8.58.5 Use Properties of Trapezoids and Kites

Bell Thinger

ANSWER 125, 125

ANSWER isosceles

2. If AX and BY intersect at point P, what kind of

quadrilateral is it?

1. What are the values of x and y?

Use the figure to answer the

questions.

8.5

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

=24

=12

The slopes of RS and OT are the same, so RS OT .

4 – 3 2 – 0

=12

8.5 Example 1

Slope of ST =2 – 4 4 – 2

=–22

= –1

Slope of OR =30

3 – 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

8.5

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

8.5

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)12= Apply Theorem 8.17.

= (12 + 28)12 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.

ANSWER yes, Theorem 8.16

8.5 Guided Practice

In Exercises 3 and 4, use the diagram of trapezoid EFGH.

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.

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 cmN P

ANSWER

( 9 + x ) = 1212

15 cm; Solve for x to find ML.

8.5

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°

8.5Exit Slip

1. Find m A, m C, m D.

ANSWER 124 , 56°, 124°

8.5Exit Slip

2. Find the length of the midsegment of the

trapezoid.

ANSWER 25

8.5Exit Slip

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 , 10 2 17

8.5

Pg 566-569

#8, 10, 15, 23, 26