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Geometry 6-6 Trapezoids and Kites
A
A trapezoid
is a quadrilateral with only one pair of
parallel sides. A trapezoid is NOT a parallelogram, so the
properties we've learned about parallelograms do not apply
here.
In a trapezoid, the parallel sides are called the bases
.
The nonparallel sides are called the legs of the trapezoid.
Angles that are formed by a base and a leg are called
base angles
. In the special case where the legs are
congruent, we have an isosceles trapezoid
.
Find each measure.
m XWZ
m XWZ = m YZW = 45 W
m WXY
m WXY = 180 - 45 = 135
XZ
XZ = WY = 15 + 10 = 25
XV
XV = VY = 10
X
15
V
10
>
B
D
Isosceles trapezoids have some special properties.
Theorem 6.21: If a trapezoid is isosceles, then each pair of
base angles is congruent.
Theorem 6.22: If a trapezoid has one pair of congruent
base angles, then it is an isosceles trapezoid.
Theorem 6.23: A trapezoid is isosceles if and only if its
diagonals are congruent.
The segment that connects the midpoints of the two legs of
a trapezoid is called the midsegment (or median)
.
Y
m YZW = 45
C
>
Z
Theorem 6.24 - Trapezoid Midsegment Theorem: The
midsegment of a trapezoid is parallel to both bases and its
measure is equal to the average of the measures of the two
bases.
A
B
C and D are midpoints.
D
C
Then AB || CD and CD || EF,
and CD = 1/2 (AB + EF).
E
F
x
12
16.8
14
x
22
Find the value of x.
1/2 (x + 16.8) = 12
1/2 x + 8.4 = 12
1/2 x = 3.6
x = 7.2
A kite
is a quadrilateral with exactly two pairs of
consecutive congruent sides. Kites are not parallelograms,
so the properties of parallelograms do not apply.
B
A
C
x = 1/2 (14 + 22)
x = 1/2 (36) = 18
Kites have a couple of interesting properties.
Theorem 6.25: If a quadrilateral is a kite, then its diagonals
are perpendicular.
Theorem 6.26: If a quadrilateral is a kite, then exactly one
pair of opposite angles is congruent.
ABCD is a kite.
D
B
If m BAD = 38 and m BCD = 50,
find m ADC.
A
38 + x + 50 + x = 360
D
2x + 88 = 360
If BG = 5 and GC = 8, find CD.
2x = 272
x = 136
52 + 82 = (BC)2
25 + 64 = (BC)2
89 = (BC)2
9.434 = BC
CD = 9.434
G
C