Math 9 properties of trapezoids and kites.ppt

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 & DA
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 BDAC ifonly 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. NM
A D
B C BC) AD
2
1
MN ,BC ll MNAD llMN ( , 

9. A B
CD
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
CD
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 BDAC

13. Theorem:
Opposite Angles of a Kite
If a quadrilateral is a kite, then exactly one
pair of opposite angles are congruentD
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 ContinuedWX = 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.
12
2
+ 20
2
= (WX)
2
144 + 400 = (WX)
2
544 = (WX)
2
12
2
+ 12
2
= (XY)
2
144 + 144 = (XY)
2
288 = (XY)
2XY =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