In trapezoid ABCD, is the median. Without writing a

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 PERSPECTIVE ON HISTORY Sketch of Thales One of the most significant contributors to the development of geometry was the Greek mathematician Thales of Miletus 625–547 B . C .. Thales is credited with being the “Father of Geometry” because he was the first person to organize geometric thought and utilize the deductive method as a means of verifying propositions theorems. It is not surprising that Thales made original discoveries in geometry. Just as significant as his discoveries was Thales’ persistence in verifying the claims of his predecessors. In this textbook, you will find that propositions such as these are only a portion of those that can be attributed to Thales: Chapter 1: If two straight lines intersect, the opposite vertical angles formed are equal. Chapter 3: The base angles of an isosceles triangle are equal. Chapter 5: The sides of similar triangles are proportional. Chapter 6: An angle inscribed in a semicircle is a right angle. Thales’ knowledge of geometry was matched by the wisdom that he displayed in everyday affairs. For example, he is known to have measured the height of the Great Pyramid of Egypt by comparing the lengths of the shadows cast by the pyramid and by his own staff. Thales also used his insights into geometry to measure the distances from the land to ships at sea. Perhaps the most interesting story concerning Thales was one related by Aesop famous for fables. It seems that Thales was on his way to market with his beasts of burden carrying saddlebags filled with salt. Quite by accident, one of the mules discovered that rolling in the stream where he was led to drink greatly reduced this load; of course, this was due to the dissolving of salt in the saddlebags. On subsequent trips, the same mule continued to lighten his load by rolling in the water. Thales soon realized the need to do something anything to modify the mule’s behavior. When preparing for the next trip, Thales filled the offensive mule’s saddlebags with sponges. When the mule took his usual dive, he found that his load was heavier than ever. Soon the mule realized the need to keep the saddlebags out of the water. In this way, it is said that Thales discouraged the mule from allowing the precious salt to dissolve during later trips to market. PERSPECTIVE ON APPLICATION Square Numbers as Sums In algebra, there is a principle that is generally “proved” by a quite sophisticated method known as mathematical induction. However, verification of the principle is much simpler when provided a geometric justification. In the following paragraphs, we:

1. State the principle 2. Illustrate the principle

3. Provide the geometric justification for the principle

Where , , or 9. Where , , or 16. The geometric explanation for this principle utilizes a wrap-around effect. Study the diagrams in Figure 4.41. 1 + 3 + 5 + 7 = 4 2 n = 4 1 + 3 + 5 = 3 2 n = 3 Where n is a counting number, the sum of the first n positive odd counting numbers is . n 2 The principle stated above is illustrated for various choices of n. Where , . Where , , or 4. 1 + 3 = 2 2 n = 2 1 = 1 2 n = 1

1 a

Figure 4.41 1 + 3 b 1 + 3 + 5 c Given a unit square one with sides of length 1, we build a second square by wrapping 3 unit squares around the