8.5 Warm Up Warm Up Lesson Quiz Lesson Quiz Lesson Presentation Lesson Presentation Use Properties of Trapezoids and Kites

8.5 Warm-Up ANSWER 125, 125 ANSWER isosceles 2. If AX and BY intersect at point P, what kind of triangle is XPY ? 1. What are the values of x and y ? Use the figure to answer the questions.

8.5 Example 1 Show that ORST is a trapezoid. SOLUTION Compare the slopes of opposite sides. Slope of RS = Slope of OT = 2 – 0 4 – 0 = 2 4 = 1 2 The slopes of RS and OT are the same, so RS OT. 4 – 3 2 – 0 = 1 2

8.5 Example 1 Slope of ST = 2 – 4 4 – 2 = –2 2 = –1 Slope of OR = – 0 0 – 0 =, which is undefined 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. ANSWER

8.5 Guided Practice 1. WHAT IF? In Example 1, suppose the coordinates of point S are (4, 5). What type of quadrilateral is ORST ? Explain. ANSWER Parallelogram; opposite pairs of sides are parallel. In Example 1, which of the interior angles of quadrilateral ORST are supplementary angles? Explain your reasoning. 2. ANSWER O and R, T and S; Consecutive Interior Angles Theorem

8.5 Example 2 The stone above the arch in the diagram is an isosceles trapezoid. Find m K, m M, and m J. Arch 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°.

8.5 Example 2 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° – 85° = 95°. STEP 3 Find m J. Because J and M are a pair of base angles, they are congruent, and m J = m M = 95°. ANSWER So, m J = 95°, m K = 85°, and m M = 95°.

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

8.5 Guided Practice In Exercises 3 and 4, use the diagram of trapezoid EFGH. 3. If EG = FH, is trapezoid EFGH isosceles? Explain. ANSWERyes, Theorem 8.16

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

8.5 Guided Practice 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 L K M 9 cm 12 cm N P ANSWER ( 9 + x ) = cm; Solve for x to find ML.

8.5 Example 4 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. Find m D in the kite shown at the right.

8.5 Example 4 m D + m F + 124° + 80° = 360° Corollary to Theorem 8.1 m D + m D + 124° + 80° = 360° 2(m D) + 204° = 360° Combine like terms. Substitute m D for m F. Solve for m D. m D = 78°

8.5 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? ANSWER 25; 75°

8.5 Lesson Quiz 1. Find m A, m C, m D. ANSWER 124°, 56°, 124°

8.5 Lesson Quiz 2. Find the length of the midsegment of the trapezoid. ANSWER 25

8.5 Lesson Quiz Use the figure to find the indicated measures. 3. If m XYZ = 80° and m XWZ = 48°, find m YZW. ANSWER 116° 4. If XO = 2, OZ = 2, YO = 6, and OW = 8, find the lengths of the sides of the kite. ANSWER 2,