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 problems

2. kitekitekitekite – a quadrilateral with exactly two pairs of congruent
consecutive nonparallel sides.
In order for a quadrilateral to be a kite, nononono sides can
be parallel and opposite sides cannot be congruent.

3. If a quadrilateral is a kite, then its diagonals are
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
X180 54 126
63
2 2
−
= = °

6. trapezoidtrapezoidtrapezoidtrapezoid – a quadrilateral with exactly one pair of parallel
sides. The parallel sides are called basesbasesbasesbases and the
nonparallel sides are the legslegslegslegs. Angles along one leg
are supplementary.
A trapezoid whose legs are congruent is called an
isosceles trapezoidisosceles trapezoidisosceles trapezoidisosceles 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 angles, then the
trapezoid is isosceles.
A trapezoid is isosceles if and only if its diagonals 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
AN
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
AN
SSSS
40°

10. Trapezoid Midsegment Theorem
The midsegment of a trapezoid is parallel to each base,
and its length is one half the sum of the lengths of the
bases.
>
>
H
A
Y
F
V
R
,AV HF AV YR
( )
1
2
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 quadrilateral, or
even from the graph, it can be tricky to classify it. 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 formula:
To show sides are parallel, use the slope formula:
Hint: You might notice that both formulas use the
differences in the x and y coordinates. Once you have
figured the differences for one formula, you can just use the
same numbers in the other formula.
( ) ( )
2 2
2 1 2 1d 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 have to show
this. To show that it is a rectangle, we look at all of the
slopes:
Two sets of equal slopes prove this is a parallelogram. Four
90° angles prove this is a rectangle.
( )
( )
2 2 4
3 6 3TO
m
− −
= =
− − −
( )
1 2 3
1 3 4OY
m
− −
= = −
− −
( )5 1 4 4
2 1 3 3YS
m
− − − −
= = =
− − −
( )
( )
2 5 3
6 2 4ST
m
− − −
= = −
− − −
opposite
reciprocals → 90°
opposite
reciprocals → 90°
equalslopes→parallellines

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 slopes, this
will be pretty straightforward:
Since it has four right angles and four congruent sides, TOYS
is a square.
2 2
3 4 5TO = + =
( )
22
4 3 5OY = + − =
( ) ( )
2 2
3 4 5YS = − + − =
( )
2 2
4 3 5ST = − + =