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

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

EXAMPLE 1 Use a coordinate plane 2 – 4 4 – 2 = –2 2 –1 Slope of ST = 3 3 – 0 0 – 0 = , which is undefined Slope of OR = 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

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. 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

EXAMPLE 2 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= 85o.

EXAMPLE 2 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 = 180o – 85o = 95o. STEP 3 Find m J. Because J and M are a pair of base angles, they are congruent, and m J = m M = 95o. ANSWER So, m J = 95o, m K = 85o, and m M = 95o.

Use the midsegment of a trapezoid EXAMPLE 3 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 (PQ + SR) 1 2 = Apply Theorem 8.17. = (12 + 28) 1 2 Substitute 12 for PQ and 28 for XU. = 20 Simplify. ANSWER The length MN is 20 inches.

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. ANSWER yes, Theorem 8.16

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

15 cm; Solve for x to find ML. 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 L K M 9 cm 12 cm N P ANSWER ( 9 + x ) = 12 1 2 15 cm; Solve for x to find ML.

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.

EXAMPLE 4 Apply Theorem 8.19 m D + m F +124o + 80o = 360o Corollary to Theorem 8.1 m D + m D +124o + 80o = 360o Substitute m D for m F. 2(m D) +204o = 360o Combine like terms. m D = 78o Solve for m D.

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

Daily Homework Quiz 1. Find m A, m C, m D. ANSWER 124o, 56o, 124o.

Daily Homework Quiz 2. Find the length of the midsegment of the trapezoid. ANSWER 25

Daily Homework Quiz Use the figure to find the indicated measures. 3. If m XYZ = 80o and m XWZ = 48o, find m YZW. ANSWER 116o 4. If XO = 2, OZ = 2, YO = 6, and OW = 8, find the lengths of the sides of the kite. ANSWER 2 , 10 68