1. Warm-Up Pg 537-539 4,10,20,23,43,49 Answers to evens: 6.Always 12. Always 36. About 6 42. About 8.3 48. 2

2. Use Properties of Trapezoids and Kites 8.5

3. Trapezoid A quadrilateral with exactly one pair of parallel sides, called bases.

4. Diagonals If a trapezoid is isosceles, then each pair of base angles is congruent. If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid. A trapezoid is isosceles if and only if its diagonals are congruent.

5. EXAMPLE 1 Use a coordinate plane Show that ORST is a trapezoid. SOLUTION Compare the slopes of opposite sides. Slope of RS = Slope of OT = 2 – 0 4 – 0 = 2 4 = 1 2 The slopes of RS and OT are the same, so RS OT. 4 – 3 2 – 0 = 1 2

6. EXAMPLE 2 Use properties of isosceles trapezoids Arch The stone above the arch in the diagram is an isosceles trapezoid. Find m K, m M, and m J. SOLUTION STEP 1 Find m K. JKLM is an isosceles trapezoid, so K and L are congruent base angles, and m K = m L= 85 o.

7. EXAMPLE 2 Use properties of isosceles trapezoids STEP 2 Find m M. Because L and M are consecutive interior angles formed by LM intersecting two parallel lines, they are supplementary. So, m M = 180 o – 85 o = 95 o. STEP 3 Find m J. Because J and M are a pair of base angles, they are congruent, and m J = m M = 95 o. ANSWER So, m J = 95 o, m K = 85 o, and m M = 95 o.

8. Midsegment The Midsegment is parallel to both bases and half the length of the sum of the bases,

9. EXAMPLE 3 Use the midsegment of a trapezoid SOLUTION Use Theorem 8.17 to find MN. In the diagram, MN is the midsegment of trapezoid PQRS. Find MN. MN (PQ + SR) 1 2 = Apply Theorem 8.17. = (12 + 28) 1 2 Substitute 12 for PQ and 28 for XU. Simplify. = 20 ANSWER The length MN is 20 inches.

10. GUIDED PRACTICE for Examples 2 and 3 In Exercises 3 and 4, use the diagram of trapezoid EFGH. 3. If EG = FH, is trapezoid EFGH isosceles? Explain. ANSWER yes, Theorem 8.16

11. GUIDED PRACTICE for Examples 2 and 3 4. If m HEF = 70 o and m FGH = 110 o, is trapezoid EFGH isosceles? Explain. SAMPLE ANSWERYes; m EFG = 70° by Consecutive Interior Angles Theorem making EFGH an isosceles trapezoid by Theorem 8.15.

12. GUIDED PRACTICE for Examples 2 and 3 5. In trapezoid JKLM, J and M are right angles, and JK = 9 cm. The length of the midsegment NP of trapezoid JKLM is 12 cm. Sketch trapezoid JKLM and its midsegment. Find ML. Explain your reasoning. J L K M 9 cm 12 cm N P ANSWER ( 9 + x ) = 12 1 2 15 cm; Solve for x to find ML.

13. Kites A quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent. Diagonals are perpendicular. Exactly one pair of opposite angles are congruent.

14. EXAMPLE 4 Apply Theorem 8.19 SOLUTION By Theorem 8.19, DEFG has exactly one pair of congruent opposite angles. Because E G, D and F must be congruent. So, m D = m F. Write and solve an equation to find m D. Find m D in the kite shown at the right.

15. m D + m F +124 o + 80 o = 360 o Corollary to Theorem 8.1 m D + m D +124 o + 80 o = 360 o 2(m D) +204 o = 360 o Combine like terms. Substitute m D for m F. Solve for m D. m D = 78 o EXAMPLE 4 Apply Theorem 8.19

16. GUIDED PRACTICE for Example 4 6. In a kite, the measures of the angles are 3x o, 75 o, 90 o, and 120 o. Find the value of x. What are the measures of the angles that are congruent? ANSWER 25; 75 o

17. Classwork Pg 546-547 4,8,12,14,18,22,26

18. Homework Pg 546-547 3-27 odd