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# Geometry Section 6-6 1112

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### Geometry Section 6-6 1112

1. 1. Section 6-6 Trapezoids and Kites Tuesday, April 29, 14
2. 2. Essential Questions • How do you apply properties of trapezoids? • How do you apply properties of kites? Tuesday, April 29, 14
3. 3. Vocabulary 1.Trapezoid: 2. Bases: 3. Legs of a Trapezoid: 4. Base Angles: 5. Isosceles Trapezoid: Tuesday, April 29, 14
4. 4. Vocabulary 1.Trapezoid: A quadrilateral with only one pair of parallel sides 2. Bases: 3. Legs of a Trapezoid: 4. Base Angles: 5. Isosceles Trapezoid: Tuesday, April 29, 14
5. 5. Vocabulary 1.Trapezoid: A quadrilateral with only one pair of parallel sides 2. Bases: The parallel sides of a trapezoid 3. Legs of a Trapezoid: 4. Base Angles: 5. Isosceles Trapezoid: Tuesday, April 29, 14
6. 6. Vocabulary 1.Trapezoid: A quadrilateral with only one pair of parallel sides 2. Bases: The parallel sides of a trapezoid 3. Legs of a Trapezoid: The sides that are not parallel in a trapezoid 4. Base Angles: 5. Isosceles Trapezoid: Tuesday, April 29, 14
7. 7. Vocabulary 1.Trapezoid: A quadrilateral with only one pair of parallel sides 2. Bases: The parallel sides of a trapezoid 3. Legs of a Trapezoid: The sides that are not parallel in a trapezoid 4. Base Angles: The angles formed between a base and one of the legs 5. Isosceles Trapezoid: Tuesday, April 29, 14
8. 8. Vocabulary 1.Trapezoid: A quadrilateral with only one pair of parallel sides 2. Bases: The parallel sides of a trapezoid 3. Legs of a Trapezoid: The sides that are not parallel in a trapezoid 4. Base Angles: The angles formed between a base and one of the legs 5. Isosceles Trapezoid: A trapezoid that has congruent legs Tuesday, April 29, 14
9. 9. Vocabulary 6. Midsegment of a Trapezoid: 7. Kite: Tuesday, April 29, 14
10. 10. Vocabulary 6. Midsegment of a Trapezoid: The segment that connects the midpoints of the legs of a trapezoid 7. Kite: Tuesday, April 29, 14
11. 11. Vocabulary 6. Midsegment of a Trapezoid: The segment that connects the midpoints of the legs of a trapezoid 7. Kite: A quadrilateral with exactly two pairs of consecutive congruent sides; Opposite sides are not parallel or congruent Tuesday, April 29, 14
12. 12. Theorems Isosceles Trapezoid 6.21: 6.22: 6.23: Tuesday, April 29, 14
13. 13. Theorems Isosceles Trapezoid 6.21: If a trapezoid is isosceles, then each pair of base angles is congruent 6.22: 6.23: Tuesday, April 29, 14
14. 14. Theorems Isosceles Trapezoid 6.21: If a trapezoid is isosceles, then each pair of base angles is congruent 6.22: If a trapezoid has one pair of congruent base angles, then it is isosceles 6.23: Tuesday, April 29, 14
15. 15. Theorems Isosceles Trapezoid 6.21: If a trapezoid is isosceles, then each pair of base angles is congruent 6.22: If a trapezoid has one pair of congruent base angles, then it is isosceles 6.23: A trapezoid is isosceles IFF its diagonals are congruent Tuesday, April 29, 14
16. 16. Theorems 6.24 - Trapezoid Midsegment Theorem: Kites 6.25: 6.26: Tuesday, April 29, 14
17. 17. Theorems 6.24 - Trapezoid Midsegment Theorem: The midsegment of a trapezoid is parallel to each base and its measure is half of the sum of the lengths of the two bases Kites 6.25: 6.26: Tuesday, April 29, 14
18. 18. Theorems 6.24 - Trapezoid Midsegment Theorem: The midsegment of a trapezoid is parallel to each base and its measure is half of the sum of the lengths of the two bases Kites 6.25: If a quadrilateral is a kite, then its diagonals are perpendicular 6.26: Tuesday, April 29, 14
19. 19. Theorems 6.24 - Trapezoid Midsegment Theorem: The midsegment of a trapezoid is parallel to each base and its measure is half of the sum of the lengths of the two bases Kites 6.25: If a quadrilateral is a kite, then its diagonals are perpendicular 6.26: If a quadrilateral is a kite, then exactly one pair of opposite angles is congruent Tuesday, April 29, 14
20. 20. Example 1 Each side of a basket is an isosceles trapezoid. If m∠JML = 130°, KN = 6.7 ft, and LN = 3.6 ft, ﬁnd each measure. a. m∠MJK Tuesday, April 29, 14
21. 21. Example 1 Each side of a basket is an isosceles trapezoid. If m∠JML = 130°, KN = 6.7 ft, and LN = 3.6 ft, ﬁnd each measure. a. m∠MJK m∠JML = m∠KLM Tuesday, April 29, 14
22. 22. Example 1 Each side of a basket is an isosceles trapezoid. If m∠JML = 130°, KN = 6.7 ft, and LN = 3.6 ft, ﬁnd each measure. a. m∠MJK m∠JML = m∠KLM 360 − 2(130) Tuesday, April 29, 14
23. 23. Example 1 Each side of a basket is an isosceles trapezoid. If m∠JML = 130°, KN = 6.7 ft, and LN = 3.6 ft, ﬁnd each measure. a. m∠MJK m∠JML = m∠KLM 360 − 2(130) = 100 Tuesday, April 29, 14
24. 24. Example 1 Each side of a basket is an isosceles trapezoid. If m∠JML = 130°, KN = 6.7 ft, and LN = 3.6 ft, ﬁnd each measure. a. m∠MJK m∠JML = m∠KLM 360 − 2(130) = 100 100/2 Tuesday, April 29, 14
25. 25. Example 1 Each side of a basket is an isosceles trapezoid. If m∠JML = 130°, KN = 6.7 ft, and LN = 3.6 ft, ﬁnd each measure. a. m∠MJK m∠JML = m∠KLM 360 − 2(130) = 100 100/2 = 50 Tuesday, April 29, 14
26. 26. Example 1 Each side of a basket is an isosceles trapezoid. If m∠JML = 130°, KN = 6.7 ft, and LN = 3.6 ft, ﬁnd each measure. a. m∠MJK m∠JML = m∠KLM 360 − 2(130) = 100 100/2 = 50 m∠MJK = 50° Tuesday, April 29, 14
27. 27. Example 1 Each side of a basket is an isosceles trapezoid. If m∠JML = 130°, KN = 6.7 ft, and LN = 3.6 ft, ﬁnd each measure. b. JL Tuesday, April 29, 14
28. 28. Example 1 Each side of a basket is an isosceles trapezoid. If m∠JML = 130°, KN = 6.7 ft, and LN = 3.6 ft, ﬁnd each measure. b. JL JN = KN Tuesday, April 29, 14
29. 29. Example 1 Each side of a basket is an isosceles trapezoid. If m∠JML = 130°, KN = 6.7 ft, and LN = 3.6 ft, ﬁnd each measure. b. JL JN = KN JL = JN + LN Tuesday, April 29, 14
30. 30. Example 1 Each side of a basket is an isosceles trapezoid. If m∠JML = 130°, KN = 6.7 ft, and LN = 3.6 ft, ﬁnd each measure. b. JL JN = KN JL = JN + LN JL = 6.7 + 3.6 Tuesday, April 29, 14
31. 31. Example 1 Each side of a basket is an isosceles trapezoid. If m∠JML = 130°, KN = 6.7 ft, and LN = 3.6 ft, ﬁnd each measure. b. JL JN = KN JL = JN + LN JL = 6.7 + 3.6 JL = 10.3 ft Tuesday, April 29, 14
32. 32. Example 2 Quadrilateral ABCD has vertices A(5, 1), B(−3, −1), C(−2, 3), and D(2, 4). Show that ABCD is a trapezoid and determine whether it is an isosceles trapezoid. Tuesday, April 29, 14
33. 33. Example 2 Quadrilateral ABCD has vertices A(5, 1), B(−3, −1), C(−2, 3), and D(2, 4). Show that ABCD is a trapezoid and determine whether it is an isosceles trapezoid. x y Tuesday, April 29, 14
34. 34. Example 2 Quadrilateral ABCD has vertices A(5, 1), B(−3, −1), C(−2, 3), and D(2, 4). Show that ABCD is a trapezoid and determine whether it is an isosceles trapezoid. x y A Tuesday, April 29, 14
35. 35. Example 2 Quadrilateral ABCD has vertices A(5, 1), B(−3, −1), C(−2, 3), and D(2, 4). Show that ABCD is a trapezoid and determine whether it is an isosceles trapezoid. x y A B Tuesday, April 29, 14
36. 36. Example 2 Quadrilateral ABCD has vertices A(5, 1), B(−3, −1), C(−2, 3), and D(2, 4). Show that ABCD is a trapezoid and determine whether it is an isosceles trapezoid. x y A B C Tuesday, April 29, 14
37. 37. Example 2 Quadrilateral ABCD has vertices A(5, 1), B(−3, −1), C(−2, 3), and D(2, 4). Show that ABCD is a trapezoid and determine whether it is an isosceles trapezoid. x y A B C D Tuesday, April 29, 14
38. 38. Example 2 Quadrilateral ABCD has vertices A(5, 1), B(−3, −1), C(−2, 3), and D(2, 4). Show that ABCD is a trapezoid and determine whether it is an isosceles trapezoid. x y A B C D Tuesday, April 29, 14
39. 39. Example 2 Quadrilateral ABCD has vertices A(5, 1), B(−3, −1), C(−2, 3), and D(2, 4). Show that ABCD is a trapezoid and determine whether it is an isosceles trapezoid. x y A B C D We need AB to be parallel with CD Tuesday, April 29, 14
40. 40. Example 2 Quadrilateral ABCD has vertices A(5, 1), B(−3, −1), C(−2, 3), and D(2, 4). Show that ABCD is a trapezoid and determine whether it is an isosceles trapezoid. x y A B C D We need AB to be parallel with CD CB ≅ ADAlso, Tuesday, April 29, 14
41. 41. Example 2 A(5, 1), B(−3, −1), C(−2, 3), and D(2, 4) Tuesday, April 29, 14
42. 42. Example 2 A(5, 1), B(−3, −1), C(−2, 3), and D(2, 4) m(AB) = −1−1 −3− 5 Tuesday, April 29, 14
43. 43. Example 2 A(5, 1), B(−3, −1), C(−2, 3), and D(2, 4) m(AB) = −1−1 −3− 5 = −2 −8 Tuesday, April 29, 14
44. 44. Example 2 A(5, 1), B(−3, −1), C(−2, 3), and D(2, 4) m(AB) = −1−1 −3− 5 = −2 −8 = 1 4 Tuesday, April 29, 14
45. 45. Example 2 A(5, 1), B(−3, −1), C(−2, 3), and D(2, 4) m(AB) = −1−1 −3− 5 = −2 −8 = 1 4 m(CD) = 4 − 3 2 − (−2) Tuesday, April 29, 14
46. 46. Example 2 A(5, 1), B(−3, −1), C(−2, 3), and D(2, 4) m(AB) = −1−1 −3− 5 = −2 −8 = 1 4 m(CD) = 4 − 3 2 − (−2) = 1 4 Tuesday, April 29, 14
47. 47. Example 2 A(5, 1), B(−3, −1), C(−2, 3), and D(2, 4) m(AB) = −1−1 −3− 5 = −2 −8 = 1 4 m(CD) = 4 − 3 2 − (−2) = 1 4 AD = (5 − 2)2 + (1− 4)2 Tuesday, April 29, 14
48. 48. Example 2 A(5, 1), B(−3, −1), C(−2, 3), and D(2, 4) m(AB) = −1−1 −3− 5 = −2 −8 = 1 4 m(CD) = 4 − 3 2 − (−2) = 1 4 AD = (5 − 2)2 + (1− 4)2 = (3)2 + (−3)2 Tuesday, April 29, 14
49. 49. Example 2 A(5, 1), B(−3, −1), C(−2, 3), and D(2, 4) m(AB) = −1−1 −3− 5 = −2 −8 = 1 4 m(CD) = 4 − 3 2 − (−2) = 1 4 AD = (5 − 2)2 + (1− 4)2 = (3)2 + (−3)2 = 9 + 9 Tuesday, April 29, 14
50. 50. Example 2 A(5, 1), B(−3, −1), C(−2, 3), and D(2, 4) m(AB) = −1−1 −3− 5 = −2 −8 = 1 4 m(CD) = 4 − 3 2 − (−2) = 1 4 AD = (5 − 2)2 + (1− 4)2 = (3)2 + (−3)2 = 9 + 9 = 18 Tuesday, April 29, 14
51. 51. Example 2 A(5, 1), B(−3, −1), C(−2, 3), and D(2, 4) m(AB) = −1−1 −3− 5 = −2 −8 = 1 4 m(CD) = 4 − 3 2 − (−2) = 1 4 AD = (5 − 2)2 + (1− 4)2 = (3)2 + (−3)2 = 9 + 9 = 18 BC = (−3+ 2)2 + (−1− 3)2 Tuesday, April 29, 14
52. 52. Example 2 A(5, 1), B(−3, −1), C(−2, 3), and D(2, 4) m(AB) = −1−1 −3− 5 = −2 −8 = 1 4 m(CD) = 4 − 3 2 − (−2) = 1 4 AD = (5 − 2)2 + (1− 4)2 = (3)2 + (−3)2 = 9 + 9 = 18 BC = (−3+ 2)2 + (−1− 3)2 = (−1)2 + (−4)2 Tuesday, April 29, 14
53. 53. Example 2 A(5, 1), B(−3, −1), C(−2, 3), and D(2, 4) m(AB) = −1−1 −3− 5 = −2 −8 = 1 4 m(CD) = 4 − 3 2 − (−2) = 1 4 AD = (5 − 2)2 + (1− 4)2 = (3)2 + (−3)2 = 9 + 9 = 18 BC = (−3+ 2)2 + (−1− 3)2 = (−1)2 + (−4)2 = 1+16 Tuesday, April 29, 14
54. 54. Example 2 A(5, 1), B(−3, −1), C(−2, 3), and D(2, 4) m(AB) = −1−1 −3− 5 = −2 −8 = 1 4 m(CD) = 4 − 3 2 − (−2) = 1 4 AD = (5 − 2)2 + (1− 4)2 = (3)2 + (−3)2 = 9 + 9 = 18 BC = (−3+ 2)2 + (−1− 3)2 = (−1)2 + (−4)2 = 1+16 = 17 Tuesday, April 29, 14
55. 55. Example 2 A(5, 1), B(−3, −1), C(−2, 3), and D(2, 4) m(AB) = −1−1 −3− 5 = −2 −8 = 1 4 m(CD) = 4 − 3 2 − (−2) = 1 4 AD = (5 − 2)2 + (1− 4)2 = (3)2 + (−3)2 = 9 + 9 = 18 BC = (−3+ 2)2 + (−1− 3)2 = (−1)2 + (−4)2 = 1+16 = 17 CB ≅ AD Tuesday, April 29, 14
56. 56. Example 2 A(5, 1), B(−3, −1), C(−2, 3), and D(2, 4) m(AB) = −1−1 −3− 5 = −2 −8 = 1 4 m(CD) = 4 − 3 2 − (−2) = 1 4 AD = (5 − 2)2 + (1− 4)2 = (3)2 + (−3)2 = 9 + 9 = 18 BC = (−3+ 2)2 + (−1− 3)2 = (−1)2 + (−4)2 = 1+16 = 17 It is a trapezoid, but not isoscelesCB ≅ AD Tuesday, April 29, 14
57. 57. Example 3 In the ﬁgure, MN is the midsegment of trapezoid FGJK. What is the value of x? Tuesday, April 29, 14
58. 58. Example 3 In the ﬁgure, MN is the midsegment of trapezoid FGJK. What is the value of x? MN = KF + JG 2 Tuesday, April 29, 14
59. 59. Example 3 In the ﬁgure, MN is the midsegment of trapezoid FGJK. What is the value of x? MN = KF + JG 2 30 = 20 + JG 2 Tuesday, April 29, 14
60. 60. Example 3 In the ﬁgure, MN is the midsegment of trapezoid FGJK. What is the value of x? MN = KF + JG 2 30 = 20 + JG 2 60 = 20 + JG Tuesday, April 29, 14
61. 61. Example 3 In the ﬁgure, MN is the midsegment of trapezoid FGJK. What is the value of x? MN = KF + JG 2 30 = 20 + JG 2 60 = 20 + JG 40 = JG Tuesday, April 29, 14
62. 62. Example 4 If WXYZ is a kite, ﬁnd m∠XYZ. Tuesday, April 29, 14
63. 63. Example 4 If WXYZ is a kite, ﬁnd m∠XYZ. m∠WXY = m∠WZY Tuesday, April 29, 14
64. 64. Example 4 If WXYZ is a kite, ﬁnd m∠XYZ. m∠WXY = m∠WZY m∠XYZ = 360 −121− 73−121 Tuesday, April 29, 14
65. 65. Example 4 If WXYZ is a kite, ﬁnd m∠XYZ. m∠WXY = m∠WZY m∠XYZ = 360 −121− 73−121 = 45° Tuesday, April 29, 14
66. 66. Example 5 If MNPQ is a kite, ﬁnd NP. Tuesday, April 29, 14
67. 67. Example 5 If MNPQ is a kite, ﬁnd NP. a2 + b2 = c2 Tuesday, April 29, 14
68. 68. Example 5 If MNPQ is a kite, ﬁnd NP. a2 + b2 = c2 62 + 82 = c2 Tuesday, April 29, 14
69. 69. Example 5 If MNPQ is a kite, ﬁnd NP. a2 + b2 = c2 62 + 82 = c2 36 + 64 = c2 Tuesday, April 29, 14
70. 70. Example 5 If MNPQ is a kite, ﬁnd NP. a2 + b2 = c2 62 + 82 = c2 36 + 64 = c2 100 = c2 Tuesday, April 29, 14
71. 71. Example 5 If MNPQ is a kite, ﬁnd NP. a2 + b2 = c2 62 + 82 = c2 36 + 64 = c2 100 = c2 100 = c2 Tuesday, April 29, 14
72. 72. Example 5 If MNPQ is a kite, ﬁnd NP. a2 + b2 = c2 62 + 82 = c2 36 + 64 = c2 100 = c2 100 = c2 c =10 Tuesday, April 29, 14
73. 73. Problem Set Tuesday, April 29, 14
74. 74. Problem Set p. 440 #1-27 odd, 35-43 odd, 49, 65, 75, 77 “Do what you love, love what you do, leave the world a better place and don't pick your nose.” - Jeff Mallett Tuesday, April 29, 14