Warm-Up Exercises. Use the figure to answer the questions. 1. What are the values of x and y? ANSWER 125, 125

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1 Warm-Up Exercises Use the figure to answer the questions. 1. What are the values of x and y? 125, If AX and BY intersect at point P, what kind of triangle is XPY? isosceles

2 EXAMPLE Warm-Up 1Exercises Use a coordinate plane Show that ORST is a trapezoid. SOLUTION Compare the slopes of opposite sides. 4 3 Slope of RS = 2 0 = 1 2 Slope of OT = = 2 4 = 1 2 The slopes of RS and OT are the same, so RS OT.

3 EXAMPLE Warm-Up 1Exercises Use a coordinate plane Slope of ST = = 2 2 = 1 Slope of OR = =, which is undefined 0 The 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.

4 Warm-Up Exercises GUIDED PRACTICE for Example 1 1. What If? In Example 1, suppose the coordinates of point S are (4, 5). What type of quadrilateral is ORST? Explain. Parallelogram; opposite pairs of sides are parallel. 2. In Example 1, which of the interior angles of quadrilateral ORST are supplementary angles? Explain your reasoning. O and R, T and S; Consecutive Interior Angles Theorem

5 EXAMPLE Warm-Up 2Exercises Use properties of isosceles trapezoids Arch The stone above the arch in the diagram is an isosceles trapezoid. Find m K, m M, and m J. SOLUTION STEP 1 Find m K. JKLM is an isosceles trapezoid, so K and L are congruent base angles, and m K = m L= 85 o.

6 EXAMPLE Warm-Up 2Exercises Use properties of isosceles trapezoids STEP 2 Find m M. Because L and M are consecutive interior angles formed by LM intersecting two parallel lines, they are supplementary. So, m M = 180 o 85 o = 95 o. STEP 3 Find m J. Because J and M are a pair of base angles, they are congruent, and m J = m M = 95 o. So, m J = 95 o, m K = 85 o, and m M = 95 o.

7 EXAMPLE Warm-Up 3Exercises Use the midsegment of a trapezoid In the diagram, MN is the midsegment of trapezoid PQRS. Find MN. SOLUTION Use Theorem 8.17 to find MN. MN 1 = (PQ + SR) 2 1 = 2 ( ) Apply Theorem Substitute 12 for PQ and 28 for XU. = 20 Simplify. The length MN is 20 inches.

8 Warm-Up Exercises GUIDED PRACTICE for Examples 2 and 3 In Exercises 3 and 4, use the diagram of trapezoid EFGH. 3. If EG = FH, is trapezoid EFGH isosceles? Explain. Yes; Theorem 8.16: A trapezoid is isosceles if and only if its diagonals are congruent.

9 Warm-Up Exercises GUIDED PRACTICE for Examples 2 and 3 4. If m HEF = 70 o and m FGH = 110 o, is trapezoid EFGH isosceles? Explain. SAMPLE Yes; m EFG = 70 by Consecutive Interior Angles Theorem making EFGH an isosceles trapezoid by Theorem 8.15.

10 Warm-Up Exercises GUIDED PRACTICE for Examples 2 and 3 5. 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. J N 9 cm 12 cm K P M L 1 15 cm; Solve ( 9 + x ) = 12 2 for x to find ML.

11 Warm-Up Exercises EXAMPLE 4 Apply Theorem 8.19 Find m D in the kite shown at the right. SOLUTION By 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.

12 EXAMPLE Warm-Up 4Exercises Apply Theorem 8.19 m D + m F +115 o + 73 o = 360 o Corollary to Theorem 8.1 m D + m D +115 o + 73 o = 360 o Substitute m D for m F. 2(m D) +188 o = 360 o Combine like terms. m D = 86 o Solve for m D.

13 Warm-Up Exercises GUIDED PRACTICE for Example 4 6. In a kite, the measures of the angles are 3x o, 75 o, 90 o, and 120 o. Find the value of x. What are the measures of the angles that are congruent? 25; 75 o

14 Warm-Up Exercises Daily Homework Quiz 1. Find m A, m C, m D. 124 o, 56 o, 124 o.

15 Daily Warm-Up Homework Exercises Quiz 2. Find the length of the midsegment of the trapezoid. 25

16 Daily Warm-Up Homework Exercises Quiz Use the figure to find the indicated measures. 3. If m XYZ = 80 o and m XWZ = 48 o, find m YZW. 116 o 4. If XO = 2, OZ = 2, YO = 6, and OW = 8, find the lengths of the sides of the kite. 2 10, 2 17

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