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6.5
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Trapezoids and Kites
Goals p Use properties of trapezoids.
p Use properties of kites.
VOCABULARY
Trapezoid A trapezoid is a quadrilateral with exactly one pair
of parallel sides.
Bases of a trapezoid The parallel sides of a trapezoid are
called bases.
Base angles of a trapezoid A trapezoid has two pairs of base
angles. Each pair shares a base as a side.
Legs of a trapezoid The nonparallel sides of a trapezoid are
called legs.
Isosceles trapezoid An isosceles trapezoid is a trapezoid with
congruent legs.
Midsegment of a trapezoid The midsegment of a trapezoid is
the segment that connects the midpoints of its legs.
Kite A kite is a quadrilateral that has two pairs of
consecutive congruent sides, but its opposite sides are not
congruent.
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THEOREM 6.14
If a trapezoid is isosceles, then each
pair of base angles is congruent .
A
aA c a B , a C c aD
B
D
C
THEOREM 6.15
If a trapezoid has a pair of congruent
base angles , then it is an isosceles
trapezoid.
A
B
D
ABCD is an isosceles trapezoid.
C
THEOREM 6.16
A trapezoid is isosceles if and only if its
diagonals are congruent .
A
ABCD is isosceles if and only
D
&* c BD
&* .
if AC
Example 1
B
C
Using Properties of Isosceles Trapezoids
WXYZ is an isosceles trapezoid.
Find maX, maY, and maZ.
W
Solution
p WXYZ is an isosceles trapezoid,
so maX ! ma W ! 110 ".
110"
Z
X
Y
p aW and aZ are consecutive interior angles formed by parallel
lines, so they are supplementary .
maW # maZ ! 180 "
Consecutive Interior Angles Theorem
110 " # maZ ! 180 "
Substitute for maW.
maZ ! 70 "
Subtract 110 ! from each side.
p WXYZ is an isosceles trapezoid, so maY ! ma Z ! 70 ".
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Example 2
Using Properties of Trapezoids
Show that HIJK is a trapezoid.
y
Compare the slopes of opposite sides.
7
8%6
2
1
&* ! $$ ! $$ ! $$
Slope of HK
4%0
4
2
5
K (4, 8)
H (0, 6)
3
3%0
3
1
** ! $$ ! $$ ! $$
Slope of IJ
9%3
6
2
J (9, 3)
1
I (3, 0)
1
&* and IJ
** are equal,
The slopes of HK
&* ! IJ
**.
so HK
5
7
9 x
6%0
6
&* ! $$ ! $$ ! %2
Slope of HI
0%3
%3
8%3
5
Slope of &*
JK ! $$ ! $$ ! %1
4%9
%5
&* and &*
&* is not parallel to &
The slopes of HI
JK are not equal, so HI
JK .
&* ! IJ
** and HI
&* is not parallel to &*,
Answer Because HK
JK HIJK is a
trapezoid .
THEOREM 6.17: MIDSEGMENT THEOREM FOR TRAPEZOIDS
The midsegment of a trapezoid is parallel
to each base and its length is one half the
sum of the lengths of the bases.
B
C
M
1
&** ! &**
&** ! BC
&* , MN ! $$(AD # BC)
MN
AD , MN
2
N
A
D
Checkpoint Complete the following exercise.
1. ABCD is an isosceles trapezoid.
Find maA, maB, and maC.
A
35"
B
D
C
maA ! 35"
maB ! maC ! 145"
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THEOREM 6.18
C
If a quadrilateral is a kite, then its
diagonals are perpendicular .
B
D
&* ∏ BD
&*
AC
A
THEOREM 6.19
C
If a quadrilateral is a kite, then
exactly one pair of opposite angles
are congruent.
B
D
A
aA c aC, aB ç aD
Example 3
Angles of a Kite
Find maS and maU.
S
STUV is a kite, so aS c a U and
maS ! ma U .
V
37"
73"
T
U
2 (maS) # ma T # ma V ! 360 "
2 (maS) # 73 " # 37 " ! 360 "
2 (maS) ! 250 "
maS ! 125 "
Sum of measures of int. A
of quad. is 360!.
Substitute.
Simplify.
Divide each side by 2 .
Answer So, maS ! ma U ! 125 ".
Checkpoint Complete the following exercise.
2. Find maX and maZ.
Y
97"
maX ! maZ ! 62"
X
139"
W
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Geometry Notetaking Guide • Chapter 6
Z