1. 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.

2. 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.

3. 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.
= ( ) 1 2
Substitute 12 for PQ and 28 for XU.
= 20
Simplify.
ANSWER
The length MN is 20 inches.

4. 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.

5. GUIDED PRACTICE
for Examples 2 and 3
4. If m HEF = 70o and m FGH = 110o, is trapezoid EFGH isosceles?Explain.
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.
ANSWER
G and F are consecutive interior angles as EF HG. Because FGH=110°, therefore EFG=70° as they are supplementary angles.

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

7. for Examples 2 and 3 GUIDED PRACTICE 1 NP (JK + ML) = 2 1 12 = (9 + x)
Apply Theorem 8.17.
= (9 + x) 1 2 12
Substitute
= 9 + x 24
Multiply each side by 2
= x 15
Simplify.