Mathematics 9

1. Trapezoids and Kites

2. Essential Questions
How do I use properties of trapezoids?
How do I use properties of kites?

3. Vocabulary
Trapezoid – a quadrilateral with exactly one
pair of parallel sides.
A trapezoid has two pairs of base angles. In
this example the base angles are A & B
and C & D
A
D C
B
leg leg
base
base

4. Base Angles Trapezoid Theorem
If a trapezoid is isosceles, then each
pair of base angles is congruent.
A
D C
B
 A   B,  C   D

5. Diagonals of a Trapezoid Theorem
A trapezoid is isosceles if and only if its
diagonals are congruent.
A
D C
B
BD
AC
if
only
and
if
isosceles
is
ABCD 

6. Example 1
PQRS is an isosceles trapezoid. Find m P,
m Q and mR.
50
S R
P Q
m R = 50 since base angles are congruent
mP = 130 and mQ = 130 (consecutive angles
of parallel lines cut by a transversal are )

7. Definition
Midline of a trapezoid – the segment
that connects the midpoints of the legs.
midsegment

8. Midline 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.
N
M
A D
B C
BC)
AD
2
1
MN
,
BC
ll
MN
AD
ll
MN (
, 


9. A B
C
D
F G
For trapezoid ABCD, F and G
Are midpoints of the legs.
If AB = 12 and DC = 24,
Find FG.
DC)
AB
2
1
FG ( 

12
24

10. A B
C
D
F G
For trapezoid ABCD, F and G
Are midpoints of the legs.
If AB = 7 and FG = 21,
Find DC.
DC)
AB
2
1
FG ( 

12
21
42 = 12 +DC
30 = DC
Multiply both sides by 2 to get rid of fraction:

11. Definition
Kite – a quadrilateral that has two pairs of
consecutive congruent sides, but opposite
sides are not congruent.

12. Theorem: Perpendicular
Diagonals of a Kite
If a quadrilateral is a kite, then its diagonals
are perpendicular.
D
C
A
B
BD
AC 

13. Theorem:
Opposite Angles of a Kite
If a quadrilateral is a kite, then exactly one
pair of opposite angles are congruent
D
C
A
B
A  C, B  D

14. Example 2
Find the side lengths of the kite.
20
12
12
12
U
W
Z
Y
X

15. Example 2 Continued
WX = 4 34
likewise WZ = 4 34
20
12
12
12
U
W
Z
Y
X
We can use the Pythagorean Theorem to
find the side lengths.
122 + 202 = (WX)2
144 + 400 = (WX)2
544 = (WX)2
122 + 122 = (XY)2
144 + 144 = (XY)2
288 = (XY)2
XY =12 2
likewise ZY =12 2

16. Example 3
Find mG and mJ.
60
132
J
G
H K
Since GHJK is a kite G  J
So 2(mG) + 132 + 60 = 360
2(mG) =168
mG = 84 and mJ = 84

17. Try This!
RSTU is a kite. Find mR, mS and mT.
x
125
x+30
S
U
R T
x +30 + 125 + 125 + x = 360
2x + 280 = 360
2x = 80
x = 40
So mR = 70, mT = 40 and mS = 125
125