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## Warm Up

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**Use Properties of Trapezoids and Kites**Warm Up Lesson Presentation Lesson Quiz**ANSWER**125, 125 2. IfAXandBYintersect at pointP,what kind of triangle isXPY? ANSWER isosceles Warm-Up Use the figure to answer the questions. 1. What are the values of xandy?**=**= Slope of RS = 4 – 3 2 – 0 Slope of OT = = 2 – 0 4 – 0 The slopes of RSand OTare the same, so RSOT . 2 1 1 2 4 2 Example 1 Show that ORSTis a trapezoid. SOLUTION Compare the slopes of opposite sides.**–2**–1 = = Slope of ST = 2 , which is undefined = Slope of OR = The slopes of ST and ORare not the same, soST is not parallel to OR . 2 – 4 3 – 0 ANSWER 4 – 2 0 – 0 Because quadrilateral ORST has exactly one pair of parallel sides, it is a trapezoid. 3 0 Example 1**2.**In Example 1, which of the interior angles of quadrilateral ORSTare supplementary angles? Explain your reasoning. Parallelogram; opposite pairs of sides are parallel. ANSWER O and R , T and S;Consecutive Interior Angles Theorem ANSWER Guided Practice 1. WHAT IF?In Example 1, suppose the coordinates of point Sare (4, 5). What type of quadrilateral is ORST? Explain.**Arch**The stone above the arch in the diagram is an isosceles trapezoid. Find mK, mM, and mJ. STEP 1 Find mK.JKLMis an isosceles trapezoid, so Kand Lare congruent base angles, and mK = mL= 85°. Example 2 SOLUTION**Find m M. Because Land M are consecutive**interior angles formed by LMintersecting two parallel lines, they are supplementary. So, mM = 180° – 85° = 95°. STEP 3 Find mJ. Because J and M are a pair of base angles, they are congruent, and mJ = mM= 95°. ANSWER So, mJ = 95°, m K = 85°, and m M = 95°. Example 2 STEP 2**In the diagram,MNis the midsegment of trapezoidPQRS. FindMN.**SOLUTION Use Theorem 8.17 to findMN. 1 =(12+ 28) 2 1 MN (PQ + SR) = 2 Example 3 Apply Theorem 8.17. Substitute 12 for PQand 28 for XU. = 20 Simplify. The length MNis 20 inches.**ANSWER**yes, Theorem 8.16 Guided Practice In Exercises 3 and 4, use the diagram of trapezoidEFGH. 3. If EG = FH, is trapezoid EFGHisosceles? Explain.**4. If mHEF = 70o and mFGH =110o, is trapezoid**EFGHisosceles? Explain. SAMPLE ANSWER Yes; mEFG =70° by Consecutive Interior Angles Theorem making EFGH an isosceles trapezoidby Theorem 8.15. Guided Practice In Exercises 3 and 4, use the diagram of trapezoidEFGH.**5. In trapezoid JKLM, Jand M are right angles,**and JK = 9 cm. The length of the midsegment NPof trapezoid JKLMis 12 cm. Sketch trapezoid JKLMand its midsegment. Find ML. Explain your reasoning. J K 9 cm ANSWER 12 cm P N M L 1 2 15 cm; Solve for x to find ML. ( 9 + x ) = 12 Guided Practice**By Theorem 8.19, DEFGhas exactly one pair of congruent**opposite angles. Because E G,Dand Fmust be congruent. So, m D = mF. Write and solve an equation to find mD. Example 4 Find m Din the kite shown at the right. SOLUTION**mD + m F + 124° + 80° = 360°**mD + m D + 124° + 80° = 360° 2(m D)+ 204° = 360° m D = 78° Substitute m Dfor m F. Solve for m D. Example 4 Corollary to Theorem 8.1 Combine like terms.**25; 75°**ANSWER Guided Practice 6. In a kite, the measures of the angles are 3x°, 75°,90°, and 120°. Find the value of x. What are the measures of the angles that are congruent?**1. Find m A, m C, m D.**124°, 56°, 124° ANSWER Lesson Quiz**2. Find the length of the midsegment of the**trapezoid. 25 ANSWER Lesson Quiz**3. If m XYZ = 80° and**m XWZ=48°, find m YZW. 10 116° 2 , ANSWER ANSWER 2 17 Lesson Quiz Use the figure to find the indicated measures. 4. If XO = 2,OZ = 2, YO = 6, and OW = 8, find the lengths of the sides of the kite.