8.5 Use Properties of Trapezoids and Kites Hubarth Geometry

A trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides are the bases. The nonparallel sides are called the legs. AB C D Leg Base Isosceles trapezoid

Theorem 8.14 Theorem 8.15 A B C D A B C D Theorem 8.16 A trapezoid is isosceles if and only if its diagonals are congruent.

Ex 1. Find Angle Measure of Trapezoid PQRS is an isosceles trapezoid. Find the missing angle measures. P Q R S 50 Solution

The midsegment of a trapezoid is the segment that connects the midpoints of its legs. The midsegment of a trapezoid is parallel to the bases. The length of the midsegment of a trapezoid is half the sum of the lengths of the bases A B C D M N

Ex 2. Midsegment of a Trapezoid C E F H 8 20 D G Solution

Theorem 8.18 If a quadrilateral is a kite, then its diagonals are perpendicular. Theorem 8.19 If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent.

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. Ex 3 Apply Theorem 8.19 m D + m F +124 o + 80 o = 360 o m D + m D +124 o + 80 o = 360 o 2(m D) +204 o = 360 o m D = 78 o

Practice 1.FGHJ is an Isosceles trapezoid. Find the missing angle measures. Find the missing angle measures. F G H J Find the length of the midsegment MN of the trapezoid. the trapezoid. a. b M N M N 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? 25; 75 o