9.5 Kites and Trapezoids

Use the properties of kites and trapezoids to solve problems
Classify quadrilaterals on a coordinate plane

1. Kites & Trapezoids
The student is able to (I can):
• Use properties of kites and trapezoids to solve pr oblems

2. kite kite kite kite –a quadrilateral with exactly two pairs of con gruent
consecutive nonparallel sides.
In order for a quadrilateral to be a kite, no nono nosides can
be parallel and opposite sides cannot be congruent.

3. If a quadrilateral is a kite, then its diagonals ar e
perpendicular.
If a quadrilateral is a kite, then exactly one pair of opposite
angles is congruent.

4. Examples
In kite NAVY, m∠YNA=54°and m∠VYX=52°. Find each
measure.
1. m∠NVY
2. m∠XYN
3. m∠NAV
N
A
V
Y
X

5. Examples
In kite NAVY, m∠YNA=54°and m∠VYX=52°. Find each
measure.
1. m∠NVY
90 –52 = 38°
2. m∠XYN
3. m∠NAV
63 + 52 = 115°
N
A
V
Y
X
180 54 126
63
2 2−
= = °

6. trapezoid trapezoid trapezoid trapezoid –a quadrilateral with exactly one pair of parallel
sides. The parallel sides are called bases bases bases basesand the
nonparallel sides are the legs legslegs legs. Angles along one leg
are supplementary.
A trapezoid whose legs are congruent is called an
isosceles trapezoid isosceles trapezoid isosceles trapezoid isosceles trapezoid.
>
>
base
base
leg
leg
base angles
base angles

7. Isosceles Trapezoid Theorems If a quadrilateral is an isosceles trapezoid, then each pair of
base angles is congruent.
If a trapezoid has one pair of congruent base angle s, then the
trapezoid is isosceles.
A trapezoid is isosceles if and only if its diagona ls are
congruent.
>
>
T
R A
P
∠R≅∠A, ∠T≅∠P
TR AP
≅
TA RP
≅

8. Examples
1. Find the value of x.
2. If NS=14 and BA=25, find SE.
140°
(5x)°
B E
A N
SSSS
40°

9. Examples
1. Find the value of x.
5x= 40
x= 8
2. If NS=14 and BA=25, find SE.
SE= 25 –14 = 11
140°
(5x)°
B E
A N
SSSS
40°

10. Trapezoid Midsegment Theorem
The midsegment of a trapezoid is parallel to each b ase,
and its length is one half the sum of the lengths o f the
bases.
>
>
H
A
Y
F
V
R
,
AV HF AV YR
N N
  
( )
12
AV HF YR
= +
>

11. Examples
is the midsegment of trapezoid OFIG.
1. If OF=22 and GI=30, find MY.
2. If OF=16 and MY=18, find GI.
>
> O
M
G
F
Y
I
MY

12. Examples
is the midsegment of trapezoid OFIG.
1. If OF=22 and GI=30, find MY.
2. If OF=16 and MY=18, find GI.
>
> O
M
G
F
Y
I
MY
( )
1
22 30 26
2
MY
= + =
16
18
+2 +2
20
GH= 20

13. Coordinate Plane Quads
If we are just given the coordinates of a quadrilat eral, or
even from the graph, it can be tricky to classify i t. It’s usually
easiest to go back to the definitions:
Parallelogram: Two pairs of parallel sides
Rectangle: Four right angles
Rhombus: Four congruent sides
Square: Rectangle and rhombus
Trapezoid: One pair of parallel sides
Kite: Two pairs of consecutive congruent sides

14. To show sides are congruent, use the distance formu la:
To show sides are parallel, use the slope formula:
Hint: You might notice that both formulas use the
differences in the xand ycoordinates. Once you have
figured the differences for one formula, you can ju st use the
same numbers in the other formula.
( ) ( )
2 2
2 1 2 1
d x x y y
= − + −
2 1 2 1
y y
m
x x
−
=
−

15. Example: What is the most specific name for the
quadrilateral formed by T(–6, –2), O(–3, 2),
Y(1, –1), and S(–2, –5)?

16. We might suspect this is a square, but we still hav e to show
this. To show that it is a rectangle, we look at a ll of the
slopes:
Two sets of equal slopes prove this is a parallelog ram. Four
90°angles prove this is a rectangle.
(
)
( )
2 2
4
3 6 3
TO
m
− −
= =
− − −
( )
1 2 3
1 3 4
OY
m
− −
= =−
− −
(
)
5 1
4 4
2 1 3 3
YS
m
− − −
−
= = =
− − −
(
)
( )
2 5
3
6 2 4
ST
m
− − −
= =−
− − −
opposite
reciprocals →90°
opposite
reciprocals →90°
equal slopes →parallel lines

17. To prove it is a square, we also have to show that all the sides
are congruent. Since we have already set up the sl opes, this
will be pretty straightforward:
Since it has four right angles and four congruent s ides, TOYS
is a square.
2 2
3 4 5
TO
= + =
( )
2
2
4 3 5
OY
= + − =
( ) ( )
2 2
3 4 5
YS
= − + − =
( )
2
2
4 3 5
ST
= − + =