Mark the figures systematically, using:

In applications of the Pythagorean Theorem, we often arrive at statements such as . Using the following property, we see that or . c = 5 c = 1 25 c 2 = 25 Technology Exploration Computer software and a calculator are needed. 1. Form a right with m C ⫽ 90⬚. 2. Measure AB, AC, and BC. 3. Show that AC 2 ⫹ BC 2 ⫽ AB 2 . Answer will probably not be “perfect.” ∠ 䉭 ABC Let x represent the length of a line segment, and let p represent a positive number. If , then x = 1p. x 2 = p SQUARE ROOTS PROPERTY The square root of p, symbolized represents the number that when multiplied times itself equals p. As we indicated earlier, because . When a square root is not exact, a calculator can be used to find its approximate value; where the symbol ⬇ means “is equal to approximately,” because . 21.9961 L 22 4.69 4.69 = 122 L 4.69 5 5 = 25 125 = 5 1p, EXAMPLE 5 Find the length of the third side of the right triangle. See the figure below. a Find c if and . b Find b if and . Solution a , so or . Then b , so or . Subtracting yields , so b = 151 L 7.14. b 2 = 51 100 = 49 + b 2 10 2 = 7 2 + b 2 c 2 = a 2 + b 2 c = 1 100 = 10. c 2 = 36 + 64 = 100 c 2 = 6 2 + 8 2 c 2 = a 2 + b 2 c = 10 a = 7 b = 8 a = 6 Exs. 12–14 C A a c b B Exercises 3.2 In Exercises 1 to 8, plan and write the two-column proof for each problem.

1. Given: and are

right Prove: 䉭ABC ⬵ 䉭ABD CA ⬵ DA ∠s ∠2 ∠1

3. Given: P

is the midpoint of both and Prove:

4. Given:

and Prove:

5. Given: and are

right Prove:

6. Given: and

Prove: 䉭RST ⬵ 䉭VST ∠3 ⬵ ∠4 ∠1 ⬵ ∠2 䉭RST ⬵ 䉭VST ∠1 ⬵ ∠2 ∠s ∠V ∠R 䉭MNP ⬵ 䉭RQP MN ⬵ QR MN 7 QR 䉭MNP ⬵ 䉭RQP NQ MR

2. Given: and are

right bisects Prove: 䉭ABC ⬵ 䉭ABD ∠CAD AB ∠s ∠2 ∠1 C B D A 1 2 Exercises 1, 2 M N P R Q Exercises 3, 4 S T R V 1 2

3 4

Exercises 5–8 쮿 Video exercises are available on DVD. For Exercises 7 and 8, use the figure on page 142.

7. Given: and

Prove:

8. Given: and are

right Prove:

9. Given: , ,

and Find: , , , and m ∠6 m ∠5 m ∠3 m ∠2 m ∠1 = m∠4 = 42° VY ⬜ XZ VY ⬜ UW UW 7 XZ 䉭RST ⬵ 䉭VST RT ⬵ VT ∠s ∠V ∠R 䉭RST ⬵ 䉭VST RT ⬵ VT SR ⬵ SV

10. Given: , ,

and Find: , , , , , and In Exercises 11 and 12, complete each proof.

11. Given: and

Prove: KJ ⬵ JL HK ⬵ HL HJ ⬜ KL m ∠6 m ∠5 m ∠4 m ∠3 m ∠2 m ∠1 m ∠2 = 6x - 3 m ∠1 = m∠4 = 4x + 3 VY ⬜ XZ VY ⬜ UW UW 7 XZ PROOF Statements Reasons 1. ? 1. Given 2. 2. ? 3. 3. ? 4. 4. ? 5. ? 5. Identity 6. ? 6. ASA 7. 7. ? In Exercises 13 to 16, first prove that triangles are congruent, and then use CPCTC.

13. Given:

and are right M is the midpoint of Prove: ∠N ⬵ ∠Q PR ∠s ∠R ∠P ∠K ⬵ ∠L ∠HJK ⬵ ∠HJL HJ ⬜ KL ∠JHK ⬵ ∠JHL

12. Given: bisects

Prove: ∠K ⬵ ∠L HJ ⬜ KL ∠KHL HJ In Exercises 17 to 22, is a right triangle. Use the given information to find the length of the third side of the triangle.

17. and

18. and b = 5

a = 12 b = 3 a = 4 䉭ABC

14. Given: M

is the midpoint of with transversals and Prove:

15. Given:

and are right H is the midpoint of Prove:

16. Given:

and Prove: EF ⬵ BA AC ⬵ FD AB 7 FE CB ⬜ AB DE ⬜ EF FG ⬵ HJ FG 7 HJ FK ∠s ∠2 ∠1 NP ⬵ QR NQ PR NP 7 RQ NQ PROOF Statements Reasons 1. and 1. ? 2. HJK and HJL 2. ? are rt. 3. 3. ? 4. ? 4. HL 5. ? 5. CPCTC HJ ⬵ HJ ∠s ∠s HK ⬵ HL HJ ⬜ KL U V W X Y Z 1 2 3 4 5 6 Exercises 9, 10 K J L H Exercises 11, 12 N P R Q M Exercises 13, 14 F H K J G 1 2 A B C F E D C A a c b B

19. and

20. and

21. and

22. and

In Exercises 23 to 25, prove the indicated relationship. c = 8 a = 7 b = 4 a = 5 c = 10 b = 6 c = 17 a = 15

35. Given: Regular pentagon ABCDE with diagonals

and Prove: HINT: First prove .

36. In the figure with regular

pentagon ABCDE, do and trisect ? HINT: . m ∠ABE = m∠AEB ∠ABC BD BE 䉭 ABE ⬵ 䉭CBD BE ⬵ BD BD BE

33. As a car moves along

the roadway in a mountain pass, it passes through a horizontal run of 750 feet and through a vertical rise of 45 feet. To the nearest foot, how far does the car move along the roadway?

34. Because of construction along the road from A to B,

Alinna drives 5 miles from A to C and then 12 miles from C to B. How much farther did Alinna travel by using the alternative route from A to B?

32. In the support system of the bridge shown, and

. Find: a b c BS m ∠ABD m ∠RST m ∠ABC = 28° AC = 6 ft

30. Given: and

Prove: HINT: First show that .

31. In the roof truss shown, and

. Find: a AH b c m ∠ADB m ∠BAD m ∠HAF = 37° AB = 8 䉭 ACE ⬵ 䉭ADB 䉭BDC ⬵ 䉭ECD AB ⬵ AE CE ⬜ DE DB ⬜ BC

26. Given: and are