Kites & Trapezoids

The student is able to (I can):

• Use properties of kites and trapezoids to solve problems

kite A quadrilateral with exactly two pairs of congruent consecutive nonparallel sides.

Note: In order for a quadrilateral to be a kite, nononono sides can be parallel and opposite sides cannot be congruent.

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.

Example In kite NAVY, m∠YNA=54º and m∠VYX=52º. Find each measure.

1. m∠NVY

90 — 52 = 38º

2. m∠XYN

3. m∠NAV

63 + 52 = 115ºN

AV

Y

X

−= = °

180 54 12663

2 2

trapezoid 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.

Note: a trapezoid whose legs are congruent is called an isosceles trapezoidisosceles trapezoidisosceles trapezoidisosceles trapezoid.

>

>base

base

leg leg

base angles

base angles

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≅

Examples 1. Find the value of x.

5x = 40

x = 8

2. If NS=14 and BA=25, find SE.

SE = 25 — 14 = 11

140º

5xº

B E

AN

SSSS

40º

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

AV HF YR2

= +

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

MY 22 30 262

= + =

( )1

18 16 GI2

= +

36 16 GI= +

GI 20=