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3.2 Theorems About Perpendicular Lines Goal Use theorems about Geo-Activity Intersecting Lines perpendicular lines. ●1 Fold a piece of ●2 Fold the paper ●3 Unfold the paper. paper to form again by lining Label the angles Key Words a line. up the first fold. as shown. • complementary angles p. 67 • perpendicular lines p. 108

4 1 3 2

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

●5 Write one or more conjectures about perpendicular lines.

The Geo-Activity above suggests the following theorems about perpendicular lines.

THEOREMS 3.1 and 3.2 Student Help Theorem 3.1 STUDY TIP Words All right angles are congruent. a ϭ Њ a ϭ Њ Symbols If m A 90 and m B 90 , B then aA c a B. A Theorem 3.2 Theorem 3.2 tells you If two lines are perpendicular, then n that if one right angle Words 1 is marked on a pair of they intersect to form four right angles. 4 2 intersecting lines, then Symbols If n ∏ m, then ma1 ϭ 90 Њ, ma2 ϭ 90 Њ, 3 the other three angles ma3 ϭ 90 Њ, and ma4 ϭ 90 Њ. are also right angles. m

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EXAMPLE 1 Perpendicular Lines and Reasoning

In the diagram, r ∏ s and r ∏ t. Determine whether enough information is given to conclude that the statement is true. Explain your reasoning. r a. a3 c a 5 1 2 3 u b. a4 c a 5 5 4 t c. a2 c a 3 s 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 a 3.

Perpendicular Lines and Reasoning

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 a 10 2. a7 c a 10 g h 6 3. a6 c a 8 4. a7 c a 11 8 e 7 9 10 5. a7 c a 9 6. a6 c a 11 f 11

Student Help THEOREMS 3.3 and 3.4 LOOK BACK Theorems 3.3 and 3.4 Theorem 3.3 refer to adjacent angles . B Words If two lines intersect to form For the definition of A adjacent angles, see adjacent congruent angles, then 1 p. 68. the lines are perpendicular. D 2 Symbols If a1 c a 2, then AC^&( ∏ BD^&( . C Theorem 3.4

Words If two sides of adjacent acute angles F G are perpendicular, then the angles are complementary. 3 4 Symbols If EF&( ∏ EH&( , then ma3 ϩ ma4 ϭ 90 Њ. E H

3.2 Theorems About Perpendicular Lines 115 Page 3 of 7

EXAMPLE 2 Use Theorems About Perpendicular Lines

Aviation In the helicopter at the right, are B aAXB and aCXB right angles? Explain.

C Solution If two lines intersect to form adjacent A X congruent angles, as AC&* and BD&* do, then the lines are perpendicular (Theorem 3.3). So, AC&* ∏ BD&* . D Because AC&* and BD&* are perpendicular, they form four right angles (Theorem 3.2). So, aAXB and aCXB are right angles.

HELICOPTERS Main rotors of a helicopter may have two to eight blades. The blades EXAMPLE 3 Use Algebra with Perpendicular Lines create the helicopter’s lift power. In the diagram at the right, F EF&*( ∏ EH&*( and maGEH ϭ 30 Њ. 6y ؇ G Find the value of y. E ؇ 30 H Solution aFEG and aGEH are adjacent acute angles and EF&*( ∏ EH&*( . So, aFEG and aGEH are complementary (Theorem 3.4).

6yЊ ϩ 30 Њ ϭ 90 Њ maFEG ϩ maGEH ϭ 90 Њ 6y ϭ 60 Subtract 30 from each side. y ϭ 10 Divide each side by 6.

ANSWER ᮣ The value of y is 10.

Use Algebra with Perpendicular Lines

Find the value of the variable. Explain your reasoning. 7. aEFG c a HFG 8. AB&( ∏ AD&*( 9. KJ&( ∏ KL&( , aJKM c a MKL

G B C M J ؇ ؇ 5x 36 z ؇ ؇ E F H 9y z ؇ A D L K

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3.23.2 Exercises Guided Practice

Vocabulary Check 1. Complete the statement: If two lines intersect to form adjacent congruent angles, then the lines are __?__.

Skill Check Write the theorem that justifies the statement about the diagram. 2. a5 and ∠6 are 3. j ∏ k 4. ma9 ϩ ma10 ϭ 90 Њ right angles. l j g f 7 5 m 8 k 6 9 10 h

In Exercises 5–7, p ∏ q. Write an equation to find the value of x. (Do not solve the equation.) 5.p 6.p 7. p x؇ 70 ؇ q ؇ ؇ ؇ x x q 45 q

Practice and Applications

Extra Practice Perpendicular Lines and Reasoning Determine whether enough See p. 679. information is given to conclude that the statement is true. Explain. 8. a c a 2 5 c 5 9. a6 c a 7 2 6 1 d 7 10. a1 c a 3 4 3 c ∏ d 11. a1 c a 5

Logical Reasoning What can you conclude about a1 and a2 using the given information? 12. BA&( ∏ BC^&( 13. n ∏ m 14. h ∏ k n h Homework Help A D Example 1: Exs. 8–11 Example 2: Exs. 12–14 1 2 m 1 2 2 k Example 3: Exs. 17–22 1 B C

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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. v v ؇ 56 x + 4) ؇ 9x ؇) w w

(x + 4)° + 90° = 56° 9x ° = 180 °

IStudent Help Using Algebra Find the value of x, given that s ∏ t. ICLASSZONE.COM 17.s 18.s 19. s HOMEWORK HELP x؇ Extra help with problem x؇ t x؇ t solving in Exs. 17–22 is 60 ؇ 55 ؇ t at classzone .com

20.s 21.s 22. s ؇ ؉ ؇ (3 x ؉ 5) ؇ 5x (2 x 20) t ؇ 40 t 10 x ؇ t

Angle Measures Find the value of x, given that AB^&( ∏ BD^&( . Then use the value of x to find maCBD . 23. 24. 25. 2x ؇ D (x ؉ 20) ؇ C C 8x ؇ A C A 7x ؇ 10 x ؇ x ؇ B D A B B D

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. You be the Judge In the diagram shown, a1 and a3 are congruent and A C complementary. Can you conclude that 2 3 BA&*( ∏ BC&*( ? Explain your reasoning. 1 4 B

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

N ؇ 40

W y E

ORIENTEERING The students above are part S of an in-school orienteering training program at Clark Montessori High School in Cincinnati, Ohio. 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&* . A B C 31. Are aDJE and aEJF complementary? Explain your reasoning. 32. If maBJC ϭ maCJD , what are H D their measures? J 33. Is there enough information to conclude that aAJG is a right angle? Explain your reasoning. G F E

Standardized Test Multiple Choice In Exercises 34 and 35, use the diagram below. Practice 34. Which of the following is true if g ∏ h?

ɕA ma1 ϩ ma2 > 180 Њ g ɕB ma1 ϩ ma2 < 180 Њ 1 2 h ɕC ma1 ϩ ma2 ϭ 180 Њ

ɕD None of these

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

ɕF 40 Њ ɕG 50 Њ ɕH 60 Њ ɕJ 140 Њ

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Mixed Review 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 ϭ 37 Њ. Find maB.

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

Vertical Angles Find the value of x. (Lesson 2.4) 41. 42. 43.

؇ 150 ؇ 80 x ؊ 30) ؇ 12 x ؇ 9) x ؉ 20) ؇ (10 x ؉ 6) ؇ 4)

Algebra Skills Decimals Evaluate. (Skills Review, p. 655) 44. 13.6 ϩ 9.8 45. 14 Ϫ 2.21 46. 7.4 ϫ 5.9 47. 79.2 Ϭ 9 48. 100 Ϫ 4.5 Ϫ 26.1 49. 30 ϫ 11.1

Quiz 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 ^&*( E to FG . A F 2. Name a line skew to BF^&(. B

3. Name a line that appears parallel to AD^&( . H 4. Name a line perpendicular to plane HGC . G D

C

Find the value of the variable, given that p ∏ q. (Lesson 3.2) 5.p 6.p 7. p

؇ (3 y ؊ 12) ؇ 57 z ؇ 23 ؇ 3x؇ q q q

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