Page 1 of 8 6.5 Trapezoids and ites What you should learn O 1 Use properties of trapezoids. O 2 Use properties of kites. Why you should learn it To solve real-life problems, such as planning the layers of a layer cake in xample 3. I O 1 USI POPTIS O TPZOIS trapezoid is a uadrilateral with exactly one pair of parallel sides. The parallel sides are the bases. trapezoid has two pairs of base angles. or instance, in trapezoid, and are one pair of base angles. The other pair is and. The nonparallel sides are the legs of the trapezoid. If the legs of a trapezoid are congruent, then the trapezoid is an isosceles trapezoid. You are asked to prove the following theorems in the exercises. leg base base isosceles trapezoid leg TOS TO 6.14 If a trapezoid is isosceles, then each pair of base angles is congruent., TO 6.15 If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid. is an isosceles trapezoid. TO 6.16 trapezoid is isosceles if and only if its diagonals are congruent. is isosceles if and only if. TOS XP 1 Using Properties of Isosceles Trapezoids PQS is an isosceles trapezoid. ind m P, m Q, and m. S 50 P SOUTIO PQS is an isosceles trapezoid, so m = m S = 50. ecause S and P are consecutive interior angles formed by parallel lines, they are supplementary. So, m P = 180 º 50 = 130, and m Q = m P = 130. 356 hapter 6 Quadrilaterals
Page 2 of 8 STUT P OWO P Visit our Web site www.mcdougallittell.com for extra examples. ITT XP 2 Show that is a trapezoid. Using Properties of Trapezoids SOUTIO ompare the slopes of opposite sides. y (4, ) 5 º 0 5 The slope of = = = º1. 0 º 5 º 5 The slope of = 4 º = º (0, 5) 3 = º1. (, 4) º 4 3 The slopes of and are eual, so. 1 The slope of = º 5 = 2 4 º 0 4 = 1 2. 1 (5, 0) The slope of = 4 º 0 = 4 =2. º 5 2 The slopes of and are not eual, so is not parallel to. So, because and is not parallel to, is a trapezoid........... x The midsegment of a trapezoid is the segment that connects the midpoints of its legs. Theorem 6.1 is similar to the idsegment Theorem for triangles. You will justify part of this theorem in xercise 42. proof appears on page 839. midsegment TO TO 6.1 idsegment Theorem for Trapezoids The midsegment of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases. TO,, = 1 ( + ) 2 XP 3 inding idsegment engths of Trapezoids Y baker is making a cake like the one at the right. The top layer has a diameter of 8 inches and the bottom layer has a diameter of 20 inches. ow big should the middle layer be? I SOUTIO Use the idsegment Theorem for Trapezoids. = 1 2 ( + ) = 1 (8 + 20) = 14 inches 2 6.5 Trapezoids and ites 35
Page 3 of 8 O 2 USI POPTIS O ITS kite is a uadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent. You are asked to prove Theorem 6.18 and Theorem 6.19 in xercises 46 and 4. The simplest of flying kites often use the geometric kite shape. TOS OUT ITS TO 6.18 If a uadrilateral is a kite, then its diagonals are perpendicular. fi TO 6.19 If a uadrilateral is a kite, then exactly one pair of opposite angles are congruent. TOS OUT ITS, xy Using lgebra XP 4 Using the iagonals of a ite WXYZ is a kite so the diagonals are perpendicular. You can use the Pythagorean Theorem to find the side lengths. WX = 2 0 2 + 1 2 2 23.32 XY = 1 2 2 + 1 2 2 16.9 ecause WXYZ is a kite, WZ = WX 23.32 and ZY = XY 16.9. W 20 X U Z Y XP 5 ngles of a ite ind m and m in the diagram at the right. 132 60 SOUTIO is a kite, so and m = m. 2(m ) + 132 + 60 = 360 Sum of measures of int. of a uad. is 360. 2(m ) = 168 Simplify. m = 84 ivide each side by 2. So, m = m = 84. 358 hapter 6 Quadrilaterals
Page 4 of 8 UI PTI Vocabulary heck oncept heck Skill heck 1. ame the bases of trapezoid. 2. xplain why a rhombus is not a kite. Use the definition of a kite. ecide whether the uadrilateral is a trapezoid, an isosceles trapezoid, a kite, or none of these. 3. 4. 5. 6. ow can you prove that trapezoid in xample 2 is isosceles? ind the length of the midsegment.. 8. 9. 3 11 5 5 PTI PPITIOS STUT P xtra Practice to help you master skills is on p. 814. STUYI TPZOI raw a trapezoid PQS with Q PS. Identify the segments or angles of PQS as bases, consecutive sides, legs, diagonals, base angles, or opposite angles. 10. Q and PS 13. QS and P 11. PQ and S. PQ and Q 14. Q and S 15. S and P II SUS ind the angle measures of. 16. 1. 18. 44 132 82 STUT P 8 OWO P xample 1: xs. 16 18 xample 2: xs. 34, 3, 38, 48 50 xample 3: xs. 19 24, 35, 39 xample 4: xs. 28 30 xample 5: xs. 31 33 II ISTS ind the length of the midsegment. 19. P 9 20. 21. P 14 P 15 9 S S 16 S 6.5 Trapezoids and ites 359
Page 5 of 8 xy USI ind the value of x. 22. x 23. 4 24. 11 8 WS The spider web above is called an orb web. lthough it looks like concentric polygons, the spider actually followed a spiral path to spin the web. OUS O PPITIOS I OTI POYOS In the diagram, is a regular dodecagon, PQ, and X is euidistant from the vertices of the dodecagon. 25. re you given enough information to prove that PQ is isosceles? xplain P your reasoning. 26. What is the measure of X? X 2. What is the measure of each interior angle of PQ? xy USI What are the lengths of the sides of the kite? ive your answer to the nearest hundredth. 28. 29. 30. 3 4 2 4 5 5 3 9 x 8 8 x 5 S O ITS is a kite. What is m? 31. 32. 33. 0 110 50 0 100 34. O YSIS student says that parallelogram is an isosceles trapezoid because and. xplain what is wrong with this reasoning. 35. ITI TII The midsegment of a trapezoid is 5 inches long. What are possible lengths of the bases? 36. OOIT OTY etermine whether the points (4, 5), (º3, 3), (º6, º13), and (6, º2) are the vertices of a kite. xplain your answer. TPZOIS etermine whether the given points represent the vertices of a trapezoid. If so, is the trapezoid isosceles? xplain your reasoning. 3. (º2, 0), (0, 4), (5, 4), (8, 0) 38. (1, 9), (4, 2), (5, 2), (8, 9) 360 hapter 6 Quadrilaterals
Page 6 of 8 OUS O S 39. Y The top layer of the cake has a diameter of 10 inches. The bottom layer has a diameter of 22 inches. What is the diameter of the middle layer? 40. POVI TO 6.14 Write a proof of Theorem 6.14. IV is an isosceles trapezoid., SIS design cakes for many occasions, including weddings, birthdays, anniversaries, and graduations. I www.mcdougallittell.com ITT I POV, Plan for Proof To show, first draw so is a parallelogram. Then show, so and. inally, show. To show, use the consecutive interior angles theorem and substitution. 41. POVI TO 6.16 Write a proof of one conditional statement of Theorem 6.16. IV TQS is an isosceles trapezoid. Q TS and QT S POV T SQ T U S 42. USTIYI TO 6.1 In the diagram below, is the midsegment of and is the midsegment of. xplain why the midsegment of trapezoid is parallel to each base and why its length is one half the sum of the lengths of the bases. STUT P SOTW P Visit our Web site www.mcdougallittell.com to see instructions for several software applications. ITT USI TOOY In xercises 43 45, use geometry software. raw points,, and segments and. onstruct a circle with center and radius. onstruct a circle with center and radius. abel the other intersection of the circles. raw and. 43. What kind of shape is? ow do you know? What happens to the shape as you drag? drag? drag? 44. easure and. What happens to the angle measures as you drag,, or? 45. Which theorem does this construction illustrate? 6.5 Trapezoids and ites 361
Page of 8 46. POVI TO 6.18 Write a two-column proof of Theorem 6.18. IV, POV fi X 4. POVI TO 6.19 Write a paragraph proof of Theorem 6.19. IV is a kite with and. POV, Plan for Proof irst show that. Then use an indirect argument to show : If, then is a parallelogram. ut opposite sides of a parallelogram are congruent. This contradicts the definition of a kite. TPZOIS ecide whether you are given enough information to conclude that is an isosceles trapezoid. xplain your reasoning. Test Preparation 48. 49. 50. 51. UTIP OI In the trapezoid at the right, P = 15. What is the value of x? 2 3 4 5 6 2x 2 15 3x 2 P hallenge XT www.mcdougallittell.com 52. UTIP OI Which one of the following can a trapezoid have? congruent bases diagonals that bisect each other exactly two congruent sides a pair of congruent opposite angles exactly three congruent angles 53. POO Prove one direction of Theorem 6.16: If the diagonals of a trapezoid are congruent, then the trapezoid is isosceles. IV PQS is a trapezoid. Q PS, P SQ POV QP S Plan for Proof raw a perpendicular segment from Q to PS and label the intersection. raw a perpendicular segment from to PS and label the intersection. Prove that QS P. Then prove that QPS SP. P S 362 hapter 6 Quadrilaterals
Page 8 of 8 IX VIW OITIO STTTS ewrite the statement in if-then form. (eview 2.1) 54. scalene triangle has no congruent sides. 55. kite has perpendicular diagonals. 56. polygon is a pentagon if it has five sides. II SUTS Use the diagram to find the side length or angle measure. (eview 6.2 for 6.6) 5. 58. 59. 60. 61. m 62. m 10 5.6 100 POS etermine whether the given points represent the vertices of a parallelogram. xplain your answer. (eview 6.3 for 6.6) 63. (º2, 8), (5, 8), (2, 0), (º5, 0) 64. P(4, º3), Q(9, º1), (8, º6), S(3, º8) QUIZ 2 Self-Test for essons 6.4 and 6.5 1. POSITIOI UTTOS The tool at the right is used to decide where to put buttons on a shirt. The tool is stretched to fit the length of the shirt, and the pointers show where to put the buttons. Why are the pointers always evenly spaced? (int: You can prove that if you know that.) (esson 6.4) etermine whether the given points represent the vertices of a rectangle, a rhombus, a suare, a trapezoid, or a kite. (essons 6.4, 6.5) 2. P(2, 5), Q(º4, 5), (2, º), S(º4, º) 3. (º3, 6), (0, 9), (3, 6), (0, º10) 4. (º5, 6), (º4, º2), (4, º1), (3, ) 5. P(º5, º3), Q(1, º2), (6, 3), S(, 9) 6. POVI TO 6.15 Write a proof of Theorem 6.15. IV is a trapezoid with. POV Plan for Proof raw so is a parallelogram. Use the Transitive Property of ongruence to show. Then, so. (esson 6.5) 6.5 Trapezoids and ites 363