# 8.5 use properties of trapezoids and kites

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8.5 Use Properties of Trapezoids and KitesBell ThingerANSWER125, 125ANSWERisosceles2.If AX and BY intersect at point P, what kind of quadrilateral is it?1.What are the values of x and y?Use the figure to answer the questions.

8.5

8.5Example 1Show that ORST is a trapezoid.

SOLUTIONCompare the slopes of opposite sides.Slope of RS =Slope of OT =2 0 4 0=24=12The slopes of RS and OT are the same, so RS OT .4 3 2 0=128.53Example 1Slope of ST =2 4 4 2=22=1 Slope of OR =303 0 0 0=,which is undefinedThe slopes of ST and OR are not the same, so ST is not parallel to OR .Because quadrilateral ORST has exactly one pair of parallel sides, it is a trapezoid.ANSWER8.54

8.5

8.5Example 2

The stone above the arch in the diagram is an isosceles trapezoid. Find m K, m M, and m J.ArchSOLUTIONSTEP 1Find m K. JKLM is an isosceles trapezoid, so K and L are congruent base angles, and m K = m L= 85.8.57Example 2STEP 2Find m M. Because L and M are consecutive interior angles formed by LM intersecting two parallel lines, they are supplementary. So, m M = 180 85 = 95.STEP 3Find m J. Because J and M are a pair of base angles, they are congruent, and m J = m M = 95.ANSWERSo, m J = 95, m K = 85, and m M = 95.8.58

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8.5Example 3SOLUTIONUse Theorem 8.17 to find MN.In the diagram, MN is the midsegment of trapezoid PQRS. Find MN.

MN (PQ + SR)12=Apply Theorem 8.17.= (12 + 28)12Substitute 12 for PQ and 28 for XU.Simplify.= 20The length MN is 20 inches.8.511Guided PracticeIn Exercises 3 and 4, use the diagram of trapezoid EFGH.3. If EG = FH, is trapezoid EFGH isosceles? Explain.

ANSWERyes, Theorem 8.168.512Guided PracticeIn Exercises 3 and 4, use the diagram of trapezoid EFGH.4. If m HEF = 70o and m FGH = 110o, is trapezoid EFGH isosceles? Explain.SAMPLE ANSWERYes;m EFG = 70 by Consecutive Interior Angles Theorem making EFGH an isosceles trapezoidby Theorem 8.15.

8.513Guided Practice5. In trapezoid JKLM, J and M are right angles, and JK = 9 cm. The length of the midsegment NP of trapezoid JKLM is 12 cm. Sketch trapezoid JKLM and its midsegment. Find ML. Explain your reasoning.JLKM9 cm12 cmNPANSWER( 9 + x ) = 121215 cm; Solve for x to find ML. 8.514

8.5Example 4SOLUTIONBy Theorem 8.19, DEFG has exactly one pair of congruent opposite angles. Because E G, D and F must be congruent. So, m D = m F. Write and solve an equation to find m D.

Find m D in the kite shown at the right.8.516Example 4m D + m F + 124 + 80 = 360Corollary to Theorem 8.1m D + m D + 124 + 80 = 3602(m D) + 204 = 360Combine like terms.Substitute m D for m F.Solve for m D. m D = 788.517Guided Practice6. 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?ANSWER25; 758.518Exit Slip1. Find m A, m C, m D.

ANSWER124, 56, 1248.5Exit Slip2. Find the length of the midsegment of thetrapezoid.

ANSWER258.5Exit SlipUse the figure to find the indicated measures.

3. If m XYZ = 80 andm XWZ = 48, findm YZW.

ANSWER1164. If XO = 2, OZ = 2, YO = 6, and OW = 8, find thelengths of the sides of the kite.ANSWER2 , 102 178.5Pg 566-569

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