An isosceles trapezoid has line symmetry; the axis of symmetry is the An isosceles trapezoid is cyclic; the center of the circle containing all four

by CPCTC. With by Identity, by SAS. Now because these are corresponding parts of and . Then trapezoid ABCD is isosceles. For several reasons, our final theorem is a challenge to prove. Looking at parallel lines a, b, and c in Figure 4.40, one sees trapezoids such as ABED and BCFE. However, the proof whose “plan” we provide uses auxiliary lines, parallelo- grams, and congruent triangles. 䉭BDC 䉭ACD AD ⬵ BC 䉭ACD ⬵ 䉭BDC DC ⬵ DC ∠ACD ⬵ ∠BDC Exs. 13–15 A B C a t m b c F E D S R Figure 4.40 If three or more parallel lines intercept congruent line segments on one transversal, then they intercept congruent line segments on any transversal. THEOREM 4.4.7 GIVEN: Parallel lines a, b, and c cut by transversal t so that ; also transversal m in Figure 4.40 PROVE: PLAN: Through D and E, draw and . In each formed, and Given it follows that By AAS, we can show ; then by CPCTC. DE ⬵ EF 䉭DER ⬵ 䉭EFS DR ⬵ ES. AB ⬵ BC, ES ⬵ BC. DR ⬵ AB ⵥ ES ‘ AB DR ‘ AB DE ⬵ EF AB ⬵ BC EXAMPLE 6 In Figure 4.40, . If and , find EF. Solution Using Theorem 4.4.7, we find that . 쮿 EF = 8.4 DE = 8.4 AB = BC = 7.2 a ‘ b ‘ c Exs. 16, 17 Exercises 4.4

1. Find the measures of the remaining angles of trapezoid

ABCD not shown if and and .

2. Find the measures of the remaining angles of trapezoid

ABCD not shown if and and .

3. If the diagonals of a trapezoid are congruent, what can

you conclude about the trapezoid?

4. If two of the base angles of a trapezoid are congruent,

what type of trapezoid is it? m ∠D = 118° m ∠B = 63° AB ‘ DC m ∠C = 125° m ∠A = 58° AB ‘ DC

5. What type of quadrilateral is formed when the midpoints

of the sides of an isosceles trapezoid are joined in order?

6. In trapezoid ABCD, is the median. Without writing a

formal proof, explain why . MN = 1 2 AB + DC MN A B C N M X W Y Z D J K L H Exercises 7–8 T V R S M P Q N Exercises 9, 10 C N M A B D Exercises 11–16 B E C D A Exercises 17, 18

7. If and

are supplementary, what type of quadrilateral is HJKL? ∠J ∠H

8. If and

are supplementary in HJKL, are and necessarily supplementary also? For Exercises 9 and 10, consider isosceles trapezoid RSTV with and midpoints M, N, P, and Q of the sides.

9. Would RSTV have symmetry with respect to

a ? b ? Í QN Í MP RS ‘ VT ∠L ∠K ∠J ∠H

18. Given: Isosceles with

; also, D and C are midpoints of and , respectively Prove: ABCD is an isosceles trapezoid

19. In isosceles trapezoid WXYZ

with bases and , , , and . Find height h the length of or . YE ZD WX = 20 YX = 10 ZY = 8 WX ZY BE AE AE ⬵ BE 䉭ABE

22. In trapezoid RSTV, ,

, and M and N are midpoints of the nonparallel sides. If , , and , how long is ?

23. Each vertical section of a suspension bridge is in the

shape of a trapezoid. For additional support, a vertical cable is placed midway as shown. If the two vertical columns shown have heights of 20 ft and 24 ft and the section is 10 ft wide, what will the height of the cable be? RN RS = 16 RV = 17 ST = 13 m ∠SRV = 90° RV ‘ ST

20. In trapezoid WXYZ with bases and

, , , , and . Find the length of base .

21. In isosceles trapezoid MNPQ with , diagonal

. If and , how long is diagonal ? MP NP = 5 PQ = 13 MP ⬜ MQ MN ‘ QP WX ZD = 8 WZ = 17 YX = 10 ZY = 12 WX ZY

11. Given: and

Find: MN

12. Given: and

Find: AB

13. Given: and

Find: DC

14. Given: , ,

and Find: x

15. Given: and

Find: MN , in terms of x

16. Given: and

Find: MN , in terms of x and y

17. Given: ABCD

is an isosceles trapezoid See figure for Exercise 18. Prove: is isosceles 䉭ABE DC = 3x + 5y - 2 AB = x + 3y + 4 DC = 8x - 1 AB = 6x + 5 MN = 5x + 3 DC = 4x - 2 AB = 7x + 5 MN = 9.5 AB = 8.2 DC = 7.5 MN = 6.3 DC = 12.1 AB = 7.3 10. a Does ? b Does ? In Exercises 11 to 16, the drawing shows trapezoid ABCD with ; also, M and N are midpoints of and , respectively. BC AD AB ‘ DC MP = 1 2 RV + ST QN = 1 2 RS + VT Z Y X E D W Exercises 19, 20 24 h 20 10 M N P Q T S M N V R

24. The state of Nevada

approximates the shape of a trapezoid with these dimensions for boundaries: 340 miles on the north, 515 miles on the east, 435 miles on the south, and 225 miles on the west. If A and B are points located midway across the north and south boundaries, what is the approximate distance from A to B?

35. In the figure for Exercise 33, suppose that and

. Find: a AB c EF b DC d Whether

36. Given:

and bisects Find:

37. In a gambrel style roof, the gable end of a barn has the

shape of an isosceles trapezoid surmounted by an isosceles triangle. If and , find: a AS b VD c CD d DE BD = 24 ft AE = 30 ft m ∠FCE ∠DCB CF CE ‘ DA m ∠A = m∠B = 56° AB ‘ DC EF = 1 2 AB + DC MF = 3.5 EM =

7.1

39. The vertical sidewall of an

in-ground pool that is 24 ft in length has the shape of a trapezoid. What is the depth of the pool in the middle?

40. For the in-ground pool shown in Exercise 39, find the

length of the sloped bottom from point D to point C.

41. In trapezoid ABCD not shown, ,

, and . Find all possible values of x.

42. In trapezoid ABCD, and . If

, , and , find the perimeter of . 䉭DAC BC = 8 AB = 6 DA = 17 BC ⬜ AC BC ⬜ AB m ∠C = x 5 + 50 m ∠B = x 3 + 50 m ∠A = x 2 + 10

38. Successive steps on a ladder form isosceles

trapezoids with the sides. and . a Find GN, the width of the bottom step b Which step is the median of the trapezoid with bases and ? GN AH BI = 2.125 ft AH = 2 ft For Exercises 34 and 35, is the median of trapezoid ABCD.

34. In the figure for Exercise 33, suppose that and

. Find: a MF c EF b EM d Whether EF = 1 2 AB + DC DC = 18.4 AB = 12.8 EF

26. In the figure, and B is the midpoint of

. If , , , and , find x and y. In Exercises 27 to 33, complete a formal proof. 27. The diagonals of an isosceles trapezoid are congruent. 28. The median of a trapezoid is parallel to each base.

29. If two consecutive angles of a quadrilateral are

supplementary, the quadrilateral is a trapezoid.

30. If two base angles of a trapezoid are congruent, the

trapezoid is an isosceles trapezoid.

31. If three parallel lines intercept congruent segments on one

transversal, then they intercept congruent segments on any transversal.

32. If the midpoints of the sides of an isosceles trapezoid

are joined in order, then the quadrilateral formed is a rhombus.

33. Given: is the median of trapezoid ABCD

Prove: HINT: Using Theorem 4.4.7, show that M is the midpoint of . For ADC and CBA, apply Theorem 4.2.5. 䉭 䉭 AC EF = 1 2 AB + DC EF EF = 5x - y + 2 DE = 2x + 3y + 3 BC = x + y + 7 AB = 2x + 3y AC a ‘ b ‘ c

25. In the figure, a b c and B is

the midpoint of . If , , and , find the length of . EF DE = 3x + 2 BC = x + 7 AB = 2x + 3 AC ‘ ‘ 435 mi 340 mi 515 mi 225 mi NEVADA A B A B C a b c F E D Exercises 25, 26 A M B C D E F Exercises 33–35 B C D E A S T V 5 ft 8 ft A E F B D C A H B I C J D K E L F M G N 3 24 A D B C 13