PERPENDICULAR AND PARALLEL LINESmoodle.monashores.net/pluginfile.php/96252/mod... · parallel lines GOAL 1 Identify relationships between lines. Identify angles formed by transversals

� Will the boats’ paths ever cross?

PERPENDICULAR AND PARALLEL LINESPERPENDICULARAND PARALLEL LINES

126

http://www.classzone.com

APPLICATION: Sailing

When you float in an innertube on a windy day, you getblown in the direction of thewind. Sailboats are designed to sail against the wind.

Most sailboats can sail at anangle of 45° to the directionfrom which the wind is blowing,as shown below. If a sailboatheads directly into the wind, thesail flaps and is useless.

You’ll learn how to analyze linessuch as the paths of sailboats inChapter 3.

Think & Discuss1. What do you think the measure of ™1 is?

Use a protractor to check your answer.

2. If the boats always sail at a 45° angle to thewind, and the wind doesn’t change direction,do you think the boats’ paths will ever cross?

Learn More About ItYou will learn more about the paths of sailboatsin Example 4 on p. 152.

APPLICATION LINK Visit www.mcdougallittell.com for more information about sailing.

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C H A P T E R

3�

127

wind

45�

45�

wind

1


128 Chapter 3

What’s the chapter about?Chapter 3 is about lines and angles. In Chapter 3, you’ll learn

• properties of parallel and perpendicular lines.

• six ways to prove that lines are parallel.

• how to write an equation of a line with given characteristics.

CHAPTER

3Study Guide

PREVIEW

Are you ready for the chapter?SKILL REVIEW Do these exercises to review key skills that you’ll apply in thischapter. See the given reference page if there is something you don’t understand.

USING ALGEBRA Solve each equation. (Skills Review, p. 789 and 790)

1. 47 + x = 180 2. 135 = 3x º 6 3. m = �2 º5 º

(º76)�

4. �12� = º5��

72�� + b 5. 5x + 9 = 6x º 11 6. 2(x º 1) + 15 = 90

Use the diagram. Write the reason that

supports the statement. (Review pp. 44–46)

7. m™1 = 90°

8. ™2 £ ™4

9. ™2 and ™3 are supplementary.

Write the reason that supports the statement. (Review pp. 96–98)

10. If m™A = 30° and m™B = 30°, then ™A £ ™B.

11. If x + 4 = 9, then x = 5.

12. 3(x + 5) = 3x + 15

PREPARE

Here’s a study strategy!STUDY

STRATEGY

� Review

• linear pair, p. 44

• vertical angles, p. 44

• perpendicular lines,p. 79

� New

• parallel lines, p. 129

• skew lines, p. 129

• parallel planes, p. 129

• transversal, p. 131

• alternate interior angles, p. 131

• alternate exterior angles, p. 131

• consecutive interior angles, p. 131

• flow proof, p. 136

KEY VOCABULARY

Write Sample Questions

Write at least six questions about topics in thechapter. Focus on the concepts that you founddifficult. Include both short-answer questions andmore involved ones. Then answer your questions.

1

243


3.1 Lines and Angles 129

Lines and AnglesRELATIONSHIPS BETWEEN LINES

Two lines are if they are coplanar and do not intersect. Lines thatdo not intersect and are not coplanar are called Similarly, two planesthat do not intersect are called

To write “AB¯̆

is parallel to CD¯̆

,” you write AB¯̆

∞ CD¯̆

. Triangles like those on AB¯̆

and CD¯̆

are used on diagrams to indicate that lines are parallel. Segments and rays are parallel if they lie on parallel lines. For example, AB

Æ∞ CDÆ

.

Identifying Relationships in Space

Think of each segment in the diagram as part of a line. Which of the lines appear to fit the description?

a. parallel to AB¯̆

and contains D

b. perpendicular to AB¯̆

and contains D

c. skew to AB¯̆

and contains D

d. Name the plane(s) that contain Dand appear to be parallel to plane ABE.

SOLUTION

a. CD¯̆

, GH¯̆

, and EF¯̆

are all parallel to AB¯̆

, but only CD¯̆

passes through D and is parallel to AB

¯̆.

b. BC¯̆

, AD¯̆

, AE¯̆

, and BF¯̆

are all perpendicular to AB¯̆

, but only AD¯̆

passes through D and is perpendicular to AB

¯̆.

c. DG¯̆

, DH¯̆

, and DE¯̆

all pass through D and are skew to AB¯̆

.

d. Only plane DCH contains D and is parallel to plane ABE.

E X A M P L E 1

parallel planes.skew lines.

parallel lines

GOAL 1

Identifyrelationships between lines.

Identify anglesformed by transversals.

� To describe andunderstand real-life objects,such as the escalator in Exs. 32–36.

Why you should learn it

GOAL 2

GOAL 1

What you should learn

3.1RE

AL LIFE

RE

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E H

G

CB

A

F

CD

A B

E

Look Back

For help identifyingperpendicular lines, see p. 79.

STUDENT HELP

U W

AB¯̆

and CD¯̆

are parallel lines.

CD¯̆

and BE¯̆

are skew lines.

Planes U and W are parallel planes.

Page 1 of 6

130 Chapter 3 Perpendicular and Parallel Lines

Notice in Example 1 that, although there are many lines through D that are skew to AB

¯̆, there is only one line through D that is parallel to AB

¯̆and there is only one

line through D that is perpendicular to AB¯̆

.

You can use a compass and a straightedge to construct the line that passesthrough a given point and is perpendicular to a given line. In Lesson 6.6, you willlearn why this construction works.

You will learn how to construct a parallel line in Lesson 3.5.

POSTULATE 13 Parallel PostulateIf there is a line and a point not onthe line, then there is exactly oneline through the point parallel tothe given line.

POSTULATE 14 Perpendicular PostulateIf there is a line and a point not on the line, then there is exactly one linethrough the point perpendicular to the given line.

PARALLEL AND PERPENDICULAR POSTULATES

P

l

P

l

A Perpendicular to a Line

Use the following steps to construct a line that passes through a given point P and is perpendicular to a given line l.

Construction

ACTIVITY

Place the compasspoint at P and drawan arc that intersectsline l twice. Labelthe intersections Aand B.

Draw an arc withcenter A. Using thesame radius, draw anarc with center B.Label the intersectionof the arcs Q.

Use a straightedgeto draw PQ

¯̆.

PQ¯̆

fi l.

321

P

lA B

P

A B

q

l

P

A B

q

l

There is exactly one linethrough P parallel to l.

There is exactly one linethrough P perpendicular to l.

APPLICATION LINKVisit our Web site

www.mcdougallittell.comfor more informationabout the parallelpostulate.

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STUDENT HELP

Page 2 of 6


IDENTIFYING ANGLES FORMED BY TRANSVERSALS

A is a line that intersects two or more coplanar lines at differentpoints. For instance, in the diagrams below, line t is a transversal. The anglesformed by two lines and a transversal are given special names.

Consecutive interior angles are sometimes called

Identifying Angle Relationships

List all pairs of angles that fit the description.

a. corresponding b. alternate exterior

c. alternate interior d. consecutive interior

SOLUTION

a. ™1 and ™5™2 and ™6™3 and ™7™4 and ™8

c. ™3 and ™6 ™4 and ™5

E X A M P L E 2

same side interior angles.

transversal

GOAL 2

b. ™1 and ™8™2 and ™7

d. ™3 and ™5 ™4 and ™6

1 243

5 687

t

HOMEWORK HELPVisit our Web site

www.mcdougallittell.comfor extra examples.

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STUDENT HELP

1 243

5 687

t

1 243

5 687

t

1 243

5 687

t

Two angles are if they occupy corresponding positions.For example, angles 1 and 5 arecorresponding angles.

corresponding angles Two angles are if they lie outside the two

lines on opposite sides of thetransversal. Angles 1 and 8 are alternate exterior angles.

anglesalternate exterior

Two angles are if they lie between

the two lines on the same side of thetransversal. Angles 3 and 5 areconsecutive interior angles.

interior anglesconsecutiveTwo angles are

if they lie between the twolines on opposite sides of thetransversal. Angles 3 and 6 arealternate interior angles.

anglesalternate interior

78

12

65

34

Page 3 of 6


1. Draw two lines and a transversal. Identify a pair of alternate interior angles.

2. How are skew lines and parallel lines alike? How are they different?

Match the photo with the corresponding description of the chopsticks.

A. skew B. parallel C. intersecting

3. 4. 5.

In Exercises 6–9, use the diagram at the right.

6. Name a pair of corresponding angles.

7. Name a pair of alternate interior angles.

8. Name a pair of alternate exterior angles.

9. Name a pair of consecutive interior angles.

LINE RELATIONSHIPS Think of each segment in the diagram as part of a

line. Fill in the blank with parallel, skew, or perpendicular.

10. DE¯̆

, AB¯̆

, and GC¯̆

are �� ?��.

11. DE¯̆

and BE¯̆

are �� ?��.

12. BE¯̆

and GC¯̆

are �� ?��.

13. Plane GAD and plane CBE are �� ?��.

IDENTIFYING RELATIONSHIPS Think of each segment in the diagram as

part of a line. There may be more than one right answer.

14. Name a line parallel to QR¯̆

.

15. Name a line perpendicular to QR¯̆

.

16. Name a line skew to QR¯̆

.

17. Name a plane parallel to plane QRS.

APPLYING POSTULATES How many lines can be drawn that fit

the description?

18. through L parallel to JK¯̆

19. through L perpendicular to JK¯̆

PRACTICE AND APPLICATIONS

GUIDED PRACTICE

STUDENT HELP

HOMEWORK HELPExample 1: Exs. 10–20,

27–36Example 2: Exs. 21–26

Extra Practice

to help you masterskills is on p. 807.

STUDENT HELP

Concept Check ✓Skill Check ✓

Vocabulary Check ✓

21

67

34

85

G C

FH

A B

D E

U

S

V

q RW

T

X

J L

K

Page 4 of 6


20. TIGHTROPE WALKING Philippe Petit sometimes uses a longpole to help him balance on the tightrope. Are the rope and the poleat the left intersecting, perpendicular, parallel, or skew?

ANGLE RELATIONSHIPS Complete the statement with corresponding,

alternate interior, alternate exterior, or consecutive interior.

21. ™8 and ™12 are �� ?�� angles.

22. ™9 and ™14 are �� ?�� angles.

23. ™10 and ™12 are �� ?�� angles.

24. ™11 and ™12 are �� ?�� angles.

25. ™8 and ™15 are �� ?�� angles.

26. ™10 and ™14 are �� ?�� angles.

ROMAN NUMERALS Write the Roman numeral that consists of the

indicated segments. Then write the base ten value of the Roman numeral.

For example, the base ten value of XII is 10 + 1 + 1 = 12.

27. Three parallel segments

28. Two non-congruent perpendicular segments

29. Two congruent segments that intersect to form only one angle

30. Two intersecting segments that form vertical angles

31. Four segments, two of which are parallel

ESCALATORS In Exercises 32–36, use the following information.

The steps of an escalator are connected to a chain that runs around a drive wheel, which moves continuously. When a step on an up-escalator reaches the top, it flips over and goes back down to the bottom. Each step is shaped like a wedge, as shown at the right. On each step, let plane A be the plane you stand on.

32. As each step moves around the escalator, is plane Aalways parallel to ground level?

33. When a person is standing on plane A, is it parallel to ground level?

34. Is line l on any step always parallel to l on any other step?

35. Is plane A on any step always parallel to plane A on any other step?

36. As each step moves around the escalator, how many positions are there atwhich plane A is perpendicular to ground level?

89

1011

1213

1415

Roman numeral I V X L M

Base ten value 1 5 10 50 1000

ground level

drive wheel

plane A

l

PHILIPPE PETITwalked more than

2000 feet up an inclinedcable to the Eiffel Tower. The photo above is from a performance in New York City.

RE

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FOCUS ONPEOPLE

Page 5 of 6


37. LOGICAL REASONING If two parallel planesare cut by a third plane, explain why the lines ofintersection are parallel.

38. Writing What does “two lines intersect” mean?

39. CONSTRUCTION Draw a horizontal line land a point P above l. Construct a line through P perpendicular to l.

40. CONSTRUCTION Draw a diagonal line m and a point Q below m.Construct a line through Q perpendicular to m.

41. MULTIPLE CHOICE In the diagram at the right, how many lines can be drawn through point Pthat are perpendicular to line l?

¡A 0 ¡B 1 ¡C 2

¡D 3 ¡E More than 3

42. MULTIPLE CHOICE If two lines intersect, then they must be �� ?��.

¡A perpendicular ¡B parallel ¡C coplanar

¡D skew ¡E None of these

ANGLE RELATIONSHIPS Complete each

statement. List all possible correct answers.

43. ™1 and ��?�� are corresponding angles.

44. ™1 and ��?�� are consecutive interior angles.

45. ™1 and ��?�� are alternate interior angles.

46. ™1 and ��?�� are alternate exterior angles.

47. ANGLE BISECTOR The ray BDÆ̆

bisects™ABC, as shown at the right. Findm™ABD and m™ABC. (Review 1.5 for 3.2)

COMPLEMENTS AND SUPPLEMENTS Find the measures of a complement

and a supplement of the angle. (Review 1.6 for 3.2)

48. 71° 49. 13° 50. 56°

51. 88° 52. 27° 53. 68°

54. 1° 55. 60° 56. 45°

WRITING REASONS Solve the equation and state a reason for each step.

(Review 2.4 for 3.2)

57. x + 13 = 23 58. x º 8 = 17 59. 4x + 11 = 31

60. 2x + 9 = 4x º 29 61. 2(x º 1) + 3 = 17 62. 5x + 7(x º 10) = º94

MIXED REVIEW

P

l

BA

JH

D E F G

C1

TestPreparation

★★Challenge

EXTRA CHALLENGE

www.mcdougallittell.com

A

B

D

C80�

Page 6 of 6


Proof and Perpendicular Lines

COMPARING TYPES OF PROOFS

There is more than one way to write a proof. The two-column proof below isfrom Lesson 2.6. It can also be written as a paragraph proof or as a flow proof. A uses arrows to show the flow of the logical argument. Each reasonin a flow proof is written below the statement it justifies.

Comparing Types of Proof

GIVEN � ™5 and ™6 are a linear pair. ™6 and ™7 are a linear pair.

PROVE � ™5 £ ™7

Method 1 Two-column Proof

Method 2 Paragraph Proof

Because ™5 and ™6 are a linear pair, the Linear Pair Postulate says that ™5 and™6 are supplementary. The same reasoning shows that ™6 and ™7 aresupplementary. Because ™5 and ™7 are both supplementary to ™6, theCongruent Supplements Theorem says that ™5 £ ™7.

Method 3 Flow Proof

E X A M P L E 1

flow proof

GOAL 1

Write differenttypes of proofs.

Prove resultsabout perpendicular lines.

� Make conclusions fromthings you see in real life,such as the reflection in Ex. 28.


GOAL 2

GOAL 1


3.2

1. ™5 and ™6 are a linear pair.™6 and ™7 are a linear pair.

2. ™5 and ™6 are supplementary.™6 and ™7 are supplementary.

3. ™5 £ ™7

Statements Reasons

1. Given

2. Linear Pair Postulate

3. Congruent Supplements Theorem

™6 and ™7 are a

linear pair.

™6 and ™7 are

supplementary.

™5 and ™6 are a

linear pair.

™5 £ ™7

™5 and ™6 are

supplementary.

Given Linear Pair Postulate

Given Linear Pair Postulate

CongruentSupplements Theorem

5 76

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Page 1 of 6

3.2 Proof and Perpendicular Lines 137

PROVING RESULTS ABOUT PERPENDICULAR LINES

You will prove Theorem 3.2 and Theorem 3.3 in Exercises 17–19.

Proof of Theorem 3.1

Write a proof of Theorem 3.1.

SOLUTION

GIVEN � ™1 £ ™2, ™1 and ™2 are a linear pair.

PROVE � g fi h

Plan for Proof Use m™1 + m™2 = 180° and m™1 = m™2 to show m™1 = 90°.

E X A M P L E 2

GOAL 2

THEOREM 3.1

If two lines intersect to form a linear pair ofcongruent angles, then the lines are perpendicular.

THEOREM 3.2

If two sides of two adjacent acute angles areperpendicular, then the angles are complementary.

THEOREM 3.3

If two lines are perpendicular, then they intersect toform four right angles.

THEOREMS

h

g

Proof

1 2

g

h

™1 and ™2 are a

linear pair.

m™1 + m™2 = 180°

™1 and ™2 are

supplementary.

Given

Def. of supplementary √

m™1 = m™2

™1 £ ™2

Given

g fi h

Def. of fi lines

Def. of £ angles

m™1 + m™1 = 180°

Substitution prop. of equality

2 • (m™1) = 180°

Distributive prop.

Def. of right angle

m™1 = 90°

Div. prop. of equality

™1 is a right ™.

Linear Pair Postulate

STUDENT HELP

Study Tip

When you write acomplicated proof, it mayhelp to write a plan first.The plan will also helpothers to understandyour proof.

g fi h

Page 2 of 6

1. Define perpendicular lines.

2. Which postulate or theorem guarantees that there is only one line that can beconstructed perpendicular to a given line from a given point not on the line?

Write the postulate or theorem that justifies the statement about the

diagram.

3. ™1 £ ™2 4. j fi k

Write the postulate or theorem that justifies the statement, given that g fi h.

5. m™5 + m™6 = 90° 6. ™3 and ™4 are right angles.

Find the value of x.

7. 8. 9.

10. ERROR ANALYSIS It is given that ™ABC £ ™CBD. A student concludesthat because ™ABC and ™CBD are congruent adjacent angles, AB

¯̆fi CB

¯̆.

What is wrong with this reasoning? Draw a diagram to support your answer.

70�x �

x �

45�x �

GUIDED PRACTICE

g

h6

5


You have now studied three types of proofs.

1. TWO-COLUMN PROOF This is the most formal type of proof. It listsnumbered statements in the left column and a reason for eachstatement in the right column.

2. PARAGRAPH PROOF This type of proof describes the logicalargument with sentences. It is more conversational than a two-column proof.

3. FLOW PROOF This type of proof uses the same statements andreasons as a two-column proof, but the logical flow connectingthe statements is indicated by arrows.

TYPES OF PROOFS

Vocabulary Check ✓Concept Check ✓

Skill Check ✓

1 2

j

k

43

g

h

CONCEPT

SUMMARY

12

Page 3 of 6


USING ALGEBRA Find the value of x.

11. 12. 13.

LOGICAL REASONING What can you conclude about the labeled angles?

14. ABÆ

fi CBÆ

15. n fi m 16. h fi k

17. DEVELOPING PARAGRAPH PROOF Fill in thelettered blanks to complete the proof of Theorem 3.2.

GIVEN � BAÆ̆

fi BCÆ̆

PROVE � ™3 and ™4 are complementary.

Because BAÆ̆

fi BCÆ̆

, ™ABC is a ��a.�� and m™ABC = ��b.��. According to the ��c.�� Postulate, m™3 + m™4 = m™ABC. So, by the substitution property of equality, ��d.�� + ��e.�� = ��f.��. By definition, ™3 and ™4 are complementary.

18. DEVELOPING FLOW PROOF Fill in the lettered blanks to complete theproof of part of Theorem 3.3. Because the lines are perpendicular, theyintersect to form a right angle. Call that ™1.

GIVEN � j fi k, ™1 and ™2 are a linear pair.

PROVE � ™2 is a right angle.

1

h

k234m

31 2

n

21

AD

CB

x �55�

x �60�

x �

xyxy


34

B C

DA

™1 and ™2 are a

linear pair.

m™1 + m™2 = 180°

��a.��

Given Given

��b.��

m™1 = 90°

j fi k

å1 is a right å.

Def. of fi lines


��c.��

��f.��d.�� Subtr. prop. of equality

90° + m™2 = 180° ��e.��

Linear Pair Postulate

Extra Practice


STUDENT HELP

1

2

j

k

STUDENT HELP

HOMEWORK HELPExample 1: Exs. 17–23Example 2: Exs. 11–19,

24, 25

Page 4 of 6


19. DEVELOPING TWO-COLUMN PROOF Fill in the blanks to complete the proof of part of Theorem 3.3.

GIVEN �™1 is a right angle.

PROVE �™3 is a right angle.

DEVELOPING PROOF In Exercises 20–23, use the following information.

Dan is trying to figure out how to prove that ™5 £ ™6 below. First he wroteeverything that he knew about the diagram, as shown below in blue.

GIVEN m fi n, ™3 and ™4 are complementary.

PROVE ™5 £ ™6

20. Write a justification for each statement Dan wrote in blue.

21. After writing all he knew, Dan wrote what he was supposed to prove in red.He also wrote ™4 £ ™6 because he knew that if ™4 £ ™6 and ™4 £ ™5,then ™5 £ ™6. Write a justification for this step.

22. How can you use Dan’s blue statements to prove that ™4 £ ™6?

23. Copy and complete Dan’s flow proof.

24. CIRCUIT BOARDS The diagramshows part of a circuit board. Write any type of proof.

GIVEN � ABÆ

fi BCÆ, BC

Æfi CD

Æ

PROVE � ™7 £ ™8

Plan for Proof Show that ™7 and ™8 are both right angles.

1

3

1. ™1 and ™3 are vertical angles.

2. �� ?��

3. m™1 = m™3

4. ™1 is a right angle.

5. �� ?��

6. �� ?��

7. �� ?��

Statements Reasons

1. Definition of vertical angles

2. Vertical Angles Theorem

3. �� ?��

4. �� ?��

5. Definition of right angle

6. Substitution prop. of equality

7. Definition of right angle

36

m

n

54

CIRCUIT BOARDSThe lines on circuit

boards are made of metaland carry electricity. Thelines must not touch eachother or the electricity willflow to the wrong place,creating a short circuit.

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FOCUS ONAPPLICATIONS

A

B C

D

7

8

m fi n ™3 and ™6 are complementary.

™3 and ™4 are complementary.

™4 and ™5 are vertical angles. ™4 £ ™5

™4 £ ™6 ™5 £ ™6


www.mcdougallittell.comfor help with writingproofs in Exs. 17–24.

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STUDENT HELP

Page 5 of 6


25. WINDOW REPAIR Cathy is fixing awindow frame. She fit two strips of woodtogether to make the crosspieces. For theglass panes to fit, each angle of thecrosspieces must be a right angle. MustCathy measure all four angles to be surethey are all right angles? Explain.

26. MULTIPLE CHOICE Which of the following is true if g fi h?

¡A m™1 + m™2 > 180°

¡B m™1 + m™2 < 180°

¡C m™1 + m™2 = 180°

¡D Cannot be determined

27. MULTIPLE CHOICE Which of the following must be true if m™ACD = 90°?

I. ™BCE is a right angle.

II. AE¯̆

fi BD¯̆

III. ™BCA and ™BCE are complementary.

¡A I only ¡B I and II only ¡C III only

¡D I, II, and III ¡E None of these

28. REFLECTIONS Ann has a full-length mirror resting against the wallof her room. Ann notices that thefloor and its reflection do not form astraight angle. She concludes that themirror is not perpendicular to thefloor. Explain her reasoning.

ANGLE MEASURES Complete the statement given that s fi t. (Review 2.6 for 3.3)

29. If m™1 = 38°, then m™4 = � ?��.

30. m™2 = � ?��

31. If m™6 = 51°, then m™1 = � ?��.

32. If m™3 = 42°, then m™1 = � ?��.

ANGLES List all pairs of angles that fit the description. (Review 3.1)

33. Corresponding angles

34. Alternate interior angles

35. Alternate exterior angles

36. Consecutive interior angles

MIXED REVIEW

3

6

s

124

5 t

231

675

4

8

TestPreparation

EXTRA CHALLENGE


A

DB C

E

21

g

h

★★Challenge

Page 6 of 6

3.3 Parallel Lines and Transversals 143

Parallel Linesand Transversals

PROPERTIES OF PARALLEL LINES

In the activity on page 142, you may have discovered the following results.

You are asked to prove Theorems 3.5, 3.6, and 3.7 in Exercises 27–29.

THEOREMS ABOUT PARALLEL LINES

GOAL 1

Prove and useresults about parallel linesand transversals.

Use properties ofparallel lines to solve real-life problems, such as estimating Earth’scircumference in Example 5.

� Properties of parallel lineshelp you understand howrainbows are formed, as in Ex. 30.


GOAL 2

GOAL 1


3.3RE

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POSTULATE 15 Corresponding Angles PostulateIf two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.

POSTULATE

THEOREM 3.4 Alternate Interior AnglesIf two parallel lines are cut by a transversal,then the pairs of alternate interior angles arecongruent.

THEOREM 3.5 Consecutive Interior AnglesIf two parallel lines are cut by a transversal,then the pairs of consecutive interior angles are supplementary.

THEOREM 3.6 Alternate Exterior AnglesIf two parallel lines are cut by a transversal,then the pairs of alternate exterior angles are congruent.

THEOREM 3.7 Perpendicular TransversalIf a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other.

j

k

h

THEOREMS ABOUT PARALLEL LINES

m™5 + m™6 = 180°

™7 £ ™8

j fi k

™1 £ ™2

™3 £ ™4

Page 1 of 7

Proving the Alternate Interior Angles Theorem

Prove the Alternate Interior Angles Theorem.

SOLUTION

GIVEN � p ∞ q

PROVE � ™1 £ ™2

Using Properties of Parallel Lines

Given that m™5 = 65°, find each measure.Tell which postulate or theorem you use.

a. m™6 b. m™7

c. m™8 d. m™9

SOLUTION

a. m™6 = m™5 = 65° Vertical Angles Theorem

b. m™7 = 180° º m™5 = 115° Linear Pair Postulate

c. m™8 = m™5 = 65° Corresponding Angles Postulate

d. m™9 = m™7 = 115° Alternate Exterior Angles Theorem

Classifying Leaves

BOTANY Some plants are classified by the arrangement of the veins in their leaves. In the diagram of the leaf, j ∞ k. What is m™1?

SOLUTION

m™1 + 120° = 180° Consecutive Interior Angles Theorem

m™1 = 60° Subtract.

E X A M P L E 3

E X A M P L E 2

E X A M P L E 1


STUDENT HELP

Study Tip

When you prove atheorem, the hypothesesof the theorem becomesthe GIVEN, and theconclusion is what youmust PROVE.

Statements Reasons

1. p ∞ q 1. Given

2. ™1 £ ™3 2. Corresponding Angles Postulate

3. ™3 £ ™2 3. Vertical Angles Theorem

4. ™1 £ ™2 4. Transitive Property of Congruence

BOTANY Botanistsstudy plants and

environmental issues suchas conservation, weedcontrol, and re-vegetation.

CAREER LINKwww.mcdougallittell.com

INTE

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RE

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RE

AL LIFE

FOCUS ONCAREERS

67 5

98

j k

1120�

p

q

Page 2 of 7


PROPERTIES OF SPECIAL PAIRS OF ANGLES

Using Properties of Parallel Lines

Use properties of parallel lines to find the value of x.

SOLUTION

m™4 = 125° Corresponding Angles Postulate

m™4 + (x + 15)° = 180° Linear Pair Postulate

125° + (x + 15)° = 180° Substitute.

x = 40 Subtract.

Estimating Earth’s Circumference

HISTORY CONNECTION Eratosthenes wasa Greek scholar. Over 2000 years ago, heestimated Earth’s circumference by using the fact that the Sun’s rays are parallel.

Eratosthenes chose a day when the Sunshone exactly down a vertical well in Syeneat noon. On that day, he measured the anglethe Sun’s rays made with a vertical stick inAlexandria at noon. He discovered that

m™2 ≈ �510� of a circle.

By using properties of parallel lines, heknew that m™1 = m™2. So he reasoned that

m™1 ≈ �510� of a circle.

At the time, the distance from Syene to Alexandria was believed to be 575 miles.

�510� of a circle ≈

Earth’s circumference ≈ 50(575 miles) Use cross product property.

≈ 29,000 miles

How did Eratosthenes know that m™1 = m™2?

SOLUTION

Because the Sun’s rays are parallel, l1 ∞ l2. Angles 1 and 2 are alternate interiorangles, so ™1 £ ™2. By the definition of congruent angles, m™1 = m™2.

575 miles��Earth’s circumference

E X A M P L E 5

E X A M P L E 4

GOAL 2

APPLICATION LINKVisit our Web site

www.mcdougallittell.comfor more informationabout Eratosthenes’estimate in Example 5.

INTE

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STUDENT HELP

125�

4(x � 15)�

UsingAlgebra

xyxy

center of Earth

shadow stick

1

∠2

well

sunlight

sunlight

Not drawn to scale

L1

L2

Page 3 of 7


1. Sketch two parallel lines cut by a transversal. Label a pair of consecutiveinterior angles.

2. In the figure at the right, j ∞ k. How many angle measures must be given inorder to find the measure of everyangle? Explain your reasoning.

State the postulate or theorem that justifies

the statement.

3. ™2 £ ™7 4. ™4 £ ™5

5. m™3 + m™5 = 180° 6. ™2 £ ™6

7. In the diagram of the feather below, lines pand q are parallel. What is the value of x?

USING PARALLEL LINES Find m™1 and m™2. Explain your reasoning.

8. 9. 10.

USING PARALLEL LINES Find the values of x and y. Explain your reasoning.

11. 12. 13.

14. 15. 16.

130� x �

y �80�

x �y �

65�x � y �

x �y �

x �

y �

109�

x �

y �

67�

1 2

118�1 2

82�

12

135�


GUIDED PRACTICE


Concept Check ✓

Skill Check ✓

STUDENT HELP

HOMEWORK HELPExample 1: Exs. 27–29Example 2: Exs. 8–17Example 3: Exs. 8–17 Example 4: Exs. 18–26Example 5: Ex. 30

Extra Practice


STUDENT HELP

5 67 8

1 23 4

5 74 6

9 118 10

j k

p q

133�x �

Page 4 of 7


17. USING PROPERTIES OF PARALLEL LINESUse the given information to find themeasures of the other seven angles in thefigure at the right.

GIVEN � j ∞ k, m™1 = 107°

USING ALGEBRA Find the value of y.

18. 19. 20.

USING ALGEBRA Find the value of x.

21. 22. 23.

24. 25. 26.

27. DEVELOPING PROOF Completethe proof of the Consecutive Interior Angles Theorem.

GIVEN � p ∞ q

PROVE � ™1 and ™2 are supplementary.

126�7(x � 7)�

94�

(13x � 5)�

89�

(5x � 24)�

135�

(12x � 9)�

(2x � 10)�70�

(3x � 14)�

xyxy

120�

6y �115� 5y �70� 2y �

xyxy

1107�

j k

2

3 4

5 6

7 8


www.mcdougallittell.comfor help with provingtheorems in Exs. 27–29.

INTE

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STUDENT HELP

Statements Reasons

1. �� ?�� 1. Given

2. ™1 £ ™3 2. �� ?��

3. �� ?�� 3. Definition of congruent angles

4. �� ?�� 4. Definition of linear pair

5. m™3 + m™2 = 180° 5. �� ?��

6. �� ?�� 6. Substitution prop. of equality

7. ™1 and ™2 are supplementary. 7. �� ?��

3 2

1

p

q

Page 5 of 7


PROVING THEOREMS 3.6 AND 3.7 In Exercises 28 and 29, complete the

proof.

28. To prove the Alternate ExteriorAngles Theorem, first show that ™1 £ ™3. Then show that ™3 £ ™2. Finally, show that ™1 £ ™2.

GIVEN � j ∞ k

PROVE � ™1 £ ™2

30. FORMING RAINBOWSWhen sunlight enters a drop of rain, different colorsleave the drop at different angles. That’s what makes a rainbow. For red light,m™2 = 42°. What is m™1? How do you know?

31. MULTI-STEP PROBLEM You are designing a lunch box like the one below.

a. The measure of ™1 is 70°. What is the measure of ™2? What is themeasure of ™3?

b. Writing Explain why ™ABC is a straight angle.

32. USING PROPERTIES OF PARALLEL LINESUse the given information to find themeasures of the other labeled angles in the figure. For each angle, tell whichpostulate or theorem you used.

GIVEN � PQÆ

∞ RSÆ,

LMÆ

fi NKÆ

, m™1 = 48°

TestPreparation

★★Challenge

EXTRA CHALLENGE


STUDENT HELP

Study Tip

When you prove atheorem you may useany previous theorem,but you may not use theone you’re proving.

1 2

L

34

5

M

N K

P R

S

q

67

1

32

j

k

29. To prove the PerpendicularTransversal Theorem, show that™1 is a right angle, ™1 £ ™2,™2 is a right angle, and finallythat p fi r.

GIVEN � p fi q, q ∞ r

PROVE � p fi r

1

2

q

r

p

shadow

sunlight

sunlight

rain

1

2

1 23

1

23

AB

C

Page 6 of 7


ANGLE MEASURES ™1 and ™2 are supplementary. Find m™2. (Review 1.6)

33. m™1 = 50° 34. m™1 = 73° 35. m™1 = 101°

36. m™1 = 107° 37. m™1 = 111° 38. m™1 = 118°

CONVERSES Write the converse of the statement. (Review 2.1 for 3.4)

39. If the measure of an angle is 19°, then the angle is acute.

40. I will go to the park if you go with me.

41. I will go fishing if I do not have to work.

FINDING ANGLES Complete the statement,

given that DEÆ̆

fi DGÆ̆

and AB¯̆

fi DCÆ̆

. (Review 2.6)

42. If m™1 = 23°, then m™2 = ��?��.

43. If m™4 = 69°, then m™3 = ��?��.

44. If m™2 = 70°, then m™4 = ��?��.

Complete the statement. (Lesson 3.1)





5. PROOF Write a plan for a proof. (Lesson 3.2)

GIVEN � ™1 £ ™2

PROVE � ™3 and ™4 are right angles.

Find the value of x. (Lesson 3.3)

6. 7. 8.

9. FLAG OF PUERTO RICO Sketch the flag of Puerto Rico shown at the right. Given that m™3 = 55°, determine the measure of ™1. Justify each step in your argument. (Lesson 3.3)

(7x � 15)�

81�

(2x � 1)�

151�2x�

138�

QUIZ 1 Self-Test for Lessons 3.1–3.3

MIXED REVIEW

1 2

3

4A B

C

E

D

G

5 687

1 243

1 23 4

1

23

Page 7 of 7


Proving Linesare Parallel

PROVING LINES ARE PARALLEL

To use the theorems you learned in Lesson 3.3, you must first know that two linesare parallel. You can use the following postulate and theorems to prove that twolines are parallel.

The following theorems are converses of those in Lesson 3.3. Remember that theconverse of a true conditional statement is not necessarily true. Thus, each of thefollowing must be proved to be true. Theorems 3.8 and 3.9 are proved inExamples 1 and 2. You are asked to prove Theorem 3.10 in Exercise 30.

GOAL 1

Prove that twolines are parallel.

Use properties ofparallel lines to solve real-life problems, such asproving that prehistoricmounds are parallel in Ex. 19.

� Properties of parallel lineshelp you predict the paths ofboats sailing into the wind, asin Example 4.


GOAL 2

GOAL 1


3.4RE

AL LIFE

RE

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POSTULATE 16 Corresponding Angles ConverseIf two lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel.

POSTULATE

j

k

j ∞ k

THEOREM 3.8 Alternate Interior Angles ConverseIf two lines are cut by a transversal so that alternate interior angles are congruent, then the lines are parallel.

THEOREM 3.9 Consecutive Interior Angles ConverseIf two lines are cut by a transversal so that consecutive interior angles are supplementary, then the lines are parallel.

THEOREM 3.10 Alternate Exterior Angles ConverseIf two lines are cut by a transversal so that alternate exterior angles are congruent, then the lines are parallel.

j

k

j

k

j

k

THEOREMS ABOUT TRANSVERSALS

If ™1 £ ™3, then j ∞ k.

If ™4 £ ™5, then j ∞ k.

If m™1 + m™2 = 180°,then j ∞ k.

Page 1 of 7

3.4 Proving Lines are Parallel 151

Proof of the Alternate Interior Angles Converse

Prove the Alternate Interior Angles Converse.

SOLUTION

GIVEN � ™1 £ ™2

PROVE � m ∞ n

. . . . . . . . .

When you prove a theorem you may use only earlier results. For example, toprove Theorem 3.9, you may use Theorem 3.8 and Postulate 16, but you may not use Theorem 3.9 itself or Theorem 3.10.

Proof of the Consecutive Interior Angles Converse

Prove the Consecutive Interior Angles Converse.

SOLUTION

GIVEN � ™4 and ™5 are supplementary.

PROVE � g ∞ h

Paragraph Proof You are given that ™4 and ™5 are supplementary. By theLinear Pair Postulate, ™5 and ™6 are also supplementary because they form alinear pair. By the Congruent Supplements Theorem, it follows that ™4 £ ™6.Therefore, by the Alternate Interior Angles Converse, g and h are parallel.

Applying the Consecutive Interior Angles Converse

Find the value of x that makes j ∞ k.

SOLUTION

Lines j and k will be parallel if the marked angles are supplementary.

x° + 4x° = 180°

5x = 180

x = 36

� So, if x = 36, then j ∞ k.

E X A M P L E 3

E X A M P L E 2

E X A M P L E 1

1. ™1 £ ™2

2. ™2 £ ™3

3. ™1 £ ™3

4. m ∞ n

Statements Reasons

1. Given

2. Vertical Angles Theorem

3. Transitive Property of Congruence

4. Corresponding Angles Converse

32

1 n

m

6

4 h

g5

4x �

j

kx �

UsingAlgebra

xyxy

Proof

Proof

Page 2 of 7


USING THE PARALLEL CONVERSES

Using the Corresponding Angles Converse

SAILING If two boats sail at a 45° angle to the wind as shown, and thewind is constant, will their paths ever cross? Explain.

SOLUTION

Because corresponding angles are congruent, the boats’ paths are parallel.Parallel lines do not intersect, so the boats’ paths will not cross.

Identifying Parallel Lines

Decide which rays are parallel.

SOLUTION

� ™BEH and ™DHG are corresponding angles, but they are notcongruent, so EB

Æ̆and HD

Æ̆are not parallel.

� ™AEH and ™CHG are congruent corresponding angles,

so EAÆ̆

∞ HCÆ̆

.

E X A M P L E 5

RE

AL LIFE

RE

AL LIFE

E X A M P L E 4

GOAL 2

GHE

AB C

D

58� 61�62� 59�

wind

45�

45�

GHE

B D

58� 61�

GHE

A C

120� 120�

a. Decide whether EBÆ̆

∞ HDÆ̆

.

m™BEH = 58°

m™DHG = 61°

m™AEH = 62°+ 58°

= 120°

m™CHG = 59° + 61°

= 120°

b. Decide whether EAÆ̆

∞ HCÆ̆

.

a. Is EBÆ̆

parallel to HDÆ̆

?

b. Is EAÆ̆

parallel to HCÆ̆

?



INTE

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STUDENT HELP

Page 3 of 7


1. What are parallel lines?

2. Write the converse of Theorem 3.8. Is the converse true?

Can you prove that lines p and q are parallel? If so, describe how.

3. 4. 5.

6. 7. 8.

9. Find the value of x that makes j ∞ k. Whichpostulate or theorem about parallel linessupports your answer?

LOGICAL REASONING Is it possible to prove that lines m and n are

parallel? If so, state the postulate or theorem you would use.

10. 11. 12.

13. 14. 15.

USING ALGEBRA Find the value of x that makes r ∞ s.

16. 17. 18.

s3x �

(2x � 20)�r

x �

r(90 � x)�

s

2x �

s

r

x �

xyxy

n

m

n

m

n

m

nmnmnm


123�

57�

p

q

105�

105�

p

q62�

62�

p

q

p

q

p

q

p

q

GUIDED PRACTICE

3x �x �

j k

Concept Check ✓Skill Check ✓


Extra Practice


STUDENT HELP

STUDENT HELP

HOMEWORK HELPExample 1: Exs. 28, 30Example 2: Exs. 28, 30Example 3: Exs. 10–18Example 4: Exs. 19, 29,

31Example 5: Exs. 20–27

Page 4 of 7


19. ARCHAEOLOGY A farm lane in Ohio crosses two long, straightearthen mounds that may have beenbuilt about 2000 years ago. Themounds are about 200 feet apart,and both form a 63° angle with thelane, as shown. Are the moundsparallel? How do you know?

LOGICAL REASONING Is it possible to prove that lines a and b are

parallel? If so, explain how.

20. 21. 22.

23. 24. 25.

LOGICAL REASONING Which lines, if any, are parallel? Explain.

26. 27.

28. PROOF Complete the proof.

GIVEN � ™1 and ™2 are supplementary.

PROVE � l1 ∞ l2

A

38�

D

B77� 114�

EC

29�

60�

a

b60�

a

60� b

60�114�

b

a

66�48�

106� 49�

ba

54�

a b

37�

b

143�

a

THE GREATSERPENT MOUND,

an archaeological moundnear Hillsboro, Ohio, is 2 to 5 feet high, and is nearly

20 feet wide. It is over �14�

mile long.

APPLICATION LINKwww.mcdougallittell.com

INTE

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RE

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RE

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FOCUS ONAPPLICATIONS

2 3 1

l1 l2

1. ™1 and ™2 are supplementary.

2. ™1 and ™3 are a linear pair.

3. �� ?��

4. �� ?��

5. l1 ∞ l2

Statements Reasons

1. �� ?��

2. Definition of linear pair

3. Linear Pair Postulate

4. Congruent Supplements Theorem

5. �� ?��

j

31�

69� 68�

32�

k mn

63�

63�

Page 5 of 7


29. BUILDING STAIRS One way to build stairs is to attach triangularblocks to an angled support, as shown at the right. If the support makes a 32° angle with the floor, what must m™1 be so the step will be parallel to the floor? The sides of the angled support are parallel.

30. PROVING THEOREM 3.10 Write a two-column proof for the Alternate Exterior AnglesConverse: If two lines are cut by a transversal sothat alternate exterior angles are congruent, thenthe lines are parallel.

GIVEN � ™4 £ ™5

PROVE � g ∞ h

Plan for Proof Show that ™4 is congruent to ™6, show that ™6 is congruentto ™5, and then use the Corresponding Angles Converse.

31. Writing In the diagram at the right, m™5 = 110° and m™6 = 110°. Explain why p ∞ q.

LOGICAL REASONING Use the information given in the diagram.

32. What can you prove about ABÆ

andCDÆ

? Explain.

PROOF Write a proof.

34. GIVEN � m™7 = 125°, m™8 = 55°

PROVE � j ∞ k

36. TECHNOLOGY Use geometry software to construct a line l, a point Pnot on l, and a line n through P parallel to l. Construct a point Q on l

and construct PQ¯̆

. Choose a pair of alternate interior angles and constructtheir angle bisectors. Are the bisectors parallel? Make a conjecture. Write aplan for a proof of your conjecture.

SOFTWARE HELPVisit our Web site

www.mcdougallittell.comto see instructions forseveral softwareapplications.

INTE

RNET

STUDENT HELP

4

h

6

5

g

A

B

E

D

C s

12

34

r

7

k8

j

1

2

ad

b

c

3

p5

6 q

triangularblock1

232�

35. GIVEN � a ∞ b, ™1 £ ™2

PROVE � c ∞ d

33. What can you prove about ™1,™2, ™3, and ™4? Explain.

Page 6 of 7


37. MULTIPLE CHOICE What is the converse of the following statement?

If ™1 £ ™2, then n ∞ m.

¡A ™1 £ ™2 if and only if n ∞ m. ¡B If ™2 £ ™1, then m ∞ n.

¡C ™1 £ ™2 if n ∞ m. ¡D ™1 £ ™2 only if n ∞ m.

38. MULTIPLE CHOICE What value of x would make lines l1 and l2 parallel?

¡A 13 ¡B 35 ¡C 37

¡D 78 ¡E 102

39. SNOW MAKING To shoot the snow as far as possible, each snowmakerbelow is set at a 45° angle. The axles of the snowmakers are all parallel. It ispossible to prove that the barrels of the snowmakers are also parallel, but theproof is difficult in 3 dimensions. To simplify the problem, think of theillustration as a flat image on a piece of paper. The axles and barrels arerepresented in the diagram on the right. Lines j and l2 intersect at C.

GIVEN � l1 ∞ l2, m™A = m™B = 45°

PROVE � j ∞ k

FINDING THE MIDPOINT Use a ruler to draw a line segment with the given

length. Then use a compass and straightedge to construct the midpoint of

the line segment. (Review 1.5 for 3.5)

40. 3 inches 41. 8 centimeters 42. 5 centimeters 43. 1 inch

44. CONGRUENT SEGMENTS Find the value of x if AB

Æ£ AD

Æand CD

Æ£ AD

Æ. Explain

your steps. (Review 2.5)

IDENTIFYING ANGLES Use the diagram to complete the statement.

(Review 3.1)

45. ™12 and �� ?�� are alternate exterior angles.

46. ™10 and �� ?�� are corresponding angles.

47. ™10 and �� ?�� are alternate interior angles.

48. ™9 and �� ?�� are consecutive interior angles.

MIXED REVIEW

★★Challenge

TestPreparation

(2x � 4)� l1

l2(3x � 9)�

45�

l1

l2

45�A

B

j k

C

5 6

12

7 8

119 10

9x � 11

A

B C

D

6x � 1

EXTRA CHALLENGE

www.mcdougallittell.comA

B

j

k

l

l

2

1

Page 7 of 7

3.5 Using Properties of Parallel Lines 157

Using Propertiesof Parallel Lines

USING PARALLEL LINES IN REAL LIFE

When a team of rowers competes, each rowerkeeps his or her oars parallel to the adjacentrower’s oars. If any two adjacent oars on thesame side of the boat are parallel, does thisimply that any two oars on that side areparallel? This question is examined below.

Example 1 justifies Theorem 3.11, and youwill prove Theorem 3.12 in Exercise 38.

Proving Two Lines are Parallel

Lines m, n, and k represent three of the oarsabove. m ∞ n and n ∞ k. Prove that m ∞ k.

SOLUTION

GIVEN � m ∞ n, n ∞ k

PROVE � m ∞ k

E X A M P L E 1

GOAL 1

Use properties ofparallel lines in real-lifesituations, such as building aCD rack in Example 3.

Construct parallellines using straightedge andcompass.

� To understand how light bends when it passesthrough glass or water, as in Ex. 42.


GOAL 2

GOAL 1


3.5RE

AL LIFE

RE

AL LIFE1. m ∞ n

2. ™1 £ ™2

3. n ∞ k

4. ™2 £ ™3

5. ™1 £ ™3

6. m ∞ k

Statements Reasons

1. Given

2. Corresponding Angles Postulate

3. Given

4. Corresponding Angles Postulate

5. Transitive Property of Congruence

6. Corresponding Angles Converse

THEOREM 3.11

If two lines are parallel to the same line, then they are parallel to each other.

THEOREM 3.12

In a plane, if two lines are perpendicular to the same line, then they are parallel to each other.

THEOREMS ABOUT PARALLEL AND PERPENDICULAR LINES

p q r

mn

k

12

3

m n

p

If p ∞ q and q ∞ r, then p ∞ r.

If m fi p and n fi p, then m ∞ n.

Page 1 of 8


Explaining Why Steps are Parallel

In the diagram at the right, each step is parallelto the step immediately below it and thebottom step is parallel to the floor. Explainwhy the top step is parallel to the floor.

SOLUTION

You are given that k1 ∞ k2 and k2 ∞ k3. By transitivity of parallel lines, k1 ∞ k3. Since k1 ∞ k3 and k3 ∞ k4, it follows that k1 ∞ k4.So, the top step is parallel to the floor.

Building a CD Rack

You are building a CD rack. You cut thesides, bottom, and top so that each corner is composed of two 45° angles. Prove thatthe top and bottom front edges of the CDrack are parallel.

SOLUTION

GIVEN � m™1 = 45°, m™2 = 45°m™3 = 45°, m™4 = 45°

PROVE � BAÆ

∞ CDÆ

E X A M P L E 3

E X A M P L E 2

m™ABC = m™1 + m™2

Angle Addition Postulate

™ABC is a

right angle.

Def. of right angle

m™BCD = m™3 + m™4

Angle Addition Postulate

m™1 = 45°

m™2 = 45°

Given

m™3 = 45°

m™4 = 45°

Given

m™ABC = 90°

Substitution property

BAÆ

fi BCÆ

Def. of fi lines

™BCD is a

right angle.

Def. of right angle

m™BCD = 90°

Substitution property

BCÆ

fi CDÆ

Def. of fi lines

BAÆ

∞ CDÆ

In a plane, 2 lines fi to the same line are ∞.

LogicalReasoning

Proof

k1

k2

k3

k4

B 1 A

C D

2

34

Page 2 of 8


CONSTRUCTING PARALLEL LINES

To construct parallel lines, you first need to know how to copy an angle.

In Chapter 4, you will learn why the Copying an Angle construction works. You can use the Copying an Angle construction to construct two congruentcorresponding angles. If you do, the sides of the angles will be parallel.

GOAL 2

Copying an Angle

Use these steps to construct an angle that is congruent to a given ™A.

A

D

C

B

E

F

A

D

C

B

E

F

A

D

C

B

E

A

D

Construction

ACTIVITY

Parallel Lines

Use these steps to construct a line that passes through a given point P and is parallel to a given line m.

P

m

Rq

T

SP

m

Rq

S

T

P

m

Rq

P

m

Rq

Construction

ACTIVITY

Draw a line. Label apoint on the line D.

1 Draw an arc withcenter A. Label B andC. With the sameradius, draw an arcwith center D. Label E.

2 Draw an arc withradius BC and centerE. Label the inter-section F.

3 Draw DFÆ̆

. ™EDF £ ™BAC.

4

Draw points Q and Ron m. Draw PQ

¯̆.

1 Draw an arc with thecompass point at Qso that it crosses QP

Æ̆

and QRÆ̆

.

2 Copy ™PQR on QP¯̆

as shown. Be surethe two angles arecorresponding. Labelthe new angle ™TPSas shown.

3 Draw PS¯̆

. Because™TPS and ™PQRare congruentcorresponding

angles, PS¯̆

∞ QR¯̆

.

4

Page 3 of 8


1. Name two ways, from this lesson, to prove that two lines are parallel.

State the theorem that you can use to prove that r is parallel to s.

2. GIVEN � r ∞ t, t ∞ s 3. GIVEN � r fi t, t fi s

Determine which lines, if any, must be parallel. Explain your reasoning.

4. 5.

6. Draw any angle ™A. Then construct ™B congruent to ™A.

7. Given a line l and a point P not on l, describe how to construct a linethrough P parallel to l.

LOGICAL REASONING State the postulate or theorem that allows you

to conclude that j ∞ k.

8. GIVEN � j ∞ n, k ∞ n 9. GIVEN � j fi n, k fi n 10. GIVEN � ™1 £ ™2

SHOWING LINES ARE PARALLEL Explain how you would show that k ∞ j.State any theorems or postulates that you would use.

11. 12. 13.

14. Writing Make a list of all the ways you know to prove that two lines are parallel.

n

j k

52�

52�

nk99�

99� j

n

j112�

112� k

kj

12

nkjk nj


m1 l1m2

l2

m1 l1m2

l2

tr st r s

GUIDED PRACTICE

Skill Check ✓Concept Check ✓

Extra Practice


STUDENT HELP

STUDENT HELP

HOMEWORK HELPExample 1: Exs. 8–24Example 2: Exs. 8–24Example 3: Exs. 8–24

if they are || to the same line, if they are fi to the same line

Page 4 of 8


SHOWING LINES ARE PARALLEL Explain how you would show that k ∞ j.

15. 16. 17.

USING ALGEBRA Explain how you would show that g ∞ h.

18. 19. 20.

NAMING PARALLEL LINES Determine which lines, if any, must be parallel.

Explain your reasoning.

21. 22.

23. 24.

CONSTRUCTIONS Use a straightedge to draw an angle that fits the

description. Then use the Copying an Angle construction on page 159 to

copy the angle.

25. An acute angle 26. An obtuse angle

27. CONSTRUCTING PARALLEL LINES Draw a horizontal line and construct aline parallel to it through a point above the line.

28. CONSTRUCTING PARALLEL LINES Draw a diagonal line and construct aline parallel to it through a point to the right of the line.

29. JUSTIFYING A CONSTRUCTION Explain why the lines in Exercise 28 areparallel. Use a postulate or theorem from Lesson 3.4 to support your answer.

x

a

w

ba bc

d

e

h kj

g

p qs

r

t80� 100�

g h

x �

(90 � x)�

(90 � x)�g

h

x �(180 � x)�

x �

g h

x �

x �

xyxy

n

j135�

k12

3

4

n

jk

35�55�

n

j k

85� 95�


www.mcdougallittell.comfor help withconstructions in Exs.25–29.

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30. FOOTBALL FIELD Thewhite lines along the long edgesof a football field are calledsidelines. Yard lines areperpendicular to the sidelinesand cross the field every fiveyards. Explain why you canconclude that the yard lines areparallel.

31. HANGING WALLPAPER When you hang wallpaper, you use a tool calleda plumb line to make sure one edge of the first strip of wallpaper is vertical.If the edges of each strip of wallpaper are parallel and there are no gapsbetween the strips, how do you know that the rest of the strips of wallpaperwill be parallel to the first?

32. ERROR ANALYSIS It is given that j fi k and k fi l.A student reasons that lines j and l must be parallel.What is wrong with this reasoning? Sketch acounterexample to support your answer.

CATEGORIZING Tell whether the statement is sometimes, always, or

never true.

33. Two lines that are parallel to the same line are parallel to each other.

34. In a plane, two lines that are perpendicular to the same line are parallel toeach other.

35. Two noncoplanar lines that are perpendicular to the same line are parallel to each other.

36. Through a point not on a line you can construct a parallel line.

37. LATTICEWORK You aremaking a lattice fence out ofpieces of wood called slats. You want the top of each slat tobe parallel to the bottom. At what angle should you cut ™1?

38. PROVING THEOREM 3.12 Rearrange the statements to write a flowproof of Theorem 3.12. Remember to include a reason for each statement.

GIVEN � m fi p, n fi p

PROVE � m ∞ n



™1 £ ™2 n fi p

m fi p m ∞ n

m n

p

1 2

l

k

j

130�

1

Page 6 of 8


39. OPTICAL ILLUSION The radiating lines make it hard to tell if the red lines are straight. Explain how you can answer the question using only a straightedge and a protractor.

a. Are the red lines straight?b. Are the red lines parallel?

40. CONSTRUCTING WITH PERPENDICULARS Draw a horizontal line l and apoint P not on l. Construct a line m through P perpendicular to l. Draw apoint Q not on m or l. Construct a line n through Q perpendicular to m. Whatpostulate or theorem guarantees that the lines l and n are parallel?

41. MULTI-STEP PROBLEM Use the information given in the diagram at the right.

a. Explain why ABÆ

∞ CDÆ

.b. Explain why CD

Æ∞ EFÆ

.c. Writing What is m™1? How do you know?

42. When light entersglass, the light bends. When it leavesglass, it bends again. If both sides of apane of glass are parallel, light leaves thepane at the same angle at which it entered.Prove that the path of the exiting light isparallel to the path of the entering light.

GIVEN � ™1 £ ™2, j ∞ k

PROVE � r ∞ s

USING THE DISTANCE FORMULA Find the distance between the two

points. (Review 1.3 for 3.6)

43. A(0, º6), B(14, 0) 44. A(º3, º8), B(2, º1) 45. A(0, º7), B(6, 3)

46. A(º9, º5), B(º1, 11) 47. A(5, º7), B(º11, 6) 48. A(4, 4), B(º3, º3)

FINDING COUNTEREXAMPLES Give a counterexample that demonstrates

that the converse of the statement is false. (Review 2.2)

49. If an angle measures 42°, then it is acute.

50. If two angles measure 150° and 30°, then they are supplementary.

51. If a polygon is a rectangle, then it contains four right angles.

52. USING PROPERTIES OF PARALLEL LINESUse the given information to find themeasures of the other seven angles in thefigure shown at the right. (Review 3.3)

GIVEN � j ∞ k, m™1 = 33°

MIXED REVIEW

SCIENCE CONNECTION

TestPreparation

★★Challenge

j

k

13 42

57 86

1

2 3

s

j

k

path oflight

r

100�

A B

C D

E F

100�

80�

1


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1. In the diagram shown at the right, determinewhether you can prove that lines j and k areparallel. If you can, state the postulate ortheorem that you would use. (Lesson 3.4)

Use the given information and the diagram to

determine which lines must be parallel. (Lesson 3.5)

2. ™1 and ™2 are right angles.

3. ™4 £ ™3

4. ™2 £ ™3, ™3 £ ™4.

5. FIREPLACE CHIMNEY In the illustration at the right, ™ABC and ™DEF are supplementary.Explain how you know that the left and right edgesof the chimney are parallel. (Lesson 3.4)

QUIZ 2 Self-Test for Lessons 3.4 and 3.5

a b

c

d

1 2

3 4

j

k

62�

118�

AB

C D

E

F

Measuring Earth’s Circumference

THENTHEN


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NOWNOW

AROUND 230 B.C., the Greek scholar Eratosthenes estimated Earth’scircumference. In the late 15th century, Christopher Columbus used asmaller estimate to convince the king and queen of Spain that his proposedvoyage to India would take only 30 days.

TODAY, satellites and other tools are used to determine Earth’s circumferencewith great accuracy.

1. The actual distance from Syene toAlexandria is about 500 miles. Use thisvalue and the information on page 145 toestimate Earth’s circumference. How close is your value to the modern daymeasurement in the table at the right?

1999

A replica of one of the shipsused by ChristopherColumbus.

Eratosthenes becomes thehead of the library inAlexandria.

Photograph of Earth fromspace.

235 B.C.

1492

Measuring Earth’s Circumference

Circumference estimated by AboutEratosthenes (230 B.C.) 29,000 mi

Circumference assumed by AboutColumbus (about 1492) 17,600 mi

Modern day measurement 24,902 mi

Page 8 of 8

3.6 Parallel Lines in the Coordinate Plane 165

Parallel Lines in the Coordinate Plane

SLOPE OF PARALLEL LINES

In algebra, you learned that the slope of anonvertical line is the ratio of the vertical change (the rise) to the horizontal change (the run). If the line passes through the points (x1, y1) and (x2, y2), then the slope is given by

Slope = �rruis

ne

�

m = .

Slope is usually represented by the variable m.

Finding the Slope of Train Tracks

COG RAILWAY A cog railway goes up theside of Mount Washington, the tallest mountainin New England. At the steepest section, the traingoes up about 4 feet for each 10 feet it goesforward. What is the slope of this section?

SOLUTION

slope = �rruis

ne

� = �140ffeeeett� = 0.4

Finding the Slope of a Line

Find the slope of the line that passes through the points (0, 6) and (5, 2).

SOLUTION

Let (x1, y1) = (0, 6) and (x2, y2) = (5, 2).

m =

= �25ºº

60�

= º�45�

� The slope of the line is º�45�.

y2 º y1�x2 º x1

E X A M P L E 2

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E X A M P L E 1

y2 º y1�x2 º x1

GOAL 1

Find slopes of lines and use slope to identifyparallel lines in a coordinateplane.

Write equations of parallel lines in acoordinate plane.

� To describe steepness inreal-life, such as the cograilway in Example 1 and thezip line in Ex. 46.


GOAL 2

GOAL 1


3.6y

x

(x2, y2)

x2 � x1

(x1, y1)

run

y2 � y1

rise

104

1

1x

y

(0, 6)5

�4

(5, 2)

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Page 1 of 7


You can use the slopes of two lines to tell whether the lines are parallel.

Deciding Whether Lines are Parallel

Find the slope of each line. Is j1 ∞ j2?

SOLUTION

Line j1 has a slope of

m1 = �42� = 2

Line j2 has a slope of

m2 = �21� = 2

� Because the lines have the same slope, j1 ∞ j2.

Identifying Parallel Lines

Find the slope of each line. Which lines are parallel?

SOLUTION

Find the slope of k1. Line k1 passes through (0, 6) and (2, 0).

m1 = �02ºº

60� = �º2

6� = º3

Find the slope of k2. Line k2 passes through (º2, 6) and (0, 1).

m2 = �01º

º(º

62)� = �0

º+

52� = º�2

5�

Find the slope of k3. Line k3 passes through (º6, 5) and (º4, 0).

m3 = �º40ºº

(5º6)� = �º4

º+5

6� = º�25

�

� Compare the slopes. Because k2 and k3 have the same slope, they are parallel.Line k1 has a different slope, so it is not parallel to either of the other lines.

E X A M P L E 4

E X A M P L E 3

POSTULATE 17 Slopes of Parallel LinesIn a coordinate plane, two nonvertical lines are parallel if and only if they have the same slope. Any two vertical lines are parallel.

POSTULATE

k1 k2

UsingAlgebra

xxyxy

y

x4

4

(0, 6)(�2, 6)

(�6, 5)

(�4, 0)

k1k2k3

(2, 0)(0, 1)

y

x1

12

4 1

2

j1 j2

Lines k1 and k2 have the same slope.

STUDENT HELP

Study Tip

To find the slope inExample 3, you can eitheruse the slope formula oryou can count units onthe graph.

Page 2 of 7


WRITING EQUATIONS OF PARALLEL LINES

In algebra, you learned that you can use the slope m of a nonvertical line to writean equation of the line in slope-intercept form.

The y-intercept is the y-coordinate of the point where the line crosses the y-axis.

Writing an Equation of a Line

Write an equation of the line through the point (2, 3) that has a slope of 5.

SOLUTION

Solve for b. Use (x, y) = (2, 3) and m = 5.

y = mx + b Slope-intercept form

3 = 5(2) + b Substitute 2 for x, 3 for y, and 5 for m.

3 = 10 + b Simplify.

º7 = b Subtract.

� Write an equation. Since m=5 and b=º7, an equation of the line is y=5x º 7.

Writing an Equation of a Parallel Line

Line n1 has the equation y = º�13�x º 1.

Line n2 is parallel to n1 and passes through the point (3, 2). Write an equation of n2.

SOLUTION

Find the slope.

The slope of n1 is º�13�. Because parallel

lines have the same slope, the slope of n2 is also º�13�.

Solve for b. Use (x, y) = (3, 2) and m = º�13�.

y = mx + b

2 = º�13�(3) + b

2 = º 1 + b

3 = b

Write an equation.

� Because m = º�13� and b = 3, an equation of n2 is y = º�

13�x + 3.

E X A M P L E 6

E X A M P L E 5

GOAL 2

slopey-intercept

y

x1

1(3, 2) n2

n1

UsingAlgebra

xxyxy

y = mx + b



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Page 3 of 7


1. What does intercept mean in the expression slope-intercept form?

2. The slope of line j is 2 and j ∞ k. What is the slope of line k?

3. What is the slope of a horizontal line? What is the slope of a vertical line?

Find the slope of the line that passes through the labeled points.

4. 5. 6.

Determine whether the two lines shown in the graph are parallel. If they

are parallel, explain how you know.

7. 8. 9.

10. Write an equation of the line that passes through the point (2, º3) and has aslope of º1.

CALCULATING SLOPE What is the slope of the line?

11. 12. 13.

CALCULATING SLOPE Find the slope of the line that passes through the

labeled points on the graph.

14. 15. 16. y

x3

3

(�8, �8)

(1, 8)

y

x2

2(3, 2)

(�2, 7)

y

x2

2

(�6, 6) (2, 6)

y

x�1�1

21

y

x�2

1

1�1

y

x2

12

3


y

x

1J K

M

L

y

x1

2

E

GF

H

y

x1

1

BA

CD

y

x2�2

(�8, �6)

(0, 4)y

x1

1

(2, 5)

(4, 1)

y

x1

1 (0, 0)

(3, 4)

GUIDED PRACTICE

Extra Practice


STUDENT HELP

Concept Check ✓

Skill Check ✓


STUDENT HELP

HOMEWORK HELPExample 1: Exs. 11–16,

23, 46, 49–52Example 2: Exs. 11–16

Page 4 of 7


IDENTIFYING PARALLELS Find the slope of each line. Are the lines parallel?

17. 18.

19. 20.

21. 22.

23. UNDERGROUND RAILROAD Thephoto at the right shows a monument inOberlin, Ohio, that is dedicated to the Underground Railroad. The slope

of each of the rails is about º�35� and the

sculpture is about 12 feet long. What is the height of the ends of the rails?Explain how you found your answer.

IDENTIFYING PARALLELS Find the slopes of AB¯̆

, CD¯̆

, and EF¯̆

. Which lines

are parallel, if any?

24. A(0, º6), B(4, º4) 25. A(2, 6), B(4, 7) 26. A(º4, 10), B(º8, 7)C(0, 2), D(2, 3) C(0, º1), D(6, 2) C(º5, 7), D(º2, 4) E(0, º4), F(1, º7) E(4, º5), F(8, º2) E(2, º3), F(6, º7)

WRITING EQUATIONS Write an equation of the line.

27. slope = 3 28. slope = �13� 29. slope = º�9

2�

y-intercept = 2 y-intercept = º4 y-intercept = 0

30. slope = �12� 31. slope = 0 32. slope = º�9

2�

y-intercept = 6 y-intercept = º3 y-intercept = º�53

�

2

2

x

y

(0, 5)(�8, 2)

(0, �4)(8, �2)

2

x

y

(5, 1)

(�2, �4)

(0, 7)

4

(�6, 2)

2

2

x

y

(2, �4)

(8, �1)

(�8, 3)

(�4, 5)

2

2

x

y

(4, 2)(0, 4)

(2, �6)

(�2, �2)

2 x

y

(1, 1)(0, 4)

(�1, �7)(�3, �8)2

4

x

y

(�1, 1) (3, 1)

(�5, 9)(0, 7)

UNDERGROUNDRAILROAD is the

name given to the network ofpeople who helped someslaves to freedom. HarrietTubman, a former slave,helped about 300 escape.

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FOCUS ONPEOPLE

STUDENT HELP


47, 48Example 5: Exs. 27–41Example 6: Exs. 42–45

Page 5 of 7


WRITING EQUATIONS Write an equation of the line that has a y-intercept

of 3 and is parallel to the line whose equation is given.

33. y = º6x + 2 34. y = x º 8 35. y = º�43�x

WRITING EQUATIONS Write an equation of the line that passes through the

given point P and has the given slope.

36. P(0, º6), m = º2 37. P(º3, 9), m = º1 38. P��32�, 4�, m = �2

1�

39. P(2, º4), m = 0 40. P(º7, º5), m = �34� 41. P(6, 1), undefined slope

USING ALGEBRA Write an equation of the line that passes through

point P and is parallel to the line with the given equation.

42. P(º3, 6), y = ºx º 5 43. P(1, º2), y = �54�x º 8 44. P(8, 7), y = 3

45. USING ALGEBRA Write an equation of a line parallel to y = �13�x º 16.

46. ZIP LINE A zip line is a taut rope or cable that you can ride down on a pulley. The zip line at the right goes from a 9 foot tall tower to a 6 foot talltower. The towers are 20 feet apart. What is the slope of the zip line?

COORDINATE GEOMETRY In Exercises 47 and 48, use the five points:

P(0, 0), Q(1, 3), R(4, 0), S(8, 2), and T(9, 5).

47. Plot and label the points. Connect every pair of points with a segment.

48. Which segments are parallel? How can you verify this?

CIVIL ENGINEERING In Exercises 49–52, use the following information.

The slope of a road is called the road’s grade. Grades are measured in percents.

For example, if the slope of a road is �210�, the grade is 5%. A warning sign is

needed before any hill that fits one of the following descriptions.

5% grade and more than 3000 feet long6% grade and more than 2000 feet long7% grade and more than 1000 feet long8% grade and more than 750 feet long9% grade and more than 500 feet long� Source: U.S. Department of Transportation

What is the grade of the hill to the nearest percent? Is a sign needed?

49. The hill is 1400 feet long and drops 70 feet.




xxyxy

xxyxy

CIVILENGINEERING

Civil engineers design andsupervise the constructionof roads, buildings, tunnels,bridges, and water supplysystems.


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FOCUS ONCAREERS

8%

9 ft6 ft

20 ft

Page 6 of 7


TECHNOLOGY Using a square viewing screen on a graphing calculator,

graph a line that passes through the origin and has a slope of 1.

53. Write an equation of the line you graphed. Approximately what angle doesthe line form with the x-axis?

54. Graph a line that passes through the origin and has a slope of 2. Write anequation of the line. When you doubled the slope, did the measure of theangle formed with the x-axis double?

55. MULTIPLE CHOICE If two different lines with equations y = m1x + b1 and y = m2x + b2 are parallel, which of the following must be true?

¡A b1 = b2 and m1 ≠ m2 ¡B b1 ≠ b2 and m1 ≠ m2

¡C b1 ≠ b2 and m1 = m2 ¡D b1 = b2 and m1 = m2

¡E None of these

56. MULTIPLE CHOICE Which of the following is an equation of a line parallel

to y º 4 = º�12�x?

¡A y = �12�x º 6 ¡B y = 2x + 1 ¡C y = º2x + 3

¡D y = �72�x º 1 ¡E y = º�

12�x º 8

57. USING ALGEBRA Find a value for k so that the line through (4, k) and

(º2, º1) is parallel to y = º2x + �32�.

58. USING ALGEBRA Find a value for k so that the line through (k, º10)

and (5, º6) is parallel to y = º�14�x + 3.

RECIPROCALS Find the reciprocal of the number. (Skills Review, p. 788)

59. 20 60. º3 61. º11 62. 340

63. �37� 64. º�

133� 65. º�

12� 66. 0.25

MULTIPLYING NUMBERS Evaluate the expression. (Skills Review, p. 785)

67. �34� • (º12) 68. º�

32� • �º�

83�� 69. º10 • �

76� 70. º�

29� • (º33)

PROVING LINES PARALLEL Can you prove that lines m and n are parallel? If

so, state the postulate or theorem you would use. (Review 3.4)

71. 72. 73. n mn mn m

MIXED REVIEW

xxyxy

xxyxy

TestPreparation

★★Challenge

Page 7 of 7


Perpendicular Lines in the Coordinate Plane

SLOPE OF PERPENDICULAR LINES

In the activity below, you will trace a piece of paper to draw perpendicular lineson a coordinate grid. Points where grid lines cross are called lattice points.

LESSON INVESTIGATION

In the activity, you may have discovered the following.

POSTULATE

Deciding Whether Lines are Perpendicular

Find each slope.

Slope of j1 = �30ºº

13� = º�3

2�

Slope of j2 = = �64� = �

32�

Multiply the slopes.

The product is �º�32

��32�� = º1, so j1 fi j2.

3 º (º3)�0 º (º4)

E X A M P L E 1

GOAL 1

Use slope toidentify perpendicular lines ina coordinate plane.

Write equations of perpendicular lines, asapplied in Ex. 46.

� Equations of perpendicularlines are used by ray tracingsoftware to create realisticreflections, as in theillustration below and in Example 6.


GOAL 2

GOAL 1


3.7RE

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Investigating Slopes of Perpendicular Lines

Put the corner of a piece of paper on a lattice point. Rotate the corner so each edge passes through another lattice point but neither edge is vertical. Trace the edges.

Find the slope of each line.

Multiply the slopes.

Repeat Steps 1–3 with the paper at a different angle.4

3

2

1

DevelopingConcepts

ACTIVITY

POSTULATE 18 Slopes of Perpendicular LinesIn a coordinate plane, two nonvertical lines areperpendicular if and only if the product of theirslopes is º1.

Vertical and horizontal lines are perpendicular.

POSTULATE

y

x1

1

(0, 3)

(3, 1)

(�4, �3)

j1 j2

product of slopes = 2�º}12}� = º1

Page 1 of 7

3.7 Perpendicular Lines in the Coordinate Plane 173


Decide whether AC¯̆

and DB¯̆

are perpendicular.

SOLUTION

Slope of AC¯̆

= = �63� = 2

Slope of DB¯̆

= = �º36� = º�

12�

The product is 2�º�12�� = º1, so AC

¯̆fi DB

¯̆.


Decide whether the lines are perpendicular.

line h: y = �34�x + 2 line j: y = º�

43�x º 3

SOLUTION

The slope of line h is �34�. The slope of line j is º�

43�.

The product is ��34��º�

43�� = º1, so the lines are perpendicular.


Decide whether the lines are perpendicular.

line r: 4x + 5y = 2 line s: 5x + 4y = 3

SOLUTION

Rewrite each equation in slope-intercept form to find the slope.

line r: line s:

4x + 5y = 2 5x + 4y = 3

5y = º4x + 2 4y = º5x + 3

y = º�45�x + �

25� y = º�

54�x + �4

3�

slope = º�45� slope = º�

54�

Multiply the slopes to see if the lines are perpendicular.

�º�45��º�

54�� = 1

� The product of the slopes is not º1. So, r and s are not perpendicular.

E X A M P L E 4

E X A M P L E 3

2 º (º1)�º1 º 5

2 º (º4)�4 º 1

E X A M P L E 2

UsingAlgebra

xxyxy

y

x1

1B (�1, 2) C (4, 2)

D (5, �1)

A (1, �4)

LogicalReasoning

Page 2 of 7


WRITING EQUATIONS OF PERPENDICULAR LINES

Writing the Equation of a Perpendicular Line

Line l1 has equation y = º2x + 1. Find an equation of the line l2 that passesthrough P(4, 0) and is perpendicular to l1. First you must find the slope, m2.

m1 • m2 = º1 The product of the slopes of fi lines is º1.

º2 • m2 = º1 The slope of l1 is º2.

m2 = �12� Divide both sides by º2.

Then use m = �12� and (x, y) = (4, 0) to find b.

y = mx + b Slope-intercept form

0 = �12�(4) + b Substitute 0 for y, }12} for m, and 4 for x.

º2 = b Simplify.

� So, an equation of l2 is y = �12�x º 2.

. . . . . . . . . .

RAY TRACING Computer illustrators use raytracing to make accurate reflections. To figureout what to show in the mirror, the computertraces a ray of light as it reflects off the mirror.This calculation has many steps. One of thefirst steps is to find the equation of a lineperpendicular to the mirror.

Writing the Equation of a Perpendicular Line

The equation y = �32�x + 3 represents a mirror.

A ray of light hits the mirror at (º2, 0). Whatis the equation of the line p that isperpendicular to the mirror at this point?

SOLUTION

The mirror’s slope is �32�, so the slope of p is º�

23�.

Use m = º�23� and (x, y) = (º2, 0) to find b.

0 = º�23�(º2) + b

º�43� = b

� So, an equation for p is y = º�23�x º �

43�.

E X A M P L E 6

E X A M P L E 5

GOAL 2

x1

1

p

ray of light

y � x � 332

y

GRAPHIC ARTSMany graphic artists

use computer software todesign images.


INTE

RNET

RE

AL LIFE

RE

AL LIFE

FOCUS ONCAREERS

STUDENT HELP

Study Tip

You can check m2 bymultiplying m1 • m2.

(º2)��12��= º1 ✓

Top view of mirror

Page 3 of 7


1. Define slope of a line.

2. The slope of line m is º�15�. What is the slope of a line perpendicular to m?

3. In the coordinate plane shown at the right, is AC¯̆

perpendicular to BD¯̆

? Explain.

4. Decide whether the lines with the equations y = 2x º 1 and y = º2x + 1 are perpendicular.

5. Decide whether the lines with the equations 5y º x = 15 and y + 5x = 2 are perpendicular.

6. The line l1 has the equation y = 3x. The line l2 is perpendicular to l1 andpasses through the point P(0, 0). Write an equation of l2.

SLOPES OF PERPENDICULAR LINES The slopes of two lines are given.

Are the lines perpendicular?

7. m1 = 2, m2 = º�12� 8. m1 = �

23�, m2 = �

32� 9. m1 = �

14�, m2 = º4

10. m1 = �57�, m2 = º�

75� 11. m1 = º�

12�, m2 = º�

12� 12. m1 = º1, m2 = 1

SLOPES OF PERPENDICULAR LINES Lines j and n are perpendicular. The

slope of line j is given. What is the slope of line n? Check your answer.

13. 2 14. 5 15. º3 16. º7

17. �23� 18. �

15� 19. º�

13� 20. º�3

4�

IDENTIFYING PERPENDICULAR LINES Find the slope of AC¯̆

and BD¯̆

.

Decide whether AC¯̆

is perpendicular to BD¯̆

.

21. 22.

23. 24.

A (�3, 0)

B (�4, �3)

C (�2, �2)

D (2, 0)

y

x

1

5

A (�2, �1)

B (�2, 3)C (4, 1)

1

y

x1

D (0, �2)

A (�1, �3)

B (�1, 1) C (3, 0)

D (1, �2)

1

y

x1

2

y

x1A (�1, �2)

B (�3, 2) C (0, 1)D (3, 0)


GUIDED PRACTICE

y

x

3

�1

A (�1, 3)

B (�2, 0)C (1, �1)

D (2, 2)

STUDENT HELP


33–37Example 3: Exs. 25–28,

47–50Example 4: Exs. 29–32Example 5: Exs. 38–41Example 6: Exs. 42–46

Extra Practice


STUDENT HELP

Vocabulary Check ✓Concept Check ✓

Skill Check ✓

Page 4 of 7


USING ALGEBRA Decide whether lines k1 and k2 are perpendicular.

Then graph the lines to check your answer.

25. line k1: y = 3x 26. line k1: y = º�45�x º 2

line k2: y = º�13�x º 2 line k2: y = �

15�x + 4

27. line k1: y = º�34�x + 2 28. line k1: y = �

13�x º 10

line k2: y = �43�x + 5 line k2: y = 3x

USING ALGEBRA Decide whether lines p1 and p2 are perpendicular.

29. line p1: 3y º 4x = 3 30. line p1: y º 6x = 2

line p2: 4y + 3x = º12 line p2: 6y º x = 12

31. line p1: 3y + 2x = º36 32. line p1: 5y + 3x = º15

line p2: 4y º 3x = 16 line p2: 3y º 5x = º33

LINE RELATIONSHIPS Find the slope of each line. Identify any parallel or

perpendicular lines.

33. 34.

35. 36.

37. NEEDLEPOINT To check whether twostitched lines make a right angle, you cancount the squares. For example, the lines atthe right are perpendicular because onegoes up 8 as it goes over 4, and the othergoes over 8 as it goes down 4. Why doesthis mean the lines are perpendicular?

WRITING EQUATIONS Line j is perpendicular to the line with the given

equation and line j passes through P. Write an equation of line j.

38. y = �12�x º 1, P(0, 3) 39. y = �

53�x + 2, P(5, 1)

40. y = º4x º 3, P(º2, 2) 41. 3y + 4x = 12, P(º3, º4)

O

P

T

œ

�2

2

y

x

R

S

A

C

R

S

1

2

y

x

Z

D

E

F

H

K2

y

xL

G

1

1

A

B V

W

y

x

P

œ

1

xxyxy

xxyxy


www.mcdougallittell.comfor help with Exs. 33–36.

INTE

RNET

STUDENT HELP

8

4 8�4

Page 5 of 7


WRITING EQUATIONS The line with the given equation is perpendicular to

line j at point R. Write an equation of line j.

42. y = º�34�x + 6, R(8, 0) 43. y = �

17�x º 11, R(7, º10)

44. y = 3x + 5, R(º3, º4) 45. y = º�25�x º 3, R(5, º5)

46. SCULPTURE Helaman Ferguson designs sculptures on a computer. Thecomputer is connected to his stone drill and tells how far he should drill atany given point. The distance from the drill tip to the desired surface of thesculpture is calculated along a line perpendicular to the sculpture.

Suppose the drill tip is at (º1, º1) and the equation

y = �14�x + 3 represents the

surface of the sculpture. Write an equation of the line that passes through the drill tip and is perpendicular to the sculpture.

LINE RELATIONSHIPS Decide whether the lines with the given equations

are parallel, perpendicular, or neither.

47. y = º2x º 1 48. y = º�12�x + 3 49. y = º3x + 1 50. y = 4x + 10

y = º2x º 3 y = º�12�x + 5 y = �

13�x + 1 y = º2x + 5

51. MULTI-STEP PROBLEM Use the diagram at the right.

a. Is l1 ∞ l2? How do you know?

b. Is l2 fi n? How do you know?

c. Writing Describe two ways to prove that l1 fi n.

DISTANCE TO A LINE In Exercises 52–54,

use the following information.

The distance from a point to a line isdefined to be the length of theperpendicular segment from the point tothe line. In the diagram at the right, thedistance d between point P and line l isgiven by QP.

52. Find an equation of QP¯̆ .

53. Solve a system of equations to find the coordinates of point Q, theintersection of the two lines.

54. Use the Distance Formula to find QP.

2

1

x

yL1 nL2

x

drill

(�1, �1)

1

1

ysculpture

y � –x � 314

2

y

x

P (2, 2)

œ

1

y � �2x � 12

L

d

HELAMANFERGUSON’S

stone drill is suspended bysix cables. The computeruses the lengths of thecables to calculate thecoordinates of the drill tip.

RE

AL LIFE

RE

AL LIFE

FOCUS ONPEOPLE

TestPreparation

EXTRA CHALLENGE


★★Challenge

Page 6 of 7


ANGLE MEASURES Use the diagram to complete the statement.


55. If m™5 = 38°, then m™8 = ��?��.

56. If m™3 = 36°, then m™4 = ��?��.

57. If ™8 £ ™4 and m™2 = 145°, then m™7 = ��?��.

58. If m™1 = 38° and ™3 £ ™5, then m™6 = ��?��.

IDENTIFYING ANGLES Use the diagram to complete the statement.






63. Writing Describe the three types of proofs you have learned so far. (Review 3.2)

Find the slope of AB¯̆

. (Lesson 3.6)

1. A(1, 2), B(5, 8) 2. A(2, º3), B(º1, 5)

Write an equation of line j2 that passes through point P and is parallel to

line j1. (Lesson 3.6)

3. line j1: y = 3x º 2 4. line j1: y = �12�x + 1

P(0, 2) P(2, º4)

Decide whether k1 and k2 are perpendicular. (Lesson 3.7)

5. line k1: y = 2x º 1 6. line k1: y º 3x = º2

line k2: y = º�12�x + 2 line k2: 3y º x = 12

7. ANGLE OF REPOSE When agranular substance is poured into apile, the slope of the pile dependsonly on the substance. For example,when barley is poured into piles,every pile has the same slope. A pileof barley that is 5 feet tall would beabout 10 feet wide. What is the slopeof a pile of barley? (Lesson 3.6)

QUIZ 3 Self-Test for Lessons 3.6 and 3.7

MIXED REVIEW

43 2

1 5 678

432

1 56 78

Not drawn to scale

10 ft

5 ft

Page 7 of 7

WHY did you learn it? Describe lines and planes in real-life objects, suchas escalators. (p. 133)

Lay the foundation for work with angles and proof.

Solve real-life problems, such as deciding howmany angles of a window frame to measure. (p. 141)

Learn to write and use different types of proof.

Understand the world around you, such as howrainbows are formed. (p. 148)

Solve real-life problems, such as predicting pathsof sailboats. (p. 152)

Analyze light passing through glass. (p. 163)

Use coordinate geometry to show that twosegments are parallel. (p. 170)

Prepare to write coordinate proofs.

Solve real-life problems, such as deciding whethertwo stitched lines form a right angle. (p. 176)

Find the distance from a point to a line. (p. 177)

179

Chapter SummaryCHAPTER

3

WHAT did you learn?Identify relationships between lines. (3.1)

Identify angles formed by coplanar linesintersected by a transversal. (3.1)

Prove and use results about perpendicular lines.(3.2)

Write flow proofs and paragraph proofs. (3.2)

Prove and use results about parallel lines andtransversals. (3.3)

Prove that lines are parallel. (3.4)

Use properties of parallel lines. (3.4, 3.5)

Use slope to decide whether lines in a coordinateplane are parallel. (3.6)

Write an equation of a line parallel to a givenline in a coordinate plane. (3.6)

Use slope to decide whether lines in a coordinateplane are perpendicular. (3.7)

Write an equation of a line perpendicular to agiven line. (3.7)

How does Chapter 3 fit into the BIGGER PICTURE of geometry?In this chapter, you learned about properties of perpendicular and parallellines. You also learned to write flow proofs and learned some importantskills related to coordinate geometry. This work will prepare you to reachconclusions about triangles and other figures and to solve real-life problemsin areas such as carpentry, engineering, and physics.

How did your studyquestions help youlearn?The study questions you wrote, following the study strategy on page 128, may resemble this one.

STUDY STRATEGY

Lines and Angles

1. If two lines do not intersect, can you concludethey are parallel?2. What is the slope of a line perpendicular to 2x – 3y = 6?3. If a transversal intersects two parallel lines,which angles are supplementary?

Page 1 of 5


Chapter ReviewCHAPTER

3

• parallel lines, p. 129• skew lines, p. 129• parallel planes, p. 129

• transversal, p. 131• corresponding angles, p. 131• alternate interior angles, p. 131

• alternate exterior angles, p. 131

• consecutive interior angles, p. 131

• same side interior angles, p. 131

• flow proof, p. 136

VOCABULARY

3.1 LINES AND ANGLES

In the figure, j ∞ k, h is a transversal, and h fi k.

™1 and ™5 are corresponding angles.

™3 and ™6 are alternate interior angles.

™1 and ™8 are alternate exterior angles.

™4 and ™6 are consecutive interior angles.

Examples onpp. 129–131

Complete the statement. Use the figure above.

1. ™2 and ™7 are ��?�� angles. 2. ™4 and ™5 are ��?�� angles.

Use the figure at the right.

3. Name a line parallel to DH¯̆

.

4. Name a line perpendicular to AE¯̆

.

5. Name a line skew to FD¯̆

.

3.2 PROOF AND PERPENDICULAR LINES

GIVEN � ™1 and ™2 are complements.

PROVE � GHÆ̆

fi GJÆ̆


6. Copy the flow proof and add a reason for each statement.

EXAMPLES

EXAMPLE

7 85 6

3 41 2

j

k

h

A

C D

EB

F

HG

H

G J

12

™1 and ™2 are

complements.

m™1 + m™2 = 90°

m™1 + m™2 = m™HGJ

™HGJ is a right ™. GHÆ̆

fi GJÆ̆

m™HGJ = 90°

��?��

��?�� ?��

��?��

��?��

��?��

Page 2 of 5

Chapter Review 181

3.3 PARALLEL LINES AND TRANSVERSALS

In the diagram, m™1 = 75°. By the Alternate Exterior Angles Theorem, m™8 = m™1 = 75°. Because ™8 and ™7 are a linear pair, m™8 + m™7 = 180°. So, m™7 = 180° º 75° = 105°.


7. Find the measures of the other five angles in the diagram above.

Find the value of x. Explain your reasoning.

8. 9. 10. (44 � 3x)�

25�(4x � 4)�92�(7x � 8)�

62�

3.4 PROVING LINES ARE PARALLEL

GIVEN � m™3 = 125°, m™6 = 125°

PROVE � l ∞ m

Plan for Proof: m™3 = 125° = m™6, so ™3 £ ™6. So, l ∞ m by the Alternate Exterior Angles Converse.


Use the diagram above to write a proof.

11. GIVEN � m™4 = 60°, m™7 = 120° 12. GIVEN � ™1 and ™7 are supplementary.

PROVE � l ∞ m PROVE � l ∞ m

3.5 USING PROPERTIES OF PARALLEL LINES

In the diagram, l fi t, m fi t, and m ∞ n.Because l and m are coplanar and perpendicular to the same line, l ∞ m. Then, because l ∞ m and m ∞ n, l ∞ n.


Which lines must be parallel? Explain.


14. ™3 £ ™6

15. ™3 and ™4 are supplements.

16. ™1 £ ™2, ™3 £ ™5

EXAMPLE

EXAMPLE

EXAMPLE

1 243

5 687

j

1

k2

l5

3

4 6

mn

1

l

23 4

5

m

67 8

lt

m

n

Page 3 of 5


3.6 PARALLEL LINES IN THE COORDINATE PLANEExamples onpp. 165–167

Find the slope of each line. Are the lines parallel?

17. 18. 19.

20. Find an equation of the line that is parallel to the line with equation y = º2x + 5 and passes through the point (º1, º4).

2

y

x

M

2

J

K N

3y

x

G

�2

E

F

H

3

y

3 xA

B

DC

3.7 PERPENDICULAR LINES IN THE COORDINATE PLANE

The slope of line j is 3. The slope of line k is º�13�.

3�º�13�� = º1, so j fi k.


In Exercises 21–23, decide whether

lines p1 and p2 are perpendicular.

21. Lines p1 and p2 in the diagram.

22. p1: y = �35�x + 2; p2: y = �

53�x º 1

23. p1: 2y º x = 2; p2: y + 2x = 4

24. Line l1 has equation y = º3x + 5. Write an equation of line l2 which is perpendicular to l1 and passes through (º3, 6).

slope of l1 = �21ºº

00� = 2

slope of l2 = �3 5º

º(º

31)

� = �42� = 2

The slopes are the same, so l1 ∞ l2.

To write an equation for l2, substitute (x, y) = (5, 3) and m = 2 into the slope-intercept form.

y = mx + b Slope-intercept form.

3 = (2)(5) + b Substitute 5 for x, 3 for y, and 2 for m.

º7 = b Solve for b.

� So, an equation for l2 is y = 2x º 7.

EXAMPLES

EXAMPLE

5

y

x

1(1, 2)

(0, 0)(3, �1)

(5, 3)

L1 L2

y

x�1

4

(�3, 2)(3, 4)

(4, �1)(0, �3)

p1

p2

Page 4 of 5

Chapter Test 183

Chapter TestCHAPTER

3

In Exercises 1–6, identify the relationship between the angles in the

diagram at the right.

1. ™1 and ™2 2. ™1 and ™4

3. ™2 and ™3 4. ™1 and ™5

5. ™4 and ™2 6. ™5 and ™6

7. Write a flow proof. 8. If l ∞ m, which angles are

GIVEN � m™1 = m™3 = 37°, BAÆ̆

fi BCÆ̆ supplementary to ™1?

PROVE � m™2 = 16°

Use the given information and the diagram at

the right to determine which lines must be parallel.

9. ∠1 £ ∠2


11. ™1 £ ™5; ™5 and ™7 are supplementary.

In Exercises 12 and 13, write an equation of the line described.

12. The line parallel to y = º�13�x + 5 and with a y-intercept of 1

13. The line perpendicular to y = º2x + 4 and that passes through the point (º1, 2)

14. Writing Describe a real-life object that has edges that are straight lines. Are any of the lines skew? If so, describe a pair.

15. A carpenter wants to cut two boards to fit snugly together. The carpenter’s squares are aligned along EF

Æ,

as shown. Are ABÆ

and CDÆ

parallel? State the theorem that justifies your answer.

16. Use the diagram to write a proof.

GIVEN � ™1 £ ™2, ™3 £ ™4

PROVE � n ∞ p

m

1

l

n

2 34

56 7

8

A

CB

13

2

5 3

14

6

2

A

B

C

DE F

m1l n

2

p

6 3q

5 74

2

m

1

l

n

p5

4

3

Page 5 of 5

186 Chapter 3

Cumulative PracticeCHAPTER

3

1. Describe a pattern in the sequence 10, 12, 15, 19, 24, . . . . Predict the next number. (1.1)

In the diagram at the right, AB¯̆

, AC¯̆

, and BC¯̆

are in plane M.

2. Name a point that is collinear with points A and D. (1.2)

3. Name a line skew to BC¯̆

. (3.1)

4. Name the ray that is opposite to GCÆ̆

. (1.2)

5. How many planes contain A, B, and C? Explain. (2.1)

MNÆ

has endpoints M(7, º5) and N(º3, º1).

6. Find the length of MNÆ

. (1.3) 7. If N is the midpoint of MPÆ

, find the coordinates of point P. (1.5)

In a coordinate plane, plot the points and sketch ™ABC. Classify the angle

as acute, right, or obtuse. (1.4)

8. A(º6, 6), B(º2, 2), C(4, 2) 9. A(2, 1), B(4, 7), C(10, 5) 10. A(2, 5), B(2, º2), C(5, 4)

Find the values of x and y. (1.5, 1.6, 3.2)

11. 12. 13.

In Exercises 14 and 15, find the area of each figure. (1.7)

14. Square with a perimeter of 40 cm 15. Triangle defined by R(0, 0), S(6, 8), and T(10, 0)

16. Construct two perpendicular lines. Bisect one of the angles formed. (1.5, 3.1)

17. Rewrite the following statement in if-then form: The measure of a straightangle is 180°. Then write the inverse, converse, and contrapositive of theconditional statement. (2.1)

In Exercises 18–21, find a counterexample that shows the statement

is false.

18. If a line intersects two other lines, then all three lines are coplanar. (2.1)

19. Two lines are perpendicular if they intersect. (2.2)

20. If AB + BC = AC, then B is the midpoint of ACÆ

. (2.5)

21. If ™1 and ™2 are supplementary, then ™1 and ™2 form a linear pair. (2.6)

22. Solve the equation 3(s º 2) = 15 and write a reason for each step. (2.4)

23. Draw a diagram of intersecting lines j and k. Label each angle with a number.Use the Linear Pair Postulate and the Vertical Angles Theorem to write truestatements about the angles formed by the intersecting lines. (2.6)

(2x � 4)�

y �(3x � 6)�

10(x � y )�

15x �3x �

A B C D

30

7y � 2 x � 3 9

BA

C

D

MG

T

Page 1 of 2

Cumulative Practice 187

Let p represent “x = 0” and let q represent “x + x = x.”

24. Write the biconditional p ¯̆ q in words. Decide whether the biconditional istrue. (2.2, 2.3)

25. If the statement p ˘ q is true and p is true, does it follow that q is true?Explain. (2.3)

Use the diagram at the right.

26. Name four pairs of corresponding angles. (3.1)

27. If AC¯̆

∞ DE¯̆

and m™2 = 55°, find m™6. (3.3)

28. If BD¯̆

∞ CF¯̆

and m™3 = 140°, find m™4. (3.3)

29. Which lines must be parallel if m™3 + m™6 = 180°?Explain. (3.4)

30. PROOF Write a proof.

GIVEN � ™6 £ ™9

PROVE � ™3 and ™4 are supplements. (3.3, 3.4)

In Exercises 31–33, use A(2, 10), B(22, º5), C(º1, 6), and D(25, º1).

31. Show that AD¯̆

is parallel to BC¯̆

. (3.6)

32. Use slopes to show that ™BAC is a right angle. (3.7)

33. Write an equation for a line through the point C and parallel to AB¯̆

. (3.6)

34. RUNNING TRACK The inside of the running track in the diagram isformed by a rectangle and two half circles. Find the distance, to the nearest yard, around the inside of the track. Then find the area enclosed by the track. (Use π ≈ 3.14.) (1.7)

35. PHOTO ENLARGEMENT A photographer took a 4-inch-by-6-inch photo and enlarged each side to 150% of the original size.

a. Find the dimensions of the enlarged photo.

b. Describe the relationship between ™1 and ™3. (1.6, 3.2)

c. Make an accurate diagram of the original photo and the enlargement, as shown. Draw DB

Æand BQ

Æ. Make a conjecture about the relationship

between ™1 and ™2. Measure the angles in your diagram to test yourconjecture. (1.1, 1.6)

36. CONSTRUCTION Two posts support a raised deck. The posts have two parallel braces, as shown.

a. If m™1 = 35°, find m™2. (3.3)

b. If m™3 = 40°, what other angle has a measure of 40°? (3.3)

c. Each post is perpendicular to the deck. Explain how this can be used to show that the posts are parallel to each other. (3.5)

1

23

4

56 7

8 9 10

A

B

C

D

E F

50 yd 50 yd

110 yd

enlarged

original

A

CD

P q

R

B1

2

3

4 in.

6 in.

12

3 4

5

Page 2 of 2

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PERPENDICULAR AND PARALLEL LINESmoodle.monashores.net/pluginfile.php/96252/mod... · parallel lines GOAL 1 Identify relationships between lines. Identify angles formed by transversals