2. Lesson 7.5 Trapezoids and Kites

3. Definition-a quadrilateral with exactly one pair of parallel sides. Leg Leg Base Base A B CD › › Trapezoid

4. A trapezoid is a quadrilateral with exactly one pair of parallel sides. Each of the parallel sides is called a base. The nonparallel sides are called legs. Base angles of a trapezoid are two consecutive angles whose common side is a base.

5. Properties of A Trapezoid Bases are parallel Leg angles are supplementary. The mid-segment or median is parallel to the bases and is located halfway between them. The length of the median (mid- segment) is the average of the length of the bases.

6. <A + <C = 180 <A + <C = 180 <B + <D = 180 <B + <D = 180 A B CD › › Leg Angles are Supplementary Property of a Trapezoid

7. Isosceles Trapezoid Definition - A trapezoid with congruent legs.

8. Isosceles Trapezoid Isosceles trapezoid – A trapezoid where the legs are . Leg Base 2 Base 1

9. Isosceles Trapezoid - Properties | | 1) Base Angles Are Congruent 2) Diagonals Are Congruent

10. The base  s of an isosceles trapezoid are . Consecutive Interior  s 1 2 34 m  1 + m  3 = 180 m  2 + m  4 = 180 m  1 = m  2 m  3 = m  4

11. Example PQRS is an isosceles trapezoid. Find m P, m Q and mR. m  R = 50  since base angles are congruent m  P = 130  and m  Q = 130  (consecutive angles of parallel lines cut by a transversal are  )

12. Find the measures of the angles in trapezoid 48 m< A = 132 m< B = 132 m< D = 48

13. Example 1: Find the length of VT S V UT SU = x + 2 VT = 2x – 1 SU = VT x + 2 = 2x – 1 x = 3 VT = 2x – 1 = 2(3) – 1 = 6 – 1 = 5

14. Example 2: Find the measure of the 3 missing angles. 156 X Y Z W m  Z = m  X = m  Y = 156 24 m  W + m  X = 180 156 + m  X = 180 m  X = 24

15. Find BE  AC = 17.5, AE = 9.6 E

16. Definition  Kite – a quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent.

17. Parts of a Kite Vertex angles: <B & <D Non-Vertex diagonal CA Non-Vertex angles: <A & <C Vertex diagonal BD

18. Properties of Kites  The diagonals of a kite are perpendicular  Non-vertex angles (a)are congruent  Vertex diagonal bisects the vertex angles  Vertex diagonal bisects the non-vertex diagonal

19. Perpendicular Diagonals of a Kite  If a quadrilateral is a kite, then its diagonals are perpendicular.

20. Non-Vertex Angles of a Kite  If a quadrilateral is a kite, then non- vertex angles are congruent  A   C,  B   D

21. Vertex diagonals bisect vertex angles If a quadrilateral is a kite then the vertex diagonal bisects the vertex angles.

22. Vertex diagonal bisects the non-vertex diagonal If a quadrilateral is a kite then the vertex diagonal bisects the non-vertex diagonal

23. Example  Find the side lengths of the kite.

24. Example Continued We can use the Pythagorean Theorem to find the side lengths. 12 2 + 20 2 = (WX) 2 144 + 400 = (WX) 2 544 = (WX) 2 12 2 + 12 2 = (XY) 2 144 + 144 = (XY) 2 288 = (XY) 2

25. Find the lengths of the sides of the kite W X Y Z 4 55 8

26. Find the lengths of the sides of kite to the nearest tenth 4 2 2 7

27. Example 3  Find mG and mJ. Since GHJK is a kite  G   J So 2(m  G) + 132  + 60  = 360  2(m  G) =168  m  G = 84  and m  J = 84 

29. Try This!  RSTU is a kite. Find mR, mS and mT. x +30 + 125 + 125 + x = 360 2x + 280 = 360 2x = 80 x = 40 So m  R = 70 , m  T = 40  and m  S = 125 

32. Try These 1. If <A = 134, find m<D 2. m<C = x +12 and m<B = 3x – 2, find x and the measures of the 2 angles m<D = 46 x = 42.5 m<C = 54.5 m<B = 125.5

33. Using Properties of Trapezoids Find the area of this trapezoid. When working with a trapezoid, the height may be measured anywhere between the two bases. Also, beware of "extra" information. The 35 and 28 are not needed to compute this area. Area of trapezoid = A = ½ * 26 * (20 + 42) A = 806

34. Using Properties of Trapezoids Find the area of a trapezoid with bases of 10 in and 14 in, and a height of 5 in. Example 2

38. Using Properties of Kites Area Kite = one-half product of diagonals

39. Using Properties of Kites Example 6 E 2 44 4 ABCD is a Kite. a) Find the lengths of all the sides. b)Find the area of the Kite.

40. Venn Diagram: http://teachers2.wcs.edu/high/rhs/staceyh/Geometry/Chapter%206%20Notes.ppt#435,22,6.2 – Properties of Parallelograms

41. Flow Chart:

42. Homework