EXAMPLE 2 Use properties of isosceles trapezoids Arch

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= 85 o.

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 = 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. ANSWER So, m J = 95 o, m K = 85 o, and m M = 95 o.

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 = 1 (PQ + SR) 2 1 = 2 (12 + 28) = 20 ANSWER Apply Theorem 8. 17. Substitute 12 for PQ and 28 for XU. Simplify. 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, trapezoid EFGH is isosceles, if and only if its diagonals are congruent. As, it is given its diagonals are congruent, therefore by theorem 8. 16 the trapezoid is isosceles.

GUIDED PRACTICE for Examples 2 and 3 4. If m HEF = 70 o and m FGH = 110 o, is trapezoid EFGH isosceles? Explain. ANSWER G and F are consecutive interior angles as EF HG. Because FGH=110°, therefore EFG=70° as they are supplementary angles. BHG=110°, as the sum of a quadrilateral is 360°. The base angles are congruent that is, 110° each therefore, the trapezoid is isosceles by theorem 8. 15.

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. ANSWER J 9 cm N 12 cm M NP is the midsegment of trapezoid JKLM. and NP = 1 ( JK + ML) 2 x cm K P L

GUIDED PRACTICE NP = 1 (JK + ML) 2 12 = 1 (9 + x) 2 for Examples 2 and 3 Apply Theorem 8. 17. Substitute 24 = 9 + x Multiply each side by 2 15 = x Simplify.