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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