1. 8.5 TRAPEZOIDS AND KITES QUADRILATERALS

2. OBJECTIVES: Use properties of trapezoids. Use properties of kites.

3. ASSIGNMENT: pp. 541 through 549 H.W problems # 4, 8, 10, 14, 16, 18, 20, 34, 36, 45 on pages 546-549 of the textbook

4. A trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides are the bases. A trapezoid has two pairs of base angles. For instance in trapezoid ABCD  D and  C are one pair of base angles. The other pair is  A and  B. The nonparallel sides are the legs of the trapezoid. USING PROPERTIES OF TRAPEZOIDS

5. If the legs of a trapezoid are congruent, then the trapezoid is an isosceles trapezoid. USING PROPERTIES OF TRAPEZOIDS

6. Theorem 8.14 If a trapezoid is isosceles, then each pair of base angles is congruent.  A ≅  B,  C ≅  D TRAPEZOID THEOREMS

7. Theorem 8.15 If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid. ABCD is an isosceles trapezoid TRAPEZOID THEOREMS

8. Theorem 8.16 A trapezoid is isosceles if and only if its diagonals are congruent. ABCD is isosceles if and only if AC ≅ BD. TRAPEZOID THEOREMS

9. PQRS is an isosceles trapezoid. Find m  P, m  Q, m  R. PQRS is an isosceles trapezoid, so m  R = m  S = 50 °. Because  S and  P are consecutive interior angles formed by parallel lines, they are supplementary. So m  P = 180 °- 50° = 130°, and m  Q = m  P = 130° EX. 1: USING PROPERTIES OF ISOSCELES TRAPEZOIDS 50 ° You could also add 50 and 50, get 100 and subtract it from 360°. This would leave you 260/2 or 130°.

10. Show that ABCD is a trapezoid. Compare the slopes of opposite sides.  The slope of AB = 5 – 0 = 5 = - 1 0 – 5 -5  The slope of CD = 4 – 7 = -3 = - 1 7 – 4 3 The slopes of AB and CD are equal, so AB ║ CD.  The slope of BC = 7 – 5 = 2 = 1 4 – 0 4 2  The slope of AD = 4 – 0 = 4 = 2 7 – 5 2 The slopes of BC and AD are not equal, so BC is not parallel to AD. So, because AB ║ CD and BC is not parallel to AD, ABCD is a trapezoid. EX. 2: USING PROPERTIES OF TRAPEZOIDS

11. The midsegment of a trapezoid is the segment that connects the midpoints of its legs. Theorem 6.17 is similar to the Midsegment Theorem for triangles. MIDSEGMENT OF A TRAPEZOID

12. The midsegment of a trapezoid is parallel to each base and its length is one half the sums of the lengths of the bases. MN ║AD, MN║BC MN = ½ (AD + BC) THEOREM 8.17: MIDSEGMENT OF A TRAPEZOID

13. LAYER CAKE A 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. How big should the middle layer be? EX. 3: FINDING MIDSEGMENT LENGTHS OF TRAPEZOIDS

14. Use the midsegment theorem for trapezoids. DG = ½(EF + CH)= ½ (8 + 20) = 14” EX. 3: FINDING MIDSEGMENT LENGTHS OF TRAPEZOIDS C D E D G F

15. A kite is a quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent. USING PROPERTIES OF KITES

16. Theorem 8.18 If a quadrilateral is a kite, then its diagonals are perpendicular. AC  BD KITE THEOREMS

17. Theorem 8.19 If a quadrilateral is a kite, then exactly one pair of opposite angles is congruent.  A ≅  C,  B ≅  D KITE THEOREMS

18. WXYZ is a kite so the diagonals are perpendicular. You can use the Pythagorean Theorem to find the side lengths. WX = √ 20 2 + 12 2 ≈ 23.32 XY = √ 12 2 + 12 2 ≈ 16.97 Because WXYZ is a kite, WZ = WX ≈ 23.32, and ZY = XY ≈ 16.97 EX. 4: USING THE DIAGONALS OF A KITE

19. EX. 5: ANGLES OF A KITE Find m  G and m  J in the diagram at the right. SOLUTION: GHJK is a kite, so  G ≅  J and m  G = m  J. 2(m  G) + 132 ° + 60° = 360°Sum of measures of int.  s of a quad. is 360° 2(m  G) = 168°Simplify m  G = 84° Divide each side by 2. So, m  J = m  G = 84° 132 ° 60 °