114 Chapter 3 Parallel and Perpendicular Lines

Goal

Use theorems aboutperpendicular lines.

Key Words

• complementary angles p. 67

• perpendicular lines p. 108

3.23.2 Theorems AboutPerpendicular Lines

�4 Do the fold lines appear perpendicular? Use a protractor to measureeach angle. How many right angles are formed?

�5 Write one or more conjectures about perpendicular lines.

Intersecting LinesIntersecting LinesIntersecting LinesGeo-Activity

Theorem 3.1

Words All right angles are congruent.

Symbols If maA � 90� and maB � 90�, then aA c aB.

Theorem 3.2

Words If two lines are perpendicular, then they intersect to form four right angles.

Symbols If n ∏ m, then ma1 � 90�, ma2 � 90�, ma3 � 90�, and ma4 � 90�. m

n1

324

THEOREMS 3.1 and 3.2

The Geo-Activity above suggests the following theorems aboutperpendicular lines.

AB

1

23

4

�1 �2 �3 Unfold the paper.Label the anglesas shown.

Fold the paperagain by liningup the first fold.

Fold a piece ofpaper to form a line.

STUDY TIP

Theorem 3.2 tells youthat if one right angle is marked on a pair ofintersecting lines, thenthe other three anglesare also right angles.

Student Help

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3.2 Theorems About Perpendicular Lines 115

In the diagram, r ∏ s and r ∏ t. Determine whether enough information is given to conclude that the statement is true. Explain your reasoning.

a. a3 c a5

b. a4 c a5

c. a2 c a3

Solution

a. Yes, enough information is given. Both angles are right angles. By Theorem 3.1, they are congruent.

b. Yes, enough information is given. Lines r and t are perpendicular. So, by Theorem 3.2, a4 is a right angle. By Theorem 3.1, all right angles are congruent.

c. Not enough information is given to conclude that a2 c a3.

EXAMPLE 111 Perpendicular Lines and Reasoning

Theorem 3.3

Words If two lines intersect to form adjacent congruent angles, then the lines are perpendicular.

Symbols If a1 c a2, then AC^&( ∏ BD^&(.

Theorem 3.4

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

Symbols If EF&( ∏ EH&(, then ma3 � ma4 � 90�.

THEOREMS 3.3 and 3.4

Perpendicular Lines and Reasoning

LOOK BACK

Theorems 3.3 and 3.4refer to adjacent angles.For the definition ofadjacent angles, see p. 68.

Student Help

In the diagram, g ∏ e and g ∏ f. Determine whether enough

information is given to conclude that the statement is true. Explain.

1. a6 c a10 2. a7 c a10

3. a6 c a8 4. a7 c a11

5. a7 c a9 6. a6 c a11

r

u

t

s

13

2

45

e6

7

11

98

f

g h

10

CD

BA

2

1

F

E H

G

43

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116 Chapter 3 Parallel and Perpendicular Lines

HELICOPTERS Main rotorsof a helicopter may have twoto eight blades. The bladescreate the helicopter’s liftpower.

Aviation In the helicopter at the right, areaAXB and aCXB right angles? Explain.

Solution

If two lines intersect to form adjacentcongruent angles, as AC&* and BD&* do, then the lines are perpendicular (Theorem 3.3). So, AC&* ∏ BD&*.

Because AC&* and BD&* are perpendicular, they form four right angles (Theorem 3.2). So, aAXB and aCXB are right angles.

B

C

D

A X

EXAMPLE 222 Use Theorems About Perpendicular Lines

In the diagram at the right,

EF&*( ∏ EH&*( and maGEH � 30�. Find the value of y.

Solution

aFEG and aGEH are adjacent acute angles and EF&*( ∏ EH&*(. So, aFEG and aGEH are complementary (Theorem 3.4).

6y � � 30� � 90� maFEG � maGEH � 90�

6y � 60 Subtract 30 from each side.

y � 10 Divide each side by 6.

ANSWER � The value of y is 10.

EXAMPLE 333 Use Algebra with Perpendicular Lines

Use Algebra with Perpendicular Lines

Find the value of the variable. Explain your reasoning.

7. aEFGc aHFG 8. AB&( ∏ AD&*( 9. KJ&( ∏ KL&(,aJKMc aMKL

F

E

H

G30�

6y �

G

HE F

5x �

B

A D

C36�

9y �

M

L K

J

z �z �

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3.2 Theorems About Perpendicular Lines 117

1. Complete the statement: If two lines intersect to form adjacentcongruent angles, then the lines are __?__.

Write the theorem that justifies the statement about the diagram.

2. a5 and ∠6 are 3. j ∏ k 4. ma9 � ma10 � 90�

right angles.

In Exercises 5–7, p ∏ q. Write an equation to find the value of x.

(Do not solve the equation.)

5. 6. 7.

Perpendicular Lines and Reasoning Determine whether enough

information is given to conclude that the statement is true. Explain.

8. a2 c a5

9. a6 c a7

10. a1 c a3

11. a1 c a5

Logical Reasoning What can you conclude about a1 and a2 using

the given information?

12. BA&( ∏ BC^&( 13. n ∏m 14. h ∏ k

Practice and Applications

Skill Check

Vocabulary Check

Guided Practice

Exercises3.23.2

Example 1: Exs. 8–11Example 2: Exs. 12–14Example 3: Exs. 17–22

Homework Help

Extra Practice

See p. 679.

l

m6

5

j

k7 8

g

h

f

109

x �

p

qx �

45�

p

q

p

qx �

70�

567

c

d2

341

m1 2

n

1

h

k221

AD

CB

c ∏ d

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Error Analysis Students were asked to set up an equation to find

the value of x, given that v ∏ w. Describe and correct any errors.

15. 16.

Using Algebra Find the value of x, given that s ∏ t.

17. 18. 19.

20. 21. 22.

Angle Measures Find the value of x, given that AB^&(∏ BD^&(. Then use

the value of x to find maCBD.

23. 24. 25.

26. Window Repair You are fixing a window frame. You fit two strips of wood together to make the crosspieces. For the glass panes to fit, each angle formed by the crosspieces must be a right angle. Do you need to measure all four angles to be sure they are all right angles? Explain.

27. In the diagram shown, a1 and a3 are congruent and complementary. Can you conclude that

BA&*( ∏ BC&*(? Explain your reasoning.

You be the Judge

A

B D

C

x �

(x � 20)�D

BA

C

10x �

8x �A

B D

C2x �

7x �

t

s

10x �

5x �

s

t

40�

(3x � 5)�

s

(2x � 20)� t

s

tx �

55�x �

60�t

s

x � t

s

A C

B1 4

2 3

118 Chapter 3 Parallel and Perpendicular Lines

w

v

(x + 4)�56�

v

w

9x�

(x + 4)° + 90° = 56° 9x° = 180°

HOMEWORK HELP

Extra help with problemsolving in Exs. 17–22 is at classzone.com

Student HelpI C L A S S Z O N E . C O M

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Sports In orienteering, a compass and a map are used to navigate

through a wilderness area. Suppose you are in an orienteering event

and you are traveling at 40� east of magnetic north, as shown below.

28. What can you conclude, given that ∠NYW and ∠SYW are congruent? Explain.

29. How many degrees do you need to turn to travel due east?

30. How many degrees do you need to turn to travel due south from the position shown on the compass?

Origami Origami is the Japanese art of folding pieces of paper into

objects.The folds on the paper shown below are the basis for many

objects. On the paper, BF&* ∏ HD&* .

31. Are aDJE and aEJF complementary?Explain your reasoning.

32. If maBJC � maCJD, what are their measures?

33. Is there enough information to conclude that aAJG is a right angle? Explain your reasoning.

Multiple Choice In Exercises 34 and 35, use the diagram below.

34. Which of the following is true if g ∏ h?

�A ma1 � ma2 > 180�

�B ma1 � ma2 < 180�

�C ma1 � ma2 � 180�

�D None of these

35. If g ∏ h and ma1 � 40�, what is ma2?

�F 40� �G 50� �H 60� �J 140�

Standardized Test

Practice

3.2 Theorems About Perpendicular Lines 119

A B C

D

EFG

H J

21

g

h

ORIENTEERINGThe people shown above are participating in aninternational orienteeringcompetition.

Sports

N

40�

W Ey

S

LAHGE_10_CSse_cp.qxd 4/5/09 2:00 PM Page 119

120 Chapter 3 Parallel and Perpendicular Lines

Classifying Angles State whether the angle appears to be acute,

right, obtuse, or straight. Then estimate its measure. (Lesson 1.6)

36. 37. 38.

Finding Complements and Supplements Find the measure of the

angle. (Lesson 2.3)

39. aA is a complement of aB, and maA � 37�. Find maB.

40. aC is a supplement of aD, and maC � 56�. Find maD.

Vertical Angles Find the value of x. (Lesson 2.4)

41. 42. 43.

Decimals Evaluate. (Skills Review, p. 655)

44. 13.6 � 9.8 45. 14 � 2.21 46. 7.4 � 5.9

47. 79.2 � 9 48. 100 � 4.5 � 26.1 49. 30 � 11.1

Think of each segment on the shopping bag as part of a line. There

may be more than one correct answer. (Lesson 3.1)

1. Name two lines perpendicular

to FG .

2. Name a line skew to BF^&(.

3. Name a line that appears parallel

to AD^&(.

4. Name a line perpendicular to plane HGC.

Find the value of the variable, given that p ∏ q. (Lesson 3.2)

5. 6. 7.

q

(3y � 12)�

p

q3x �

57�

pp

qz � 23�

Quiz 1

Algebra Skills

12x �(10x � 6)�(9x � 30)�150�

(4x � 20)�80�

Mixed Review

A

B

C

D

EF

G

H

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