and DF = 3 and and Given: ; ;

PROOF Statements Reasons 1. ? 1. Given 2. 2. ? 3. ? 3. ?

18. Given:

, See figure for Exercise 17. Prove: PROOF Statements Reasons 1. ? 1. Given 2. G and J are right 2. ? 3. 3. ? 4. 4. ? 5. ? 5. ?

19. Given:

Prove: ∠N ⬵ ∠R RQ NM = RS NP = QS MP ∠HKJ ⬵ ∠GKF ∠G ⬵ ∠J ∠s ∠s 䉭HJK 䉭FGK HG ⬜ FG HJ ⬜ JF ∠HKJ ⬵ ∠FKG PROOF Statements Reasons 1. ? 1. ? 2. and 2. ? 3. ? 3. AA 4. ? 4. ?

22. Given:

, Prove: AB DC = BC CE AC 7 DE AB 7 DC ∠S ⬵ ∠U ∠R ⬵ ∠V R P M S Q N E F D H G U R V T S C E D A B B A C D E Exercises 27, 28 E C D B A Exercises 23–26 PROOF Statements Reasons 1. ? 1. Given 2. ? 2. SSS 3. ? 3. CASTC

20. Given:

Prove: ∠DGH ⬵ ∠E DG DE = DH DF PROOF Statements Reasons 1. ? 1. ? 2. 2. ? 3. 3. ? 4. ? 4. ?

21. Given:

Prove: RT VT = RS VU RS 7 UV 䉭DGH 䉭DEF ∠D ⬵ ∠D PROOF Statements Reasons 1. 1. ? 2. ? 2. If 2 lines are cut by a trans. corr. are 3. ? 3. Given 4. 4. ? 5. 5. ? 6. ? 6. ? In Exercises 23 to 26, . 䉭ABC 䉭DBE 䉭ACB 䉭DEC ∠ACB ⬵ ∠E ⬵ ∠s 7 AB 7 DC

23. Given: , ,

Find: EB HINT: Let , and solve an equation.

24. Given: ,

E is the midpoint of Find: DE

25. Given: , ,

Find: DB

26. Given: , ,

Find: DB

27. with .

If , , and , find EB. CE = 6 DA = 8 CD = 10 ∠CDE ⬵ ∠B 䉭CDE 䉭CBA AD = 5 CE = 4 CB = 12 AD = 4 DE = 8 AC = 10 CB CB = 12 AC = 10 EB = x CB = 6 DE = 6 AC = 8 A C D B F E G

28. with .

If , , and , find CE. EB = 12 CA = 16 CD = 10 ∠CDE ⬵ ∠B 䉭CDE 䉭CBA

35. Use the result of Exercise 11

to do the following problem. In , and . The length of altitude to side is 6. Find the length of altitude from Q to .

36. Use the result of Exercise 11

to do the following problem. In , and . The length of altitude to side is 5. Find the length of altitude from A to .

37. The distance across a pond is to be measured indirectly by

using similar triangles. If , , , and , find XT. WZ = 50 ft TY = 120 ft YW = 40 ft XY = 160 ft DC AE BC AF BC = 12 AB = 7 ⵥABCD PN QS MN QR QM = 9 QP = 12 ⵥMNPQ Exercises 29, 30 In Exercise 31, provide a two-column proof.

31. Given: ,

Prove: 䉭ABC 䉭EFG BD 7 FG AB 7 DF In Exercise 32, provide a paragraph proof.

32. Given: ,

Prove: 䉭BSR 䉭BCA CB ⬜ AC RS ⬜ AB E F G D N P Q M T U S R Y X Z A B F C D E X Y W Z T Pond Pond Pond A D B C S B A C R

33. Use a two-column proof to prove the following theorem:

“The lengths of the corresponding altitudes of similar triangles have the same ratio as the lengths of any pair of corresponding sides.” Given: ; and are altitudes Prove: DG MQ = DE MN MQ DG 䉭DEF 䉭MNP

34. Provide a paragraph proof for the following problem.

Given: , Prove: RS ZX = ZY RT RU 7 XZ RS 7 YZ Q P N M R S

38. In the figure, . Find AB if and

. DC = 6 AD = 2 ∠ABC ⬵ ∠ADB

39. Prove that the altitude drawn to the hypotenuse of a right

triangle separates the right triangle into two right triangles that are similar to each other and to the original right triangle.

40. Prove that the line segment joining the midpoints of two

sides of a triangle determines a triangle that is similar to the original triangle. see the figure for Exercise 27.

29. with

obtuse angles at vertices D and F as indicated. If , and , find x.

30. with

obtuse angles at vertices D and F. If and : , find . m ∠A 1:3 m ∠CDB = m ∠A m ∠B = 44° 䉭ABF 䉭CBD m ∠AFB = 4x m ∠C = x m ∠B = 45° 䉭ABF 䉭CBD A D E F B C