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8.5 Trapezoids
and Kites
Using properties of trapezoids A trapezoid is a
quadrilateral with exactly one pair of parallel sides.
The parallel sides are the bases.
A trapezoid has two pairs of base angles.
For instance in trapezoid ABCD D and C are one pair of base angles. The other pair is A and B. The nonparallel sides are the legs of the trapezoid.
base
base
legleg
A B
D C
Using properties of trapezoids
If the legs of a
trapezoid are
congruent, then
the trapezoid is
an isosceles
trapezoid.
Trapezoid Theorems
Theorem 8.14
If a trapezoid is
isosceles, then
each pair of base
angles is congruent.
A ≅ B, C ≅ D
A B
D C
Trapezoid Theorems
Theorem 8.15
If a trapezoid has a
pair of congruent
base angles, then it
is an isosceles
trapezoid.
ABCD is an isosceles
trapezoid
A B
D C
Trapezoid Theorems
Theorem 8.16
A trapezoid is isosceles
if and only if its
diagonals are
congruent.
ABCD is isosceles if and
only if AC ≅ BD.
A B
D C
Ex. 1: Using properties of Isosceles
Trapezoids
PQRS is an isosceles trapezoid. Find mP, mQ, mR.
PQRS is an isosceles trapezoid, so mR = mS = 50°.
Because S and P are consecutive interior angles formed by parallel lines, they are supplementary.
So mP = 180°- 50° = 130°, and mQ = mP = 130°
m PS = 2.16 cm
m RQ = 2.16 cm
S R
P Q
50°
You could also add 50 and 50,
get 100 and subtract it from
360°. This would leave you
260/2 or 130°.
Ex. 2: Using properties of trapezoids Show that ABCD is a trapezoid.
Compare the slopes of opposite sides.
The slope of AB = 5 – 0 = 5 = - 1
0 – 5 -5
The slope of CD = 4 – 7 = -3 = - 1
7 – 4 3
The slopes of AB and CD are equal,
so AB ║ CD.
The slope of BC = 7 – 5 = 2 = 1
4 – 0 4 2
The slope of AD = 4 – 0 = 4 = 2
7 – 5 2
The slopes of BC and AD are not equal, so BC is not parallel to AD.
So, because AB ║ CD and BC is not parallel to AD, ABCD is a trapezoid.
8
6
4
2
5 10 15A(5, 0)
D(7, 4)
C(4, 7)
B(0, 5)
Midsegment of a trapezoid
The midsegment of a
trapezoid:
the segment that
connects the
midpoints of its legs.
Theorem 8.17 is
similar to the
Midsegment Theorem
for triangles.
midsegment
B C
DA
Theorem 8.17: Midsegment of a
trapezoid
The midsegment of
a trapezoid is
parallel to each
base and its length
is one half the sums
of the lengths of the
bases.
MN║AD, MN║BC
MN = ½ (AD + BC)
NM
A D
CB
Ex. 3: Finding Midsegment lengths of
trapezoids A baker is making a
cake like the one at the right. The top layer has a diameter of 8 inches and the bottom layer has a diameter of 20 inches. How big should the middle layer be?
Ex. 3: Finding Midsegment lengths of
trapezoids
Use the midsegment theorem for trapezoids.
DG = ½(EF + CH)=
½ (8 + 20) = 14”
C
D
E
D
G
F
Using properties of kites
A kite is a quadrilateral
that has two pairs of
consecutive congruent
sides, but opposite
sides are not
congruent.
Kite Theorems
Theorem 8.18
If a quadrilateral is
a kite, then its
diagonals are
perpendicular.
AC BD
B
C
A
D
Kite Theorems
Theorem 8.19
If a quadrilateral is a
kite, then exactly
one pair of opposite
angles is congruent.
A ≅ C, B ≅ D
B
C
A
D
Ex. 4: Using the diagonals of a kite
WXYZ is a kite so the diagonals are perpendicular.
You can use the Pythagorean Theorem to find the side lengths.
WX = √202 + 122 ≈ 23.32
XY = √122 + 122 ≈ 16.97
Because WXYZ is a kite, WZ = WX ≈ 23.32, and ZY = XY ≈ 16.97
12
12
20
12
U
X
Z
W Y
Ex. 5: Angles of a kite
Find mG and mJ in the
diagram.
SOLUTION:
GHJK is a kite, so G ≅ J and mG = mJ.
2(mG) + 132° + 60° = 360° Sum of measures of int. s
of a quad. is 360°
2(mG) = 168° Simplify
mG = 84° Divide each side by 2.
So, mJ = mG = 84°
J
G
H K
132° 60°
Homework
Chapter 8.5
3, 7, 9, 12, 14, 16, 19, 26,
34, 37