6.2.3 Properties of Kites and Trapezoids (continued)
The measure of an angle in a kite, PQRS , is determined in this example. The measures of ∠ RQW and ∠ RSW are given. By the definition of a kite, the short sides are congruent, as are the long sides. By the properties of kites, ∠ QRS and ∠ QPS are congruent. Thus, △ QRS ≅ △ QPS by Side-Angle-Side Congruence. It follows from CPCTC that ∠ PQW ≅ ∠ RQW and ∠ PSW ≅ ∠ RSW (and therefore that QS bisects both ∠ PQR and ∠ PSR ). Therefore, m ∠ PQW = 49° and m ∠ PSW = 20°. By the Triangle Sum Theorem,
m ∠ QPS + m ∠ PQW + m ∠ PSW = 180°. Substituting the known angles yields m ∠ QPS = 180° − (49° + 20°) = 111°.
The measure of an angle in a kite, PQRS , is determined in this example. The measures of ∠ RQW and ∠ RSW are given. Since QS and PR are perpendicular by the properties of kites, △ RWQ is a right triangle. Therefore, m ∠ QRW + m ∠ RQW = 90°. Substitute 49° for m ∠ RQW to find m ∠ QRW . m ∠ QRW = 90° − 49° = 41°
Example 3 Using Properties of Isosceles Trapezoids
This is a property of an isosceles trapezoid. If a quadrilateral is an isosceles trapezoid, then each pair of base angles is congruent.
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