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Geometry 6-6 Trapezoids and Kites A A trapezoid is a quadrilateral with only one pair of parallel sides. A trapezoid is NOT a parallelogram, so the properties we've learned about parallelograms do not apply here. In a trapezoid, the parallel sides are called the bases . The nonparallel sides are called the legs of the trapezoid. Angles that are formed by a base and a leg are called base angles . In the special case where the legs are congruent, we have an isosceles trapezoid . Find each measure. m XWZ m XWZ = m YZW = 45 W m WXY m WXY = 180 - 45 = 135 XZ XZ = WY = 15 + 10 = 25 XV XV = VY = 10 X 15 V 10 > B D Isosceles trapezoids have some special properties. Theorem 6.21: If a trapezoid is isosceles, then each pair of base angles is congruent. Theorem 6.22: If a trapezoid has one pair of congruent base angles, then it is an isosceles trapezoid. Theorem 6.23: A trapezoid is isosceles if and only if its diagonals are congruent. The segment that connects the midpoints of the two legs of a trapezoid is called the midsegment (or median) . Y m YZW = 45 C > Z Theorem 6.24 - Trapezoid Midsegment Theorem: The midsegment of a trapezoid is parallel to both bases and its measure is equal to the average of the measures of the two bases. A B C and D are midpoints. D C Then AB || CD and CD || EF, and CD = 1/2 (AB + EF). E F x 12 16.8 14 x 22 Find the value of x. 1/2 (x + 16.8) = 12 1/2 x + 8.4 = 12 1/2 x = 3.6 x = 7.2 A kite is a quadrilateral with exactly two pairs of consecutive congruent sides. Kites are not parallelograms, so the properties of parallelograms do not apply. B A C x = 1/2 (14 + 22) x = 1/2 (36) = 18 Kites have a couple of interesting properties. Theorem 6.25: If a quadrilateral is a kite, then its diagonals are perpendicular. Theorem 6.26: If a quadrilateral is a kite, then exactly one pair of opposite angles is congruent. ABCD is a kite. D B If m BAD = 38 and m BCD = 50, find m ADC. A 38 + x + 50 + x = 360 D 2x + 88 = 360 If BG = 5 and GC = 8, find CD. 2x = 272 x = 136 52 + 82 = (BC)2 25 + 64 = (BC)2 89 = (BC)2 9.434 = BC CD = 9.434 G C

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