8.5 Use Properties of Trapezoids
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1 8.5 Use Properties of Trapezoids and Kites Goal p Use properties of trapezoids and kites. Your otes VOULY Trapezoid ases of a trapezoid ase angles of a trapezoid Legs of a trapezoid Isosceles trapezoid idsegment of a trapezoid Kite opyright Holt cougal. ll rights reserved. Lesson 8.5 Geometry otetaking Guide 221
2 8.5 Use Properties of Trapezoids and Kites Goal p Use properties of trapezoids and kites. Your otes VOULY Trapezoid trapezoid is a quadrilateral with exactly one pair of parallel sides. ases of a trapezoid The parallel sides of a trapezoid are the bases. ase angles of a trapezoid 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 the legs. Isosceles trapezoid n isosceles trapezoid is a trapezoid in which the legs are congruent. idsegment of a trapezoid The midsegment of a trapezoid is the segment that connects the midpoints of its legs. Kite kite is a quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent. opyright Holt cougal. ll rights reserved. Lesson 8.5 Geometry otetaking Guide 221
3 Example 1 Use a coordinate plane how that EF is a trapezoid. ompare the slopes of opposite sides. 1 y (1, 3) E(4, 4) F(6, 2) lope of } E 5 5 (0, 0) 4 x lope of } F The slopes of } E and } F are the same, so } E } F. lope of } EF lope of } The slopes of } EF and } are not the same, so } EF is to }. ecause quadrilateral EF has exactly one pair of, it is a trapezoid. THEOE 8.14 If a trapezoid is isosceles, then each pair of base angles is. If trapezoid is isosceles, then > and >. THEOE 8.15 If a trapezoid has a pair of congruent, then it is an isosceles trapezoid. If > (or if > ), then trapezoid is isosceles. THEOE 8.16 trapezoid is isosceles if and only if its diagonals are. Trapezoid is isosceles if and only if >. 222 Lesson 8.5 Geometry otetaking Guide opyright Holt cougal. ll rights reserved.
4 Example 1 Use a coordinate plane how that EF is a trapezoid. y (1, 3) E(4, 4) ompare the slopes of opposite sides. 1 (0, 0) 4 lope of } E 5 4 } } 1 3 lope of } F 5 2 } } } 1 3 The slopes of } E and } F are the same, so } E i } F. lope of } EF 5 2 } } lope of } } } The slopes of } EF and } are not the same, so } EF is not parallel to }. ecause quadrilateral EF has exactly one pair of parallel sides, it is a trapezoid. F(6, 2) x THEOE 8.14 If a trapezoid is isosceles, then each pair of base angles is congruent. If trapezoid is isosceles, then > and >. THEOE 8.15 If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid. If > (or if > ), then trapezoid is isosceles. THEOE 8.16 trapezoid is isosceles if and only if its diagonals are congruent. Trapezoid is isosceles if and only if } > }. 222 Lesson 8.5 Geometry otetaking Guide opyright Holt cougal. ll rights reserved.
5 Example 2 Use properties of isosceles trapezoids Kitchen shelf fitting into a cupboard in the corner of a kitchen is an isosceles trapezoid. Find m, m L, and m. K 508 L tep 1 Find m. KL is an, so and are congruent base angles, and m 5 m 5. tep 2 Find m L. ecause K and L are consecutive interior angles formed KL intersecting two parallel lines, they are. o, m L tep 3 Find m. ecause and are a pair of base angles, they are congruent, and m 5 m 5. o, m 5, m L 5, and m 5. heckpoint omplete the following exercises. 1. In Example 1, suppose the coordinates of point E are (7, 5). What type of quadrilateral is EF? Explain. 2. Find m, m, and m in the trapezoid shown opyright Holt cougal. ll rights reserved. Lesson 8.5 Geometry otetaking Guide 223
6 Example 2 Use properties of isosceles trapezoids Kitchen shelf fitting into a cupboard in the corner of a kitchen is an isosceles trapezoid. Find m, m L, and m. K 508 L tep 1 Find m. KL is an isosceles trapezoid, so and K are congruent base angles, and m 5 m K tep 2 Find m L. ecause K and L are consecutive interior angles formed KL intersecting two parallel lines, they are supplementary. o, m L tep 3 Find m. ecause and L are a pair of base angles, they are congruent, and m 5 m L o, m 5 508, m L , and m heckpoint omplete the following exercises. 1. In Example 1, suppose the coordinates of point E are (7, 5). What type of quadrilateral is EF? Explain. Parallelogram; opposite pairs of sides are parallel. 2. Find m, m, and m in the trapezoid shown. m , m 5 458, m opyright Holt cougal. ll rights reserved. Lesson 8.5 Geometry otetaking Guide 223
7 THEOE 8.17: IEGET THEOE FO TPEZOI The midsegment of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases. If } is the midsegment of trapezoid, then } i, } i, and 5 ( 1 ). Example 3 Use the midsegment of a trapezoid In the diagram, } is the midsegment P of trapezoid PQ. Find. Use Theorem 8.17 to find. 5 ( 1 ) pply Theorem in. 9 in. Q 5 ( 1 ) ubstitute for PQ and for. 5 implify. The length is inches. heckpoint omplete the following exercise. 3. Find in the trapezoid at the right. P Q 12 ft 30 ft 224 Lesson 8.5 Geometry otetaking Guide opyright Holt cougal. ll rights reserved.
8 THEOE 8.17: IEGET THEOE FO TPEZOI The midsegment of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases. If } is the midsegment of trapezoid, then } i }, } i }, and 5 ( 1 ). 1 } 2 Example 3 Use the midsegment of a trapezoid In the diagram, } is the midsegment P of trapezoid PQ. Find. 16 in. Q Use Theorem 8.17 to find } 2 1 } 2 ( PQ 1 ) pply Theorem in. ( ) ubstitute 16 for PQ and 9 for implify. The length is 12.5 inches. heckpoint omplete the following exercise. 3. Find in the trapezoid at the right. P Q 12 ft 5 21 ft 30 ft 224 Lesson 8.5 Geometry otetaking Guide opyright Holt cougal. ll rights reserved.
9 THEOE 8.18 If a quadrilateral is a kite, then its diagonals are. If quadrilateral is a kite, then. THEOE 8.19 If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent. If quadrilateral is a kite and } > }, then and. Example 4 pply Theorem 8.19 Find m T in the kite shown at the right. Q 708 y Theorem 8.19, QT has exactly one 888 pair of opposite angles. T ecause Q À, and T must be congruent. o, m 5 m T. Write and solve an equation to find m T. m T 1 m orollary to Theorem 8.1 m T 1 m T ubstitute m T for m. (m T) 1 5 ombine like terms. m T 5 olve for m T. Homework heckpoint omplete the following exercise. 4. Find m G in the kite shown G H 858 at the right. J 758 I opyright Holt cougal. ll rights reserved. Lesson 8.5 Geometry otetaking Guide 225
10 THEOE 8.18 If a quadrilateral is a kite, then its diagonals are perpendicular. If quadrilateral is a kite, then } }. THEOE 8.19 If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent. If quadrilateral is a kite and } > }, then > and À. Example 4 pply Theorem 8.19 Find m T in the kite shown at the right. Q 708 y Theorem 8.19, QT has exactly one pair of congruent opposite angles. ecause Q À, and T must be congruent. o, m 5 m T. Write and solve an equation to find m T. m T 1 m m T 1 m T T 888 orollary to Theorem 8.1 ubstitute m T for m. 2 (m T) ombine like terms. m T olve for m T. Homework heckpoint omplete the following exercise. 4. Find m G in the kite shown G H 858 at the right. m G J I opyright Holt cougal. ll rights reserved. Lesson 8.5 Geometry otetaking Guide 225
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