1. 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

2. 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 .

3. 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

4. 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

5. 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.

6. 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.

7. 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.

8. 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

9. 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.

10. 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.

11. 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 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.

13. 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

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

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

16. 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 , 10 68