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Use the figure to answer the questions.
1. What are the values of x and y?
ANSWER
125, 125
2. If AX and BY intersect at point P, what kind of
triangle is XPY?
ANSWER
isosceles
EXAMPLE 1
Use a coordinate plane
Show that ORST is a trapezoid.
SOLUTION
Compare the slopes of
opposite sides.
4–3
Slope of RS = 2 – 0 =
2–0
Slope of OT = 4 – 0 =
1
2
1
2
=
2
4
The slopes of RS and OT are the same, so RS
OT .
EXAMPLE 1
Slope of ST =
Use a coordinate plane
2 – 4 –2 –1
4–2= 2 =
3 ,
Slope of OR = 30 –– 00 =
0
which is undefined
The slopes of ST and OR are not the same, so ST is not
parallel to OR .
ANSWER
Because quadrilateral ORST has exactly one pair of
parallel sides, it is a trapezoid.
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.
2. In Example 1, which of the interior angles of
quadrilateral ORST are supplementary angles?
Explain your reasoning.
ANSWER
O and R , T and S;
Consecutive Interior Angles Theorem
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.
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.
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, Theorem 8.16
GUIDED PRACTICE
4.
for Examples 2 and 3
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.
GUIDED PRACTICE
5.
for Examples 2 and 3
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
K
P
L
1
15 cm; Solve 2 ( 9 + x ) = 12 for x to find ML.
EXAMPLE 4
Find m
Apply Theorem 8.19
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.
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
2(m
D) +204o = 360o
m
D = 78o
D for m
Combine like terms.
Solve for m
D.
F.
GUIDED PRACTICE
6.
for Example 4
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
Homework:
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