T F FRow 3 F T TRow 4 F F T

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My Notes

Truth Tables ACTIVITY 1.4continued The Truth of the MatterThe Truth of the Matter

Th e symbol ~p is the negation of p and can be read as “not p.” Th is truth table shows the conditions under which ~p is true or false. When p is true, ~p is false, and when p is false, ~p is true.

5. Let p represent the simple statement “Triangles are convex,” which is a true statement. Write the statement denoted by ~p and state whether it is true or false.

6. Create a simple statement p that you know to be false. Write the statement ~p and state whether is true or false.

7. Let p represent “you are not in the band” and let q represent “you can go on the trip.” Write the compound statement represented by ~p → q.

EXAMPLE

Make a truth table for ~p → q.

Step 1 Step 2Write down all Add a columnpossible T and F for ~p andcombinations for negate p.p and q.

Step 3Add a column for ~p → q and evaluate ~p → q.

SUGGESTED LEARNING STRATEGIES: Close Reading, Create Representations, Think/Pair/Share

p ~pT FF T

p qT TT FF TF F

p q ~pT T FT F FF T TF F T

p q ~p ~p → qT T F TT F F TF T T TF F T F

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My Notes

ACTIVITY 1.4continued

Truth TablesThe Truth of the MatterThe Truth of the Matter

SUGGESTED LEARNING STRATEGIES: Close Reading, Think/Pair/Share, Create Representations, Identify a Subtask, Activating Prior Knowledge, Interactive Word Wall, Vocabulary Organizer

8. Follow the steps and complete the truth table for ~(p → q).

Step 1Find all possible T and F combinations for p and q.

Step 2Evaluate p → q.

Step 3Negate p → q.

9. Write the steps and complete the truth table for ~q → ~p.

10. Th e symbol ↔ represents a biconditional. You read it as “if and only if.” Let p represent “A chord in a circle contains the center,” and let q represent “Th e chord is a diameter.” Write the statement that is represented by p ↔ q.

Step 1 Step 2 Step 3p q p → q ~(p → q)

Step 1 Step 2 Step 3p q ~p ~q ~q → ~p

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My NotesMy Notes

Truth Tables ACTIVITY 1.4continued The Truth of the MatterThe Truth of the Matter

Th e truth table for p ↔ q isshown to the right. Notice thatp ↔ q is true only whenp and q are both true or both false.

11. Construct a truth table for (p → q) ↔ ~q.

12. Th e symbol ∧ represents a conjunction. You read it as “and.” Let p represent “the fi gure is a rectangle,” and let q represent “the fi gure is a rhombus.”

a. Write the statement that is represented by p ∧ q.

b. Write the statement that is represented by p ∧ ~q.

Th e truth table for p ∧ q is shown to the right. Notice that p ∧ q is only true when both p and q are true.

SUGGESTED LEARNING STRATEGIES: Close Reading, Think/Pair/Share, Create Representations, Identify a Subtask, Discussion Group, Group Presentation, Interactive Word Wall, Vocabulary Organizer

p q p ↔ qT T TT F FF T FF F T

p q p ∧ qT T TT F FF T FF F F

A rectangle is a parallelogram with a right angle.

A rhombus is a parallelogram with consecutive congruent sides.

MATH TERMS

Because biconditionals are true only when both parts have the same truth value, many defi ni-tions are written in the form of a biconditional.

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My Notes

ACTIVITY 1.4continued

Truth TablesThe Truth of the MatterThe Truth of the Matter

SUGGESTED LEARNING STRATEGIES: Close Reading, Create Representations, Identify a Subtask

13. Th e symbol ∨ represents a disjunction. You read it as “or.” Let p represent “the fi gure is not a rectangle,” and let q represent “the fi gure is a rhombus.”

a. Write the statement that is represented by p ∨ q.

b. Write the statement that is represented by ~p ∨ q.

Th e truth table for p ∨ q is shown to the right. Notice that p ∨ q is only false when both p and q are false.

14. Construct a truth table for p ∨ ( p ∧ q).

p q p ∨ qT T TT F TF T TF F F

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My Notes

Truth Tables ACTIVITY 1.4continued The Truth of the MatterThe Truth of the Matter

Write your answers on notebook paper. Show your work.

CHECK YOUR UNDERSTANDING

Write your answers on notebook paper. Show your work.

Construct a truth table for the compound statements. Identify which statements, if any, are tautologies.

1. ~p ∧ q

2. q → (p ∨ ~p)

3. p ∨ (q → p)

4. (p → q) ∧ p

5. q ∧ (p → ~q)

6. (p → q) ↔ (~q → ~p)

7. MATHEMATICAL R E F L E C T I O N

How could a truth table help you to analyze a

campaign promise made by a person running for offi ce? Th ink of a conditional statement starting, “If I am elected, …”

ACTIVITY 1.4continued

A tautology is a statement that is true for all cases of p and q. Th e corresponding truth table will have only T’s in the last column.

15. Construct a truth table for (p → q) ↔ (p ∧ q). Is this a tautology?

16. Construct a truth table for ~(p ∨ q) ↔ (~p ∧ ~q). Is this a tautology?

SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Identify a Subtask, Create Representations, Identify a Subtask

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Unit 1 • Proof, Parallel and Perpendicular Lines 37

Embedded Assessment 1 Conditional Statements and Logic

HEALTHY HABITS

Statements given in advertisements oft en involve conditional statements. Analyze the advertisement below.

“If you want to feel your very best, take one SBV tablet every day. People who take SBV vitamins care about their health. You owe it to those close to you to take care of yourself. Begin taking SBV today!”

Write your answers on notebook paper. Show your work.

1. Consider the fi rst statement in the advertisement: “If you want to feel your very best, take one SBV tablet every day.”

a. Identify the hypothesis and conclusion of the statement.

b. Write the converse of the statement. Discuss the validity of the converse and why it might be used to infl uence people to buy the vitamin.

2. Consider the second statement: “People who take SBV vitamins care about their health.”

a. Rewrite the statement in if-then form.

b. Write the inverse of the statement. Discuss the validity of the inverse and why it might be used to infl uence people to buy the vitamin.

According to current guidelines, people are advised to maintain cholesterol levels below 200 units. Four friends; Juana, Bea, Les, and Hugh, all have cholesterol levels above the recommended limit. Th ey have all heard cholesterol-lowering strategies discussed on talk shows and in commercials. Th ose strategies included: taking two red yeast rice tablets daily; eating oatmeal daily; practicing meditation for 1 __ 2 hour each day; and getting at least twenty minutes of aerobic exercise daily.

3. For one month, they each used strategies daily. Th ey had their cholesterol rechecked at the end of the month. Juana meditated and ate oatmeal. Her cholesterol level fell 5 points. Hugh exercised and meditated. His cholesterol lowered 10 points. Les took red yeast rice tablets and meditated. His cholesterol level stayed the same. Bea took red yeast rice tablets and ate oatmeal. Her cholesterol level went down 7 points. Which technique(s) appear to be most eff ective in lowering cholesterol levels? Explain your reasoning.

Use after Activity 1.4.

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38 SpringBoard® Mathematics with Meaning™ Geometry

Embedded Assessment 1 Use after Activity 1.4.

Conditional Statements and LogicHEALTHY HABITS

Exemplary Profi cient Emerging

Math Knowledge#a, b; 2a, b

The student:• Correctly

identifi es the hypothesis and conclusion. (1a)

• Writes the correct converse of the statement. (1b)

• Writes a correct if-then form and inverse of the statement. (2a, b)

The student:• Correctly

identifi es the hypothesis or conclusion but not both.

• Writes a correct if-then form but not the correct inverse.

The student:• Correctly

identifi es neither the hypothesis nor the conclusion.

• Writes an incorrect converse of the statement.

• Writes neither a correct if-then form nor a correct inverse.

Problem Solving#3

The student correctly identifi es the seemingly most effective technique(s). (3)

The student identifi es a technique that is somewhat effective.

The student identifi es a technique that is not effective.

Communication#1b, 2b, 3

The student:• Gives a complete

discussion of the validity of the converse. (1b)

• Gives a complete discussion of the validity of the inverse. (2b)

• Gives logical reasoning for his/her choice(s) of technique(s). (3).

The student:• Gives an

incomplete discussion of the validity of the converse.

• Gives an incomplete discussion of the validity of the inverse.

• Gives incomplete reasoning for the choice.

The student:• Gives an irrelevant

discussion of the validity of the converse.

• Gives an irrelevant discussion of the validity of the inverse.

• Gives illogical reasoning for the choice.

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Unit 1 • Proof, Parallel and Perpendicular Lines 39

My Notes

ACTIVITYSegment and Angle MeasurementIt All Adds UpSUGGESTED LEARNING STRATEGIES: Close Reading

If two points are no more than one foot apart, you can fi nd the distance between them by using an ordinary ruler. (Th e inch rulers below have been reduced to fi t on the page.)

1inches

2 3 4 5 6 7

BA

In the fi gure, the distance between point A and point B is 5 inches. Of course, there is no need to place the zero of the ruler on point A. In the fi gure below, the 2-inch mark is on point A. In this case, AB, measured in inches, is |7 - 2| = |2 - 7| = 5, as before.

2 3 4 5 6 7 8 9

BA

Th e number obtained as a measure of distance depends on the unit of measure. On many rulers one edge is marked in centimeters. Using the centimeter scale, the distance between the points A and B above is about 12.7 cm.

1. Determine the length of each segment in centimeters.

D FE

a. DE = b. EF = c. DF =

2. Determine the length of each segment in centimeters.

G KH

a. KH = b. HG = c. GK =

The Ruler Postulate

a. To every pair of points there corresponds a unique positive number called the distance between the points.

b. The points on a line can be matched with the real numbers so that the distance between any two points is the absolute value of the difference of their associated numbers.

MATH TERMS

1.5

READING MATH

AB denotes the distance between points A and B. If A and B are the endpoints of a segment (

_ AB ), then AB denotes

the length of _

AB .

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40 SpringBoard® Mathematics with Meaning™ Geometry

My Notes

Segment and Angle MeasurementACTIVITY 1.5continued It All Adds UpIt All Adds Up

SUGGESTED LEARNING STRATEGIES: Look for a Pattern, Think/Pair/Share, Vocabulary Organizer, Interactive Word Wall, Create Representations

3. Using your results from Items 1 and 2, describe any patterns that you notice.

4. Given that N is a point between endpoints M and P of line segment MP, describe how to determine the length of

__ MP , without measuring,

if you are given the lengths of __

MN and __

NP .

5. Use the Segment Addition Postulate and the given information to complete each statement.

a. If B is between C and D, BC = 10 in., and BD = 3 in., then CD = _______.

b. If Q is between R and T, RT = 24 cm, and QR = 6 cm, then QT = _______.

c. If P is between L and A, PL = x + 4, PA = 2x - 1, and LA = 5x - 3, then x = _______ and LA = _______.

For each part of Item 5, make a sketch so that you can identify the parts of the segment.

Item 4 and your answer together form a statement of the Segment Addition Postulate.

MATH TERMS

CONNECT TO APAP

You will frequently be asked to fi nd the lengths of horizontal, vertical, and diagonal segments in the coordinate plane in AP Calculus.

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Unit 1 • Proof, Parallel and Perpendicular Lines 41

My Notes

ACTIVITY 1.5continued

Segment and Angle MeasurementIt All Adds UpIt All Adds Up

Th e midpoint of a segment is the point on the segment that divides it into two congruent segments. For example, if B is the midpoint of

__ AC ,

then __

AB � __

BC .

A CB3 3

6. Given: M is the midpoint of __

RS . Complete each statement.

a. If RS = 10, then SM = _______.

b. If RM = 12, then MS = _______, and RS = _______.

You can use the defi nition of midpoint and properties of algebra to determine the length of a segment.

EXAMPLE 1

If Q is the midpoint of __

PR , PQ = 4x - 5, and QR = 11 + 2x, determine the length of

__ PQ .

1. __

PQ � __

QR 1. Defi nition of midpoint2. PQ = QR 2. Defi nition of congruent segments3. 4x - 5 = 11 + 2x 3. Substitution Property4. 2x - 5 = 11 4. Subtraction Property of Equality5. 2x = 16 5. Addition Property of Equality6. x = 8 6. Division Property of Equality7. PQ = 4(8) - 5 = 27 7. Substitution Property

SUGGESTED LEARNING STRATEGIES: Vocabulary Organizer, Interactive Word Wall

ACADEMIC VOCABULARY

The midpoint of a segment is a point on the segment that divides it into two congruent segments.

Congruent segments are segments that have equal length. To indicate congruence, you can write

__ AB �

__ BC .

Use __

AB when you talk about segment AB.

Use AB when you talk about the measure, or length, of

__ AB .

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42 SpringBoard® Mathematics with Meaning™ Geometry

My Notes

Segment and Angle MeasurementACTIVITY 1.5continued It All Adds UpIt All Adds Up

SUGGESTED LEARNING STRATEGIES: Create Representations, Think/Pair/Share

EXAMPLE 2

If Y is the midpoint of __

WZ , YZ = x + 3 and WZ = 3x - 4, determine the length of

__ WZ .

1. WZ = WY + YZ 1. Segment Addition Postulate2.

__ WY �

__ YZ 2. Defi nition of midpoint

3. WY = YZ 3. Defi nition of congruent segments4. WZ = YZ + YZ 4. Substitution Property5. 3x - 4 = (x + 3) + (x + 3) 5. Substitution Property6. x - 4 = 6 6. Subtraction Property of Equality7. x = 10 7. Addition Property of Equality8. WZ = 3(10) - 4 = 26 8. Substitution Property

TRY THESE A

Given: M is the midpoint of __

RS . Use the given information to fi nd the missing values.

a. RM = x + 3 and MS = 2x - 1 b. RM = x + 6 and RS = 5x + 3x = _____ and RM = _____ x = _____ and SM = _____

You measure angles with a protractor. Th e number of degrees in an angle is called its measure.

90

16020

180017010

1503014040

13050

12060

11070

10080

8010070

11060

12050130

40140

30150

20160

10 170

0 180

O AE

B

C

D

7. Determine the measure of each angle.

a. m∠AOB = 50° b. m∠BOC = ____ c. m∠AOC = ____

d. m∠EOD = ____ e. m∠BOD = ____ f. m∠BOE = ____

CONNECT TO ASTRONOMYASTRONOMY

The astronomer Claudius Ptolemy (about 85–165 CE) based his observations of the solar system on a unit that resulted from dividing the distance around a circle into 360 parts. This later became known as a degree.

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Unit 1 • Proof, Parallel and Perpendicular Lines 43

My Notes

ACTIVITY 1.5continued

Segment and Angle MeasurementIt All Adds UpIt All Adds Up

SUGGESTED LEARNING STRATEGIES: Look for a Pattern, Quickwrite, Vocabulary Organizer, Interactive Word Wall, Think/Pair/Share

8. Use a protractor to determine the measure of each angle.

R

TP

Q

a. m∠TQP = b. m∠TQR = c. m∠RQP =

9. Using your results from Items 7 and 8, describe any patterns that you notice.

10. Given that point D is in the interior of ∠ABC, describe how to determine the measure of ∠ABC, without measuring, if you are given the measures of ∠ABD and ∠DBC.

11. Use the angle addition postulate and the given information to complete each statement.

a. If P is in the interior of ∠XYZ, m∠XYP = 25°, and m∠PYZ = 50°, then m∠XYZ = .

b. If M is in the interior of ∠RTD, m∠RTM = 40°, and m∠RTD = 65°, then m∠MTD = .

c. If H is in the interior of ∠EFG, m∠EFH = 75°, and m∠HFG = (10x)°, and m∠EFG = (20x - 5)°, then x = and m∠HFG = .

The Protractor Postulate

a. To each angle there corresponds a unique real number between 0 and 180 called the measure of the angle.

b. The measure of an angle formed by a pair of rays is the absolute value of the difference of their associated numbers.

MATH TERMS

Item 10 and your answer together form a statement of the Angle Addition Postulate.

MATH TERMS

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My Notes

Segment and Angle MeasurementACTIVITY 1.5continued It All Adds UpIt All Adds Up

SUGGESTED LEARNING STRATEGIES: Vocabulary Organizer, Interactive Word Wall, Create Representations

11d. Lines DB and EC intersect at point F. If m∠BFC = 44° and m∠AFB = 61°, then

m∠AFC =

m∠AFE =

m∠EFD =

Th e bisector of an angle is a ray that divides the angle into two congruent adjacent angles. For example, in the fi gure to the left , if �� � BD bisects ∠ABC, then ∠ABD � ∠DBC.

12. Given: �� � AH bisects ∠MAT. Determine the missing measure.

a. m∠MAT = 70°, m∠MAH =

b. m∠HAT = 80°, m∠MAT =

You can use defi nitions, postulates, theorems, and properties of algebra to determine the measures of angles.

EXAMPLE 3

If � � QP bisects ∠DQL, m∠DQP = 5x - 7 and m∠PQL = 11 + 2x, determine the measure of ∠DQL.

1. ∠DQP � ∠PQL 1. Defi nition of Angle Bisector 2. m∠DQP = m∠PQL 2. Defi nition of congruent angles 3. 5x - 7 = 11 + 2x 3. Substitution Property 4. 3x - 7 = 11 4. Subtraction Property of Equality 5. 3x = 18 5. Addition Property of Equality 6. x = 6 6. Division Property of Equality 7. m∠DQP = 5(6) - 7 = 23 7. Substitution Property 8. m∠PQL = 11 + 2(6) = 23 8. Substitution Property 9. m∠DQL = m∠DQP +

m∠PQL 9. Angle Addition Postulate

10. m∠DQL = 23 + 23 = 46° 10. Substitution Property

BD

F

E A

C

BD

F

E A

C

Adjacent angles are coplanar angles that share a vertex and side, but no interior points. In the fi gure above, ∠ABD is adjacent to ∠DBC.

Congruent angles are angles that have the same measure. To indicate congruence, you can either write m∠ABD = m∠DBC. or ∠ABD � ∠DBC.

MATH TERMS

C

D

A

B30°

30°

ACADEMIC VOCABULARY

angle bisector

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Unit 1 • Proof, Parallel and Perpendicular Lines 45

My Notes

ACTIVITY 1.5continued

Segment and Angle MeasurementIt All Adds UpIt All Adds Up

SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Think/Pair/Share, Group Presentation

GUIDED EXAMPLE

Complete the missing statements and reasons.

∠A and ∠B are complementary, m∠A = 3x + 7, and m∠B = 6x + 11. Determine the measure of each angle.

Statements Reasons1. m∠A + m∠B = 90° 1. Defi nition of 2. + = 90° 2. Substitution Property3. 9x + 18 = 90 3. Substitution Property4. 9x = 72 4. Property of Equality5. x = 8 5. Property of Equality6. m∠A = 3(8) + 7 = 31 6. Substitution Property7. m∠B = 6( ) +11 = 7. Substitution Property

TRY THESE B

Given: � � FL bisects ∠AFM. Determine each missing value.

a. m∠LFM = 11x + 4 and m∠AFL = 12x - 2

x = , m∠LFM = , and m∠AFM =

b. m∠AFM = 6x - 2 and m∠AFL = 4x - 10

x = and m∠LFM =

In this diagram, lines AC and DB intersect as shown. Determine the missing values.

c. x =

m∠AEB =

m∠CEB = .

d. ∠P and ∠Q are complementary. m∠P = 2x + 25 and m∠Q = 4x + 11. Determine the measure of each angle.

BDE

A

C

(5x + 68)° (9x)°

(Diagram is not drawn to scale.)

BDE

A

C

(5x + 68)° (9x)°

(Diagram is not drawn to scale.)

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46 SpringBoard® Mathematics with Meaning™ Geometry

Segment and Angle MeasurementACTIVITY 1.5continued It All Adds UpIt All Adds Up

Write your answers on notebook paper. Show your work.

CHECK YOUR UNDERSTANDING

Write your answers on notebook paper. Show your work.

1. Given: Point K is between points H and J, HK = x - 5, KJ = 5x - 12 and HJ = 25. Find the value of x.

a. 7 b. 2 c. 6 d. 9

2. If K is the midpoint of ___

HJ , HK = x + 6, and HJ = 5x - 6, then KJ = ? .

a. 6 b. 3 c. 12 d. 9

3. Point D is in the interior of ∠ABC, m∠ABC = 10x - 7, m∠ABD = 6x + 5, and m∠DBC = 36°. What is m∠ABD?

a. 3° b. 23° c. 17° d. 77°

4. If ∠P and ∠Q are complementary, m∠P = 5x + 3, and m∠Q = x + 3, then x = ?

a. 30 b. 29 c. 0 d. 14

5. Ray QS bisects ∠PQR. If m∠PQS = 5x and m∠RQS = 2x + 6, then m∠PQR = ? .

a. 10° b. 20° c. 6° d. 2°

Use the given diagram and information to determine the missing measures. Show all work.

6. Given: m∠1 = 4x + 30 and m∠3 = 2x + 48

a. x =

b. m∠3 =

c. m∠2 =

7. Given: � � EA ⊥ � � ED , � � EB bisects ∠AEC, m∠AEB = 4x + 1, and m∠CED = 3x.

a. x =

b. m∠BEC =

8. Given: m∠1 = 2x + 8, m∠2 = x + 4, and m∠3 = 3x + 18

1

2

3

4

a. x =

b. m∠4 =

c. Is ∠4 complementary to ∠2? Explain.

9. MATHEMATICAL R E F L E C T I O N

How could you model two angles with a common

vertex and a common side that are not adjacent angles?

12

3

12

3

E D

A B

C

E D

A B

C

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Unit 1 • Proof, Parallel and Perpendicular Lines 47

My Notes

ACTIVITY

1.6Parallel and Perpendicular LinesPatios by Madeline

Matt works for Patios by Madeline. A new customer has asked him to design a brick patio and walkway. Th e customer requires a blueprint showing both the patio and the walkway. Th e customer would like the rows of bricks in the completed patio to be parallel to the walkway. To help create his blueprint, Matt will attach a string between two stakes on either side of the patio as shown in the diagram below, and continue the pattern with parallel strings. In his blueprint, Matt must include a line to indicate the path of the underground gas line, which is located below the patio. (Matt drew in the location of the gas line from information supplied by the local gas company in order to avoid accidents during construction of the patio.)

Stake

Stake

House

Gas

Lin

e

String Line

Walkway

SUGGESTED LEARNING STRATEGIES: Summarize/Paraphrase/Retell

ACADEMIC VOCABULARY

Parallel lines are coplanar lines that do not intersect.

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My Notes

Parallel and Perpendicular LinesACTIVITY 1.6continued Patios by MadelinePatios by Madeline

SUGGESTED LEARNING STRATEGIES: Visualization, Create Representations

CONNECT TO OTHER OTHER GEOMETRIESGEOMETRIES

Euclidean geometry is based on fi ve postulates proposed by Euclid (about 300 BCE). His fi fth postulate, “Through a point outside a line, there is exactly one line parallel to the given line” was a source of contention for many mathematicians. Their thought was that this fi fth postulate could be proven from the fi rst four postulates and should therefore be called a theorem. Since Euclid could not prove it, he kept it as a postulate. Some mathematicians thought that since the postulate could not be proven they should be able to replace it with a contrary postulate. This created other “non-Euclidean” geometries.

Continue designing the patio using the blueprint Matt has already begun. Matt’s beginning blueprint is on the next page. Add information to the blueprint as you work through this Activity.

Matt has placed the gas line on the blueprint from the information he received from the gas company. To assure that the bricks will be parallel to the walkway, Matt extended the string line along the edge of the walkway and labeled points X, A, and Y. He also labeled additional points B, C, and D where the string lines will cross the gas line.

1. Follow these steps to draw a line parallel to ___

XY .

Step A Determine the measure of ∠HAX.

Step B Locate the third stake on the left edge of the patio at point P by drawing

___

BP so that m∠HBP = m∠HAX.

Step C Locate the fourth stake at point J on the right edge of the patio by extending

___

BP in the opposite direction.

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Unit 1 • Proof, Parallel and Perpendicular Lines 49

My Notes

ACTIVITY 1.6continued

Parallel and Perpendicular LinesPatios by MadelinePatios by Madeline

SUGGESTED LEARNING STRATEGIES: Visualization; Create Representations

Matt’s Blueprint

X

Y

H

A

B

C

D

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My Notes

Parallel and Perpendicular LinesACTIVITY 1.6continued Patios by MadelinePatios by Madeline

SUGGESTED LEARNING STRATEGIES: Interactive Word Wall, Quickwrite, Create Representations

A transversal is a line that intersects two or more coplanar lines in different points.

MATH TERMS

∠HAX and ∠HBP are called corresponding angles because they are in corresponding positions on the same side of the transversal (gas line). According to the Corresponding Angles Postulate, when parallel lines are cut by a transversal, the corresponding angles will always have the same measure.

2. Postulates and theorems are oft en stated in if-then form.

a. State the Corresponding Angles Postulate in if-then form.

b. State the converse of the Corresponding Angles Postulate.

c. How does the statement in part b ensure that Matt’s string lines are parallel?

d. State the inverse of the Corresponding Angles Postulate.

e. Determine the validity of the inverse of the Corresponding Angles Postulate. Support your answer with illustrations.

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Unit 1 • Proof, Parallel and Perpendicular Lines 51

My Notes

ACTIVITY 1.6continued

Parallel and Perpendicular LinesPatios by MadelinePatios by Madeline

SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Quickwrite, Self/Peer Revision

3. Use the diagram shown to complete the table.

a. Fill in the corresponding angles column in the table below.

1 2

3456

7 8

1012

911

l

m

n

Angle Corresponding Angles

∠1

∠2

∠3

∠4

b. What conditions are necessary in order to apply the Corresponding Angles Postulate?

c. What conditions are necessary in order to apply the inverse of the Corresponding Angles Postulate?

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52 SpringBoard® Mathematics with Meaning™ Geometry

My Notes

Parallel and Perpendicular LinesACTIVITY 1.6continued Patios by MadelinePatios by Madeline

SUGGESTED LEARNING STRATEGIES: Quickwrite, Group Presentation

4. Other types of special angle pairs formed when lines are intersected by a transversal are alternate interior angles. In the diagram below, ∠3 and ∠6 are alternate interior angles.

1 2

3456

7 8

l

m

a. Explain why you think ∠3 and ∠6 are alternate interior angles.

b. Lines l and m in the diagram are parallel. What appears to be true about the measures of ∠3 and ∠6?

c. Name the remaining pair of alternate interior angles in the diagram. What appears to be true about them?

d. Using what you know about corresponding and vertical angles, write a convincing argument to prove your response to part b.

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Unit 1 • Proof, Parallel and Perpendicular Lines 53

My Notes

ACTIVITY 1.6continued

Parallel and Perpendicular LinesPatios by MadelinePatios by Madeline

SUGGESTED LEARNING STRATEGIES: Interactive Word Wall, Quickwrite, Create Representations

5. Use your results from Item 4 to state a theorem in if-then form about alternate interior angles.

6. State the inverse of the Alternate Interior Angles Th eorem you wrote in Item 5.

7. Determine the validity of the inverse of the Alternate Interior Angles Th eorem. Support your answer with illustrations.

8. State the converse of the Alternate Interior Angles Th eorem.

9. Matt realizes that he is not limited to using corresponding angles to draw parallel lines. Use a protractor to draw another parallel string line through point C on the blueprint, using alternate interior angles. Mark the angles that were used to draw this new parallel line.

10. How does the converse of the Alternate Interior Angles Th eorem ensure that Matt’s string lines are parallel?

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54 SpringBoard® Mathematics with Meaning™ Geometry

My Notes

Parallel and Perpendicular LinesACTIVITY 1.6continued Patios by MadelinePatios by Madeline

SUGGESTED LEARNING STRATEGIES: Think/Pair/Share

11. Another angle pair relationship is same-side interior angles. In the diagram below, ∠3 and ∠5 are same-side interior angles.

1 2

3456

7 8

l

m

a. Explain why you think ∠3 and ∠5 are same-side interior angles.

b. Lines l and m in the diagram are parallel. What appears to be true about the measures of ∠3 and ∠5?

c. Name the remaining pair of same-side interior angles in the diagram. What appears to be true?

d. Provide a convincing argument to prove your response to part b.

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Unit 1 • Proof, Parallel and Perpendicular Lines 55

My Notes

ACTIVITY 1.6continued

Parallel and Perpendicular LinesPatios by MadelinePatios by Madeline

SUGGESTED LEARNING STRATEGIES: Interactive Word Wall, Quickwrite, Create Representations

12. Use your results from Item 11 to state a theorem in if-then form, about same-side interior angles.

13. State the inverse of the Same-Side Interior Angles Th eorem you wrote in Item 12.

14. Determine the validity of the inverse of the Same-Side Interior Angles Th eorem. Support your answer with illustrations.

15. State the converse of the Same-Side Interior Angles Th eorem.

16. Use a protractor to draw another parallel string line through point D on the blueprint using same-side interior angles. Mark the angles you used to draw this line.

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56 SpringBoard® Mathematics with Meaning™ Geometry

My Notes

Parallel and Perpendicular LinesACTIVITY 1.6continued Patios by MadelinePatios by Madeline

SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge

This theorem guarantees that the line you draw in Item 17 is perpendicular to line XY:

In a given plane containing a line, there is exactly one line perpendicular to the line through a given point not on the line.

A second customer of Madeline’s company has commissioned them to build a patio with bricks that are perpendicular to the walkway. To help plan this design, Matt will attach a string between two stakes on either side of the patio as he did for the previous diagram. Continue drawing the blueprint for the patio using the diagram below. To assure that the bricks will be perpendicular to the walkway, Matt extended the string line along the edge of the walkway and labeled points X and Y. He also attached a string to another stake labeled point W as shown below.

X

W

Y

House

17. Use a protractor to draw the line perpendicular to ‹

___ ›

XY that passes through W. Label the point at which the line intersects

___ ›

XY point Z.

18. Use your protractor and your knowledge of parallel lines to draw a line, across the patio, parallel to

___ ›

WZ . Explain how you know the lines are parallel.

ACADEMIC VOCABULARY

Perpendicular fi gures intersect to form right angles.

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Unit 1 • Proof, Parallel and Perpendicular Lines 57

My Notes

ACTIVITY 1.6continued

Parallel and Perpendicular LinesPatios by MadelinePatios by Madeline

SUGGESTED LEARNING STRATEGIES: RAFT

19. Describe the relationship between the newly drawn line and ‹

___ ›

XY .

20. Complete the statement to form a true conditional statement.

“If a transversal is perpendicular to one of two parallel lines, then it is to the other line also.’’

21. Matt’s boss Madeline is so impressed with Matt’s work that she asks him to write an instruction guide for creating rows of bricks parallel to patio walkways. Madeline asks that the guide provide enough direction so that a bricklayer can use any pair of angles to determine the location of each pair of stakes since not all patios will allow for measuring the same pair of angles for each string line. She also requests that Matt include directions for creating rows of bricks that are perpendicular to the walkway. Use the space below to write Matt’s instruction guide, following Madeline’s directions.

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58 SpringBoard® Mathematics with Meaning™ Geometry

Parallel and Perpendicular LinesACTIVITY 1.6continued Patios by MadelinePatios by Madeline

CHECK YOUR UNDERSTANDING

Write your answers on notebook paper. Show your work.

Use this fi gure for Items 1-14.

1 2

3456

7 8

1012

911

l

m

n

1. List all pairs of alternate interior angles.

2. List all pairs of same-side interior angles.

3. If l ‖ m and m∠5 = 80°, then what is m∠2?

a. 100° c. 80°

b. 50° d. Not enough information

4. If l ‖ m and m∠6 = 110°, then what is m∠3?

a. 70° c. 80°

b. 110° d. Not enough information

5. If l ‖ m, m∠4 = 12x + 5, and m∠5 = 8x + 17, then what is m∠2?

a. 139° c. 3°

b. 7.9° d. 41°

6. If l ‖ m, which two angles are supplementary?

a. ∠1 and ∠6 c. ∠1 and ∠8

b. ∠3 and ∠5 d. ∠4 and ∠5

7. If l ‖ m, m∠4 = 15x − 7, and m∠6 = 20x + 12, then m∠8 = ?

a. 112° c. 68°

b. 5° d. 3°

In the diagram, l ‖ m and m ∦ n.

Determine whether each statement is true or false. Justify your response with the appropriate postulate or theorem.

8. ∠5 is supplementary to m∠3

9. m∠9 = m∠6

10. ∠3 � ∠8

11. ∠4 � ∠10

12. ∠4 � ∠6

13. m∠8 + m∠10 = 180°

14. ∠4 is supplementary to ∠9.

15. MATHEMATICAL R E F L E C T I O N

How could the types of angles you studied in this

activity be useful to a city planner?

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Unit 1 • Proof, Parallel and Perpendicular Lines 59

My Notes

ACTIVITY

1.7Two-Column ProofsNow I'm ConvincedSUGGESTED LEARNING STRATEGIES: Close Reading, Activating Prior Knowledge, Think/Pair/Share

A proof is an argument, a justifi cation, or a reason that something is true. A proof is an answer to the question “why?” when the person asking wants an argument that is indisputable.

Th ere are three basic requirements for constructing a good proof.

1. Awareness and knowledge of the defi nitions of the terms related to what you are trying to prove.

2. Knowledge and understanding of postulates and previous proven theorems related to what you are trying to prove.

3. Knowledge of the basic rules of logic.

To write a proof, you must be able to justify statements. Th e statements in the Examples 1–5 are based on the diagram to the right in which lines AC, EG, and DF all intersect at point B. Each of the statements is justifi ed using a property, postulate, or defi nition.

EXAMPLE 1

Statement Justifi cationIf ∠ABE is a right angle,then m∠ABE = 90°. Defi nition of right angle

EXAMPLE 2

Statement Justifi cationIf ∠2 � ∠1 and ∠1 � ∠5,then ∠2 � ∠5. Transitive Property

EXAMPLE 3

Statement Justifi cationGiven: B is the midpoint of

___ AC .

Prove: ___

AB � ___

BC Defi nition of midpoint

CA

E

B5

6

12

GD

F

CA

E

B5

6

12

GD

F

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60 SpringBoard® Mathematics with Meaning™ Geometry

My Notes

Two-Column Proofs ACTIVITY 1.7continued Now I’m ConvincedNow I’m Convinced

EXAMPLE 4

Statement Justifi cationm∠2 + m∠ABE = m∠DBE Angle Addition Postulate

EXAMPLE 5

Statement Justifi cationIf ∠1 is supplementary to ∠FBG,then m∠1 + m∠FBG = 180°.

Defi nition of supplementary angles

TRY THESE A

Using the diagram from the previous page, reproduced to the left , name the property, postulate, or defi nition that justifi es each statement.

a. Statement Justifi cation

EB + BG = EG

b. Statement Justifi cationIf ∠5 � ∠6, then � � BF bisects ∠EBC.

c. Statement Justifi cationIf m∠1 + m∠6 = 90, then∠1 is complementary to ∠6.

d. Statement Justifi cationIf m∠1 + m∠5 = m∠6 + m∠5, then m∠1 = m∠6.

e. Statement Justifi cation

Given: � � � AC ⊥ � � � EG Prove: ∠ABG is a right angle.

SUGGESTED LEARNING STRATEGIES: Think/Pair/Share

CA

E

B5

6

12

GD

F

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Unit 1 • Proof, Parallel and Perpendicular Lines 61

My Notes

ACTIVITY 1.7continued

Two-Column Proofs Now I’m ConvincedNow I’m Convinced

One way to learn how to write proofs is to start by studying completed proofs and then practicing simple proofs before moving on to more challenging proofs.

EXAMPLE 6

Th eorem: Vertical angles are congruent.

Given: ∠1 and ∠2 are vertical angles.Prove: ∠1 � ∠2

Statements Reasons1. m∠1 + m∠3 = 180° 1. Angle Addition Postulate2. m∠2 + m∠3 = 180° 2. Angle Addition Postulate3. m∠1 + m∠3 = m∠2 + m∠3 3. Substitution Property4. m∠1 = m∠2 4. Subtraction Property of Equality5. ∠1 � ∠2 5. Defi nition of congruent angles

EXAMPLE 7

Th eorem: Th e sum of the angles of a triangle is 180°.

Given: � ABCProve: m∠2 + m∠1 + m∠3 = 180°

Statements Reasons

1. Th rough point A, draw � � � AD , so that � � � AD || � � � BC .

1. Th rough any point not on a line, there is exactly one line || to the given line.

2. m∠5 + m∠1 + m∠4 = 180° 2. Angle Addition Postulate3. ∠5 � ∠2; ∠4 � ∠3 3. If parallel lines are cut by a

transversal, then alternate interior angles are congruent.

4. m∠5 = m∠2; m∠4 = m∠3 4. Defi nition of congruent angles5. m∠2 + m∠1 + m∠3 = 180° 5. Substitution Property

31

231

2

3CB

AD

15 4

2 3CB

AD

15 4

2

SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Vocabulary Organizer

Line AD is an example of an auxiliary line. An auxiliary line is a line that is added to a fi gure to aid in the completion of a proof.

MATH TERMS

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62 SpringBoard® Mathematics with Meaning™ Geometry

My NotesMy Notes

Two-Column Proofs ACTIVITY 1.7continued Now I’m ConvincedNow I’m Convinced

GUIDED EXAMPLE

Supply the missing statements and reasons.

Th eorem: In a plane, two lines perpendicular to the same line are parallel.

Given: a ⊥ b, c ⊥ bProve: a || c

Statements Reasons1. 1. Given information2. ∠1 and ∠2 are right angles. 2. Defi nition of 3. m∠1 = 90°; m∠2 = 90° 3. Defi nition of 4. m∠1 = m∠2 4. 5. ∠1 � ∠2 5. 6. a || c 6.

TRY THESE B

a. Complete the proof.

Given: a || b, c || dProve: ∠2 � ∠16

Statements Reasons1. c || d 1.

2. ∠2 � ∠7 2.

3. a || b 3.

4. ∠7 � ∠16 4.

5. ∠2 � ∠16 5.

b

a

c

1

2

b

a

c

1

2

ba

c

d

12

34

67

85

912

1110

1514

1316

ba

c

d

12

34

67

85

912

1110

1514

1316

SUGGESTED LEARNING STRATEGIES: Vocabulary Organizer

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Unit 1 • Proof, Parallel and Perpendicular Lines 63

My Notes

ACTIVITY 1.7continued

Two-Column Proofs Now I’m ConvincedNow I’m Convinced

SUGGESTED LEARNING STRATEGIES: Group Presentation

TRY THESE B (continued)

b. Write a two-column proof for the following.

ba

c

d

12

34

6 78

5

912

1110

1514

1316

Given: a || b, c || d

Prove: ∠1 � ∠15

Statements Reasons

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64 SpringBoard® Mathematics with Meaning™ Geometry

Two-Column Proofs Now I’m ConvincedNow I’m Convinced

ACTIVITY 1.7continued

Write your answers on notebook paper. Show your work.

CHECK YOUR UNDERSTANDING

Write your answers on notebook paper. Show your work.

Lines CF, DH, and EA intersect at point B. Use this fi gure for Items 1–8. Write the defi nition, postulate, property, or theorem that justifi es each statement.

H

A

D

C

56

3 41 2

F

GE

B

1. ∠1 � ∠5

2. If ∠2 is supplementary to ∠CBE, then m∠2 + m∠CBE = 180°.

3. If ∠2 � ∠3, then � � BF bisects ∠GBE.

4. CB + BF = CF

5. If ∠DBF is a right angle, then � � � HD ⊥ � � � CF .

6. If m∠3 = m∠6, then m∠3 + m∠2 = m∠6 + m∠2.

7. If ___

AB � ___

BE , then B is the midpoint of ___

AE .

8. m∠2 + m∠3 = m∠EBG

9. Supply the Statements and Reasons.

Given: ∠1 is complementary to ∠2; � � BE bisects ∠DBC.

Prove: ∠1 is complementary to ∠3.

B

D

E1 23

A

C

Statements Reasons

1. � � BE bisects ∠DBC. 1. 2. ∠2 � ∠3 2. 3. 3. Defi nition of

congruent angles4. ∠1 is complementary

to ∠2.4.

5. m∠1 + m∠2 = ____ 5. 6. m∠1 + m∠3 = 90° 6. 7. 7. Defi nition of

complementary angles

10. MATHEMATICAL R E F L E C T I O N

Choose one of the theorems about angles formed when

parallel lines are cut by a transversal(Activity 1–6: Patios by Madeline). Explain how you would set up and prove the theorem in two-column format.

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Unit 1 • Proof, Parallel and Perpendicular Lines 65

Embedded Assessment 2 Use after Activity 1.7.

Angles and Parallel LinesTHROUGH THE LOOKING GLASS

Light rays are bent as they pass through glass. Since a block of glass is a rectangular prism, the opposite sides are parallel and a ray is bent the same amount entering a piece of glass as exiting the glass.

I

B G

R E

F

AS

1. Extend � � IR and � � EF through the block of glass. Label the points of intersection with

__ SB and

__ AG ; X and Y, respectively.

2. Name a pair of same side interior angles formed by the transversal __

RE .

3. Identify the geometric term that describes the relationship between the pair of angles ∠IRE and ∠REF.

4. Explain why a ray exits the glass in a path parallel to the path in which it entered the glass.

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66 SpringBoard® Mathematics with Meaning™ Geometry

Embedded Assessment 2 Use after Activity 1.7.

Angles and Parallel LinesTHROUGH THE LOOKING GLASS

Exemplary Profi cient Emerging

Math Knowledge#2, 3

The student:• Gives the correct

name for the pair of angles. (2)

• Gives the correct geometric term for the relationship between the pair of angles. (3)

The student:• Gives the name

of a pair of angles that are not same side interior angles.

• Gives an incorrect geometric term for the relationship.

Representations#1

The student correctly extends the rays and labels the points of intersection correctly. (1)

The student correctly extends the rays and labels some, but not all of the points of intersection correctly.

The student extends the rays, but does not label the points of intersection correctly.

Communication#4

The student gives a complete and correct explanation. (4)

The student gives an incomplete explanation that does not contain mathematical errors.

The student gives an explanation that is not correct.

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Unit 1 • Proof, Parallel and Perpendicular Lines 67

My Notes

ACTIVITY

1.8Equations of Parallel andPerpendicular LinesSkateboard GeometrySUGGESTED LEARNING STRATEGIES: Look for a Pattern

Th e new ramp at the local skate park is shown above. In addition to the wooden ramp, an aluminum rail (not shown) is mounted to the edge of the ramp. While the image of the ramp may conjure thoughts of kickfl ips, nollies, and nose grinds, there are mathematical forces at work here as well.

1. Use the diagram of the ramp to complete the chart below:

Describe two parts of the ramp that appear to be parallel.

Describe two parts of the ramp that appear to be perpendicular.

Describe two parts of the ramp that appear to be neither parallel nor perpendicular.

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68 SpringBoard® Mathematics with Meaning™ Geometry

My Notes

Equations of Parallel and Perpendicular Lines ACTIVITY 1.8continued Skateboard GeometrySkateboard Geometry

SUGGESTED LEARNING STRATEGIES: Look for a Pattern, Work Backward

x

CAFE

Y X WV

UT S

J

IH

G

B

D

ML

NP

O

K

R

–12 –8 –4 4

4

8

8

2. Th e diagram above shows a cross-section of the skate ramp transposed onto a coordinate grid. Find the slopes of the following segments.

a. __

AB

b. __

XT

c. __

WG

d. _

TJ

e. __

AD

3. Compare and contrast the slopes of the segments. Make conjectures about how the slopes of these segments relate to the positions of the segments on the graph.

CONNECT TO ALGEBRAALGEBRA

Slope =

� y ____

� x =

y2 − y1 ______ x2 − x1

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Unit 1 • Proof, Parallel and Perpendicular Lines 69

My Notes

ACTIVITY 1.8continued

Equations of Parallel and Perpendicular Lines Skateboard GeometrySkateboard Geometry

SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Predict and Confirm, Look for a Pattern, Identify a Subtask

4. Th e equation of the line containing __

BD is y = − 2 __ 3 x + 4.

Identify the slope and the y-intercept of this line.

5. Based on your conjectures in Item 3, what would be the slope of the line containing

__ JM ? Use the formula for slope to verify your answer

or provide an explanation of the method you used to determine the slope.

6. Identify the y-intercept of the line containing __

JM . Use the slope and the y-intercept to write the equation of this line.

7. If __

JM ⊥ _

RJ , identify the slope, y-intercept, and the equation for the line containing

_ RJ .

8. Is __

BD ⊥ _

RJ ? Explain your reasoning.

CONNECT TO ALGEBRAALGEBRA

The equation of a line inslope-intercept form isy = mx + b, where m represents the slope and b represents the y-intercept of the line.

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70 SpringBoard® Mathematics with Meaning™ Geometry

Equations of Parallel and Perpendicular Lines ACTIVITY 1.8continued Skateboard GeometrySkateboard Geometry

Write your answers on notebook paper. Show your work.

CHECK YOUR UNDERSTANDING

Write your answers on notebook paper. Show your work.

1. What is the slope of a line parallel to the line with equation y = 4 __ 5 x - 7?

2. What is the slope of a line perpendicular to the line with equation y = -2x + 1?

3. Find the equation of the line that has a y-intercept of -4 and is parallel to the line with equation y = 9x + 11.

4. Triangle ABC has vertices at the pointsA(-2, 3), B (7, 4), and C(5, 22). What kind of triangle is �ABC? Explain your answer.

5. Consider the lines described below:

line a: a line with the equation y = 3 __ 5 x - 2

line b: a line containing the points (-6, 1) and (3, -14)

line c: a line containing the points (-1, 8) and (4, 11)

line d: a line with the equation -5x + 3y = 6

Use the vocabulary from the lesson to write a paragraph that describes the relationships among lines a, b, c, and d.

6. MATHEMATICAL R E F L E C T I O N

What is the slope of a line parallel to the line with

equation y = -3? What is the slope of a line perpendicular to the line with equation y = -3?

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Unit 1 • Proof, Parallel and Perpendicular Lines 71

My Notes

ACTIVITY

1.9Distance and MidpointWe � DescartesSUGGESTED LEARNING STRATEGIES: Predict and Confirm, Visualization, Simplify the Problem

If mathematics concept popularity contest were ever held, the Pythagorean Th eorem might be the runaway victor. With well over 300 known proofs, people from Plato to U. S. President James A. Garfi eld have spent at least part of their lives studying various proofs and applications of the theorem.

While some of these proofs involve a level of complexity comparable to fi guring out how to land a remote control car on the surface of Mars, the following is a very simple example of how the Pythagorean Th eorem actually works.

b

c a

1. Suppose you painted the large square extending from the hypotenuse of the triangle above. If it required exactly one can of paint to cover the area of the large square, make a conjecture as to how many cans you would need to paint the other two squares. Explain your reasoning.

The Pythagorean Theorem: The square of the hypotenuse of a right triangle is equal to the sum of the squares of the legs of the right triangle.

MATH TERMS

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72 SpringBoard® Mathematics with Meaning™ Geometry

My Notes

Distance and Midpoint ACTIVITY 1.9continued We We �� Descartes Descartes

SUGGESTED LEARNING STRATEGIES: Simplify the Problem, Identify a Subtask, Create Representations

You can use a common Pythagorean triple to explore the idea of the painted squares further.

5 cm3 cm

4 cm

From the information in the fi gure above, you can derive that the area of the large square is (5 cm)2 = 25 cm2, and that the areas of the smaller squares are (3 cm)2 = 9 cm2 and (4 cm)2 = 16 cm2, respectively.

a = 3 b = 4 c = 5

Area = a2 = 32 = 9 Area = b2 = 42 = 16 Area = c2 = 52 = 25

Th erefore,

52 = 32 + 42 25 = 9 + 16 c2 = a2 + b2

Pythagorean triples are sets of whole numbers that represent the side lengths of a right triangle. Some common Pythagorean triples are (3, 4, 5), (5, 12, 13), (7, 24, 25), and (8, 15, 17).

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Unit 1 • Proof, Parallel and Perpendicular Lines 73

My Notes

ACTIVITY 1.9continued

Distance and Midpoint We We �� Descartes Descartes

SUGGESTED LEARNING STRATEGIES: Create Representations, Identify a Subtask, Simplify the Problem

You can use the Pythagorean theorem to determine the distance between two points on a coordinate grid by drawing auxiliary lines to create right triangles.

EXAMPLE 1

What is the distance between the points with coordinates (1, 4) and (13, 9)?

Step 1: Start by plotting the points and connecting them.

x

(13, 9)

(1, 4)

Step 2: Add some auxiliary line segments to create a right triangle.

x

(13, 9)

(1, 4)

The Cartesian Coordinate System was developed by the French philosopher and mathematician René Descartes in 1637 as a way to specify the position of a point or object on a plane.

MATH TERMS

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74 SpringBoard® Mathematics with Meaning™ Geometry

My Notes

Distance and Midpoint ACTIVITY 1.9continued We We �� Descartes Descartes

SUGGESTED LEARNING STRATEGIES: Create Representations, Identify a Subtask, Simplify the Problem

EXAMPLE 1 (continued)

Step 3: Count units along the horizontal and vertical legs to determine the coordinates of the vertex of the right angle.

Th e coordinates of the vertex of the right angle are (13, 4).

x

(13, 9)

12 units

(1, 4)5 units

Step 4: Use the Pythagorean theorem to fi nd the distance between the two points (the endpoints of the hypotenuse of the right triangle).

hypotenuse 2 = leg 2 + leg 2

c2 = 52 + 122

√__

c2 = √________

25 + 144 √

__ c2 = √

____ 169

c = 13Solution: Th e distance between the points with coordinates (1, 4) and

(13, 9) is 13 units.

TRY THESE A

Plot each pair of points on grid paper, and demonstrate how you can use the Pythagorean Th eorem to fi nd the distance between the points.

a. (-3, 6) and (3, 14) b. (-8, 5) and (7, -3)

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Unit 1 • Proof, Parallel and Perpendicular Lines 75

My Notes

ACTIVITY 1.9continued

Distance and Midpoint We We �� Descartes Descartes

SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Simplify the Problem, Identify a Subtask

While this method for fi nding the distance between two points will always work, it may not be practical to plot points on a coordinate grid and draw auxiliary lines each time you want to fi nd the distance between them.

In Example 1, you determined that the lengths of the legs of the right triangle were 5 and 12 units, respectively. You could also have determined that the slope of the segment between the points (1, 4) and (13, 9) is 5 ___ 12 .

To understand why getting the numbers 5 and 12 is more than a coincidence, recall some ways in which slope can be defi ned.

Slope = rate of change = Δy

___ Δx =

y2 - y1 ______ x2 - x1

To fi nd the length of the horizontal leg, you simply counted to get 12 units. However, if you think of the x-coordinates of the points as x-values on a horizontal number line, the distance between x-values = (Δx) = 13 − 1 = 12.

1 13

In other words, the distance between the x-values is the same as the length of the horizontal leg of the right triangle.

Likewise, the distance between the y-values is the same as the length of the vertical leg of the right triangle: (Δy) = 9 − 4 = 5

4 9

CONNECT TO SCIENCESCIENCE

The Greek letter Δ (delta) is used in mathematical formulas to denote a change in quantities such as value, magnitude, or position. The expression Δ x wouldrepresent a change in x. Similarly, in Physics, the expression Δv (where v = velocity) wouldrepresent a change in velocity.

Remember that distances are always measured in positive units.

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76 SpringBoard® Mathematics with Meaning™ Geometry

My Notes

Distance and Midpoint ACTIVITY 1.9continued We We �� Descartes Descartes

SUGGESTED LEARNING STRATEGIES: Identify a Subtask, Simplify the Problem, Create Representations

You can use algebraic methods to derive a formula for determining the distance between two points.

From Example 1, the Pythagorean equation looked like this:

hypotenuse 2 = leg 2 + leg 2 c2 = 52 + 122

c2 = 25 + 144 √

__ c2 = √

________ 25 + 144

√__

c2 = √____

169 13 = c

Now, redefi ne the variables based on exploration of slope on the previous page.

a = horizontal leg = Δx = x2 - x1 b = vertical leg = Δy = y2 - y1

c = hypotenuse = distance between two points = d

Use substitution:

hypotenuse 2 = leg 2 + leg 2

c2 = a2 + b2

distance2 = (Δx)2 + (Δy)2

d 2 = (x2 - x1)2 + (y2 - y1)2

2. Finish simplifying the equation to fi nd the formula for determining the distance d between two points.

3. In the space to the right, create a Venn diagram that compares and contrasts the mathematical operations performed in the Pythagorean theorem and the distance formula.

CONNECT TO APAP

The coordinate geometry formulas introduced in this activity are used frequently in AP Calculus in a wide variety of applications.

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Unit 1 • Proof, Parallel and Perpendicular Lines 77

My Notes

ACTIVITY 1.9continued

Distance and Midpoint We We �� Descartes Descartes

SUGGESTED LEARNING STRATEGIES: Guess and Check, Create Representations, Work Backward, Visualization

TRY THESE B

Use the distance formula to fi nd the distance between the points with the coordinates shown.

a. (-4, 6) and (15, 6)

b. (12, -5) and (-3, 7)

c. (8, 1) and (14, -2)

In Activity 1–5, you defi ned midpoint as the point on the segment that divides it into two congruent segments.

4. Explore various methods, and explain how you could determine the midpoint of the following segments:

a.

b. 0 12–3

c.

5 175 17

x

(0, 0)

(9, 12)

x

(0, 0)

(9, 12)

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78 SpringBoard® Mathematics with Meaning™ Geometry

My Notes

Distance and Midpoint ACTIVITY 1.9continued We We �� Descartes Descartes

SUGGESTED LEARNING STRATEGIES: Identify a Subtask, Simplify the Problem, Create Representations

5. Consider the right triangle below:

x

a. Using the defi nition of midpoint, identify and label the midpoints of the legs of the triangle

b. Draw an auxiliary line vertically from the midpoint of the horizontal leg through the hypotenuse

c. Draw an auxiliary line horizontally from the midpoint of the vertical leg through the hypotenuse.

d. Identify the coordinates of the point of intersection of the two auxiliary lines.

Th is point represents the midpoint of the hypotenuse!

TRY THESE C

Find the midpoint of the segment whose endpoints have the given coordinates.

a. (2, 3) and (-5, 11)

b. (-8, -1) and (-7, -5)

c. (6, -11) and (22, 16)

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Unit 1 • Proof, Parallel and Perpendicular Lines 79

My Notes

ACTIVITY 1.9continued

Distance and Midpoint We We �� Descartes Descartes

SUGGESTED LEARNING STRATEGIES: Identify a Subtask, Simplify the Problem

As was the case with the distance formula, it is not always practical to plot the points on a coordinate grid and use auxiliary lines to fi nd the midpoint of a segment. Mathematically, the coordinates of the midpoint of a segment are simply the averages of the x- and y-coordinates of the endpoints.

EXAMPLE 2

What are the coordinates of the midpoint of __

AC with endpoints A(1,4) and C(11,10)?

Step 1: Find the average of the x-coordinates:x-coordinate of point A = 1x-coordinate of C = 11Average = 1 + 11 ______ 2 = 6

Step 2: Find the average of the y-coordinates.y-coordinate of point A = 4y-coordinate of C = 10Average = 4 + 10 ______ 2 = 7

Solution: Th e coordinates of the midpoint of ___

AC are (6, 7).

TRY THESE D

a. Find the midpoint of __

QR with endpoints Q(-3, 14) and R(7, 5).

b. Find the midpoint of S(13, 7) and T(-2, -7)

c. Find the midpoint of each side of the right triangle with verticesD(-1, 2), E(8, 2), and F(8, 14).

6. Let (x1, y1) and (x2, y2) represent the coordinates of two points. Write a formula for fi nding the coordinates of midpoint M of the segment connecting those two points.

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80 SpringBoard® Mathematics with Meaning™ Geometry

Distance and Midpoint ACTIVITY 1.9continued We We �� Descartes Descartes

Write your answers on notebook paper. Show your work.

CHECK YOUR UNDERSTANDING

Write your answers on notebook paper or grid paper. Show your work.

1. Find the distance between the points A(-8, -6) and B(4, 10).

2. Find the distance between the points C(5, 14) and D(-3, -9)

3. Use your Venn diagram from Item 3 to write a RAFT .

Role: Teacher

Audience: A classmate who was absent for the lesson on distance between two points

Format: Personal note

Topic: Explain the mathematical similarities and diff erences between the Pythagorean theorem and the distance formula.

4. Find the midpoint of the segment with endpoints E(4, 11) and F(-11, -5).

5. Find and explain the errors that were made in the following calculation of midpoint. Th en fi x the errors, and determine the correct answer.

Find the midpoint of the segment with endpoints R(-2, 3) and S(13, -7).

( -2 + 13 ________ 2 , 3 - 7 _____ 2 )

= ( -15 ____ 2 , -4 ___ 2 ) = (-7.5, -2)

6. Th e vertices of �XYZ are X(-3, -6), Y(21, -6) and Z(21, 4).

a. Find the length of each side of the triangle.

b. Find the coordinates of the midpoint of each side of the triangle.

7. MATHEMATICAL R E F L E C T I O N

Segment RS is graphed on a coordinate grid. Explain

how you would determine the coordinates of the point on the segment that is one fourth of the distance from R to S.

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Unit 1 • Proof, Parallel and Perpendicular Lines 81

Embedded Assessment 3 Use after Activity 1.9.

Slope, Distance, and MidpointGRAPH OF STEEL

Th e fi rst hill of the Steel Dragon 2000 roller coaster in Nagashima, Japan, drops riders from a height of 318 ft . A portion of this fi rst hill has been transposed onto a coordinate plane and is shown to the right.

Write your answers on notebook paper or grid paper. Show your work.

1. Th e structure of the supports for the hill consists of steel beams that run parallel and perpendicular to one another. Th e endpoints of the longer of the two support beams highlighted in Quadrant I are (0, 150) and (120, 0). If the endpoints of the other highlighted support beam are (0, 125) and (100, 0), verify and explain why the two beams are parallel.

2. Determine the equations of the lines containing the beams from Item 1, and explain how the equations of the lines can help you determine that the beams are parallel.

3. Th e equation of a line containing another support beam is y = 4 __ 5 x + 150. Determine whether this beam is parallel or

perpendicular to the other two beams, and explain your reasoning.

4. A linear portion of the fi rst drop is also highlighted in the photo and has endpoints of (62, 258) and (110, 132). To the nearest foot, determine the distance between the endpoints of the linear section of the track. Justify your result by showing your work.

5. A camera is being installed at the midpoint of the linear portion of the track described in Item 4. Determine the coordinates where this camera should be placed.

6. Explain how you could use the distance formula to verify that the coordinates you determined for the midpoint are correct.

x

x

1–6: See below.

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82 SpringBoard® Mathematics with Meaning™ Geometry

Embedded Assessment 3 Use after Activity 1.9.

Slope, Distance, and MidpointGRAPH OF STEEL

Exemplary Profi cient Emerging

Math Knowledge#2, 3, 4, 5

The student:• Gives the correct

equations for the parallel lines. (2)

• Determines the correct relationship among the beams. (3)

• Determines the correct distance between the points. (4)

• Correctly determines the coordinates of the midpoint of the linear section. (5)

The student:• Gives the correct

equation for one, but not both, of the parallel lines.

• Uses a correct method to determine the distance but makes a computational error.

• Uses a correct method to determine the coordinates of the midpoint but makes a computational error.

The student:• Gives incorrect

linear equations.• Does not determine

the correct relationship among the beams.

• Uses an incorrect method to determine the distance.

• Uses an incorrect method to determine the coordinates of the midpoint.

Representations#2

The student gives the correct equations for the parallel lines. (2)

The student gives the correct equation for only one of the lines.

The student gives the correct equation for neither of the lines.

Communication#1, 2, 3, 4, 6

The student:• Verifi es and writes a

correct explanation for the reason the two beams are parallel. (1)

• Gives a correct explanation for the connection between the equations and the parallel relationship of the lines. (2)

• Gives a complete explanation for the parallelism and perpendicularity of the beams. (3)

• Shows work that is complete. (4)

• Gives a complete explanation of how the distance formula could be used to verify the coordinates of the midpoint. (6)

The student:• Verifi es and writes

an incomplete explanation for the reason the two beams are parallel.

• Gives an incomplete explanation for the connection between the equations and the parallel relationship of the lines.

• Gives a complete explanation for the parallelism or the perpendicularity of the beams but not both.

• Shows incomplete work that contains no mathematical errors.

• Gives an incomplete explanation of how the distance formula could be used for verifi cation.

The student:• Says the beams

are parallel, but does not give an explanation.

• Gives an incorrect explanation for the connection.

• Gives an incorrect explanation for the relationship among the beams.

• Shows incorrect work

• Gives an incorrect explanation for the use of the distance formula.

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UNIT 1Practice

Unit 1 • Proof, Parallel and Perpendicular Lines 83

ACTIVITY 1.1

1. Which is the correct name for this line?

G E M

a. �� � G b. ____

GM c. � ��

MG d. �� � ME

2. Use the diagram to name each of the following.

L NM

P RQ

a. parallel lines b. perpendicular lines

3. In this diagram, m∠SUT = 25°.

P S

R

U

T

Q

a. Name another angle that has measure 25°. b. Name a pair of complementary angles. c. Name a pair of supplementary angles.

Use this circle Q for Items 4–7.

WR

J

Q

T

U

4. Name the radii of circle Q. 5. Name the diameter(s) in circle Q. 6. Name the chord(s) in circle Q. 7. Which statement below must be true about

circle Q? a. The distance from U to W is the same as the

distance from R to T. b. The distance from U to W is the same as the

distance from Q to J. c. The distance from R to T is half the distance

from Q to R. d. The distance from R to T is twice the

distance from Q to J.

ACTIVITY 1.2

8. Use inductive reasoning to determine the next two terms in the sequence.

a. 1, 3, 7, 15, 31, … b. 3, -6, 12, -24, 48, …

9. Write the fi rst fi ve terms of two diff erent sequences for which 24 is the third term.

10. Generate a sequence using the description: the fi rst term in the sequence is 2 and the terms increased by consecutive odd numbers beginning with 3.

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UNIT 1 Practice

84 SpringBoard® Mathematics with Meaning™ Geometry

11. Use this picture pattern.

a. Draw the next shape in the pattern, b. Write a sequence of numbers that could be

used to express the pattern, c. Verbally describe the pattern of the

sequence.

12. Use expressions for odd integers to confi rm the conjecture that the product of two odd integers is an odd integer.

ACTIVITY 1.3

13. Identify the property that justifi es the statement: 5(x – 3) = 5x – 15

a. multiplication b. transitive c. subtraction d. distributive

14. Complete the prove statement and write a two column proof for the equation:

Given: 5(x – 2) = 2x – 4 Prove: 15. Write the statement in if-then form. Th e sum of two even numbers is even.

Use this statement for Items 16–19.

If today is Th ursday, then tomorrow is Friday. 16. State the conclusion of the statement. 17. Write the converse of the statement. 18. Write the inverse of the statement. 19. Write the contrapositive of the statement. 20. Given the false conditional statement, “If a

vehicle is built to fl y, then it is an airplane”, write a counter example.

21. Give an example of a statement that is false and logically equivalent to its inverse.

ACTIVITY 1.4

Construct a truth table for each of the following compound statements.

22. p → ˜q 23. p ∧ ˜q 24. ˜(˜p ∨ q) 25. p ∨ (q → p)

ACTIVITY 1.5

26. If Q is between A and M and MQ = 7.3 and AM = 8.5, then QA =

a. 5.8 b. 1.2 c. 7.3 d. 14.6

27. Given: K is between H and J, HK = 2x - 5, KJ = 3x + 4, and HJ = 24. What is the value of x?

a. 9 b. 5 c. 19 d. 3

28. If K is the midpoint of __

HJ , HK = x + 6, and HJ = 4x - 6, then KJ = ? .

a. 15 b. 9 c. 4 d. 10

29. P lies in the interior of ∠RST. m∠RSP = 40° and m∠TSP = 10°. m∠RST = ?

a. 100° b. 50° c. 30° d. 10°

30. �� � QS bisects ∠PQR. If m∠PQS = 3x and m∠RQS = 2x + 6, then m∠PQR = ? .

a. 18° b. 36° c. 30° d. 6°

31. ∠P and ∠Q are supplementary. m∠P = 5x + 3 and m∠Q = x + 3. x = ?

a. 14 b. 0 c. 29 d. 30

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UNIT 1Practice

Unit 1 • Proof, Parallel and Perpendicular Lines 85

In this fi gure, � � � AE ⊥ � � � CG , m∠BHC = 20°, and m∠DHE = 60°. Use this fi gure determine the angle measures for Items 32–35.

G C

A

H

D

E

BF

32. m∠CHD 33. m∠AHF 34. m∠GHF 35. m∠DHA 36. In this fi gure, m∠3 = x + 18, m∠4 = x + 15,

and m∠5 = 4x + 3. Show all work for parts a, b, and c.

1 5

43

a. What is the value of x? b. What is the measure of ∠1 c. Is ∠3 complementary to ∠1? Explain.

37. In this fi gure, m∠1 = 4x + 50 and m∠3 = 2x + 66. Show all work for parts a, b, and c.

1

2

3

a. What is the value of x? b. What is the measure of m∠3? c. What is the measure of m∠2?

38. Given: � � AE ⊥ � � AB , �� � AD bisects ∠EAC, m∠CAB = 2x - 4 and m∠CAE = 3x + 14. Show all work for parts a and b.

B

E

A

C

D

a. What is the value of x? b. What is the measure of m∠DAC?

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UNIT 1 Practice

86 SpringBoard® Mathematics with Meaning™ Geometry

ACTIVITY 1.6

Use this diagram for Items 39–50.

l

t

m

54

3

12

9

86

7

14

1312

11

16

15

10

n

39. List all angles in the diagram that form a corresponding angle pair with ∠5.

40. List all angles in the diagram that form an alternate interior angle pair with ∠14.

41. List all angles in the diagram that form a same side interior angle pair with ∠14.

42. Given l ‖ n. If m∠6 = 120°, then m∠16 = ? . 43. Given l ‖ n. If m∠12 = 5x +10 and

m∠14 = 6x - 4, then x = ? and m∠4 = ? .

44. Given l ‖ n. If m∠14 = 75°, then m∠2 = ? . 45. Given l ‖ n. If m∠9 = 3x + 12 and

m∠15 = 2x + 27, then x = ? and m∠10 = ? .

46. Given l ‖ n. If m∠3 = 100°, then m∠2 = ? . 47. Given l ‖ n. If m∠16 = 5x + 18 and

m∠15 = 3x+ 2, then x = ? and m∠8 = ? .

48. If m∠10 = 135°, m∠12 = 75°, and l ‖ n determine the measure of each angle. Justify each answer.

a. m∠16 b. m∠15 c. m∠8 d. m∠14 e. m∠3

49. If l ‖ n and m ⊥ n, then m∠3 = ? 50. If l ‖ n and m ⊥ l, then m∠13 = ? .

ACTIVITY 1.7

Use this diagram to identify the property, postulate, or theorem that justifi es each statement in Items 51–55.

A E

PD

Q1

23

4

R

51. PQ + QR = PR a. Angle addition postulate b. Addition property c . Definition of congruent segments d. Segment addition postulate

52. If Q is the midpoint of ___

PR , then ___

PQ � ___

QR . a. Definition of midpoint b. Definition of congruent segments c. Definition of segment bisector d. Segment addition postulate

53. ∠3 � ∠4 a. Definition of linear pair b. Definition of congruent angles c. Vertical angles are congruent. d. Definition of angle bisector

54. If ∠1 is complementary to ∠2, then m∠1 + m∠2 = 90

a. Angle Addition Postulate b. Addition property c. Definition of perpendicular d. Definition of complementary

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UNIT 1Practice

Unit 1 • Proof, Parallel and Perpendicular Lines 87

55. If m∠3 = 90, then ∠3 is a right angle. a. Definition of perpendicular b. Definition of complementary angles c. Definition of right angle d. Vertical angles are congruent. 56. Write a two-column proof. Given: a ‖ b, c ‖ d Prove: ∠5 � ∠12

a

c

d

b

12

34

912

1110

1514

1316

67

85

ACTIVITY 1.8

57. Determine the slope of the line that contains the points with coordinates (1, 5) and (-2, 7).

58. Which of the following is NOT an equation for a line parallel to y = 1 __ 2 x - 6?

a. y = 2 __ 4 x + 6 b. y = 0.5x - 3

c. y = 1 __ 2 x + 1 d. y = 2x - 4

59. Determine the slope of a line perpendicular to the line with equation 6x - 4y = 30.

60. Line m contains the points with coordinates (-4, 1) and (5, 8), and line n contains the points with coordinates (6, -2) and (10, 7). Are the lines parallel, perpendicular, or neither? Justify your answer.

ACTIVITY 1.9

61. Calculate the distance between the points A(-4, 2) and B(15, 6).

62. Calculate the distance between the points R(1.5, 7) and S(-2.3, -8).

63. Kevin and Clarice both live on a street that runs through the center of town. If the police station marks the midpoint between their houses, at what point is the police station on the number line?

Kevin’sHouse

-11 1

Clarice’sHouse

64. Determine the coordinates of the midpoint of the segment with endpoints R(3, 16) and S(7, -6).

65. Determine the coordinates of the midpoint of the segment with endpoints W(-5, 10.2) and X(12, 4.5).

66. Two explorers on an expedition to the Arctic Circle have radioed their coordinates to base camp. Explorer A is at coordinates (-26, -15). Explorer B is at coordinates (13, 21). Th e base camp is located at the origin.

a. Determine the linear distance between the two explorers.

b. Determine the midpoint between the two explorers.

c. Determine the distance between the midpoint of the explorers and the base camp.

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UNIT 1 Reflection

88 SpringBoard® Mathematics with Meaning™ Geometry

An important aspect of growing as a learner is to take the time to refl ect on your learning. It is important to think about where you started, what you have accomplished, what helped you learn, and how you will apply your new knowledge in the future. Use notebook paper to record your thinking on the following topics and to identify evidence of your learning.

Essential Questions

1. Review the mathematical concepts and your work in this unit before you write thoughtful responses to the questions below. Support your responses with specifi c examples from concepts and activities in the unit.

Why are properties, postulates, and theorems important in mathematics? How are angles and parallel and perpendicular lines used in real-world settings?

Academic Vocabulary

2. Look at the following academic vocabulary words: angle bisector counterexample perpendicular complementary angles deductive reasoning postulate conditional statement inductive reasoning proof congruent midpoint of a segment supplementary angles conjecture parallel theorem

Choose three words and explain your understanding of each word and why each is important in your study of math.

Self-Evaluation

3. Look through the activities and Embedded Assessments in this unit. Use a table similar to the one below to list three major concepts in this unit and to rate your understanding of each.

Unit Concepts

Is Your Understanding Strong (S) or Weak (W)?

Concept 1

Concept 2

Concept 3

a. What will you do to address each weakness?

b. What strategies or class activities were particularly helpful in learning the concepts you identifi ed as strengths? Give examples to explain.

4. How do the concepts you learned in this unit relate to other math concepts and to the use of mathematics in the real world?

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Unit 1 • Proof, Parallel and Perpendicular Lines 89

Unit 1

Math Standards Review

1. Keisha walked from her house to Doris’ house. Th is can best be represented by which of the following:

A. a ray C. a line

B. a segment D. a point

2. Sam wanted to go fi shing. He took his fi shing pole and walked 6 blocks due south to pick up his friend. Together, they walked 8 blocks due east before they reached the lake. Th ere is a path that goes directly from Sam’s house to the lake. If he had taken that path, how many blocks would he have walked?

3. To amuse her brother, Shirka was making triangular formations out of rocks. She made the following pattern:

How many rocks will be in the next triangular formation in the pattern?

1. Ⓐ Ⓑ Ⓒ Ⓓ

2.

○‒ ⊘⊘⊘⊘○• ○• ○• ○• ○• ○•⓪⓪⓪⓪⓪⓪①①①①①①②②②②②②③③③③③③④④④④④④⑤⑤⑤⑤⑤⑤⑥⑥⑥⑥⑥⑥⑦⑦⑦⑦⑦⑦⑧⑧⑧⑧⑧⑧⑨⑨⑨⑨⑨⑨

3.

○‒ ⊘⊘⊘⊘○• ○• ○• ○• ○• ○•⓪⓪⓪⓪⓪⓪①①①①①①②②②②②②③③③③③③④④④④④④⑤⑤⑤⑤⑤⑤⑥⑥⑥⑥⑥⑥⑦⑦⑦⑦⑦⑦⑧⑧⑧⑧⑧⑧⑨⑨⑨⑨⑨⑨

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90 SpringBoard® Mathematics with Meaning™ Geometry

Unit 1 (continued)

Math Standards Review

4. When camping, Sudi and Raul did not want to get lost. Th ey made a coordinate grid for their camp site. Camp was at the point (0, 0). On their grid, Sudi went to the point (-3, 10) and Raul went to the point (0, 6).

Part A: On the grid below, draw a diagram of the camp site and label the campsite location, Sudi’s location, and Raul’s location.

–12

–10

–8

–6

–4

4

10864–2

2

12

2 12–10 –8 –6 –4–12 –2

y

x

10

8

6

Part B: Find the distance between Sudi and Raul.

Part C: Find the midpoint between Sudi and Raul.

Read

Explain

Solve

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