Download 8.5 Use Properties of Trapezoids and Kites

8.5 Use Properties of Trapezoids
and Kites
Hubarth
Geometry
A trapezoid is a quadrilateral with exactly one pair of parallel sides.
The parallel sides are the bases. The nonparallel sides are called the legs.
A trapezoid has two pairs of base angles. In trapezoid ABCD, C
and D are one pair of base angles. A andB are the other pair.
If the legs of a trapezoid are congruent, then the trapezoid is an
isoscles trapezoid.
A
Base
Leg
D
B
Leg
Base
C
Isosceles trapezoid
Theorem 8.14
Words
Symbols
If a trapezoid is isosceles, then each
pair of base angles are congruent.
A
In the isosceles trapezoid ABCD,
A  B and C  D.
B
D
C
Theorem 8.15
Words
Symbols
If a trapezoid has a pair of congruent base
angles, then it is isosceles.
In Trapezoid ABCD, if C  D
then ABCD is isosceles.
Theorem 8.16
A trapezoid is isosceles if and only if its diagonals are congruent.
A
B
D
C
Ex 1. Find Angle Measure of Trapezoid
PQRS is an isosceles trapezoid.
Find the missing angle measures.
P
S
Solution
P  Q  50
Because P and S are same-side interior angles
P+S=180, 50+S  180, S  130
S  R  130
Q
50
R
The midsegment of a trapezoid is the segment that connects the midpoints of its legs.
The midsegment of a trapezoid is parallel to the bases.
The length of the midsegment of a trapezoid is half the
sum of the lengths of the bases
C
B
M
1
MN  ( AD  BC )
2
N
A
D
Ex 2. Midsegment of a Trapezoid
E
Find the length of the midsegment DG of
trapezoid CEFH.
1
( EF  CH )
2
1
= (8  20)
2
1
= (28)
2
=14
DG 
F
G
D
C
Solution
8
20
H
Theorem 8.18
If a quadrilateral is a kite, then its diagonals are perpendicular.
Theorem 8.19
If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent.
Ex 3 Apply Theorem 8.19
Find m
D in the kite shown at the right.
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.
m
D+m
F +124o + 80o = 360o
m
D+m
D +124o + 80o = 360o
2(m
D) +204o = 360o
m
D = 78o
F
Practice
62
G
1. FGHJ is an Isosceles trapezoid.
Find the missing angle measures.
H
G  118, H=118, J=62
J
2. Find the length of the midsegment MN of
the trapezoid.
8
a.
M
N
14
1
MN  (8  14)
2
1
MN  (22)
2
MN  11
M
b.
24
N
1
MN  (18  24)
2
18
1
MN  (42)
2
MN  21
3. 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?
25; 75o