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Mtgnt_06 trans wk bk 9/29/03 3:22 PM 6.5 Page 131 Trapezoids and Kites Goals p Use properties of trapezoids. p Use properties of kites. VOCABULARY Trapezoid A trapezoid is a quadrilateral with exactly one pair of parallel sides. Bases of a trapezoid The parallel sides of a trapezoid are called bases. Base angles of a trapezoid A trapezoid has two pairs of base angles. Each pair shares a base as a side. Legs of a trapezoid The nonparallel sides of a trapezoid are called legs. Isosceles trapezoid An isosceles trapezoid is a trapezoid with congruent legs. Midsegment of a trapezoid The midsegment of a trapezoid is the segment that connects the midpoints of its legs. Kite A kite is a quadrilateral that has two pairs of consecutive congruent sides, but its opposite sides are not congruent. Lesson 6.5 • Geometry Notetaking Guide 131 Mtgnt_06 trans wk bk 9/29/03 3:22 PM Page 132 THEOREM 6.14 If a trapezoid is isosceles, then each pair of base angles is congruent . A aA c a B , a C c aD B D C THEOREM 6.15 If a trapezoid has a pair of congruent base angles , then it is an isosceles trapezoid. A B D ABCD is an isosceles trapezoid. C THEOREM 6.16 A trapezoid is isosceles if and only if its diagonals are congruent . A ABCD is isosceles if and only D &* c BD &* . if AC Example 1 B C Using Properties of Isosceles Trapezoids WXYZ is an isosceles trapezoid. Find maX, maY, and maZ. W Solution p WXYZ is an isosceles trapezoid, so maX ! ma W ! 110 ". 110" Z X Y p aW and aZ are consecutive interior angles formed by parallel lines, so they are supplementary . maW # maZ ! 180 " Consecutive Interior Angles Theorem 110 " # maZ ! 180 " Substitute for maW. maZ ! 70 " Subtract 110 ! from each side. p WXYZ is an isosceles trapezoid, so maY ! ma Z ! 70 ". 132 Geometry Notetaking Guide • Chapter 6 Mtgnt_06 trans wk bk 9/29/03 3:22 PM Page 133 Example 2 Using Properties of Trapezoids Show that HIJK is a trapezoid. y Compare the slopes of opposite sides. 7 8%6 2 1 &* ! $$ ! $$ ! $$ Slope of HK 4%0 4 2 5 K (4, 8) H (0, 6) 3 3%0 3 1 ** ! $$ ! $$ ! $$ Slope of IJ 9%3 6 2 J (9, 3) 1 I (3, 0) 1 &* and IJ ** are equal, The slopes of HK &* ! IJ **. so HK 5 7 9 x 6%0 6 &* ! $$ ! $$ ! %2 Slope of HI 0%3 %3 8%3 5 Slope of &* JK ! $$ ! $$ ! %1 4%9 %5 &* and &* &* is not parallel to & The slopes of HI JK are not equal, so HI JK . &* ! IJ ** and HI &* is not parallel to &*, Answer Because HK JK HIJK is a trapezoid . THEOREM 6.17: MIDSEGMENT THEOREM FOR TRAPEZOIDS The midsegment of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases. B C M 1 &** ! &** &** ! BC &* , MN ! $$(AD # BC) MN AD , MN 2 N A D Checkpoint Complete the following exercise. 1. ABCD is an isosceles trapezoid. Find maA, maB, and maC. A 35" B D C maA ! 35" maB ! maC ! 145" Lesson 6.5 • Geometry Notetaking Guide 133 Mtgnt_06 trans wk bk 9/29/03 3:22 PM Page 134 THEOREM 6.18 C If a quadrilateral is a kite, then its diagonals are perpendicular . B D &* ∏ BD &* AC A THEOREM 6.19 C If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent. B D A aA c aC, aB ç aD Example 3 Angles of a Kite Find maS and maU. S STUV is a kite, so aS c a U and maS ! ma U . V 37" 73" T U 2 (maS) # ma T # ma V ! 360 " 2 (maS) # 73 " # 37 " ! 360 " 2 (maS) ! 250 " maS ! 125 " Sum of measures of int. A of quad. is 360!. Substitute. Simplify. Divide each side by 2 . Answer So, maS ! ma U ! 125 ". Checkpoint Complete the following exercise. 2. Find maX and maZ. Y 97" maX ! maZ ! 62" X 139" W 134 Geometry Notetaking Guide • Chapter 6 Z