Lesson 5.3Trapezoids and Kites

Homework: 5.3/1-8,19

QUIZ Wednesday 5.1 – 5.4

1. OPEN TEXTBOOKS2. Tools – patty paper(2), protractor, ruler3. INVESTIGATIONs 1 & 2 – ALL steps4. Complete the 4 kite conjectures & the 3 trapezoid conjectures

PROCEDURES for today:

DefinitionKite – a quadrilateral that has two pairs of

consecutive congruent sides, but opposite sides are not congruent.

Perpendicular Diagonals of a Kite

If a quadrilateral is a kite, then its diagonals are perpendicular.

D

C

A

B

BDAC

Non-Vertex Angles of a Kite

If a quadrilateral is a kite, then non-vertex angles are congruent

D

C

A

B

A C, B D

Vertex diagonals bisect vertex angles

D

C

A

B

If a quadrilateral is a kite then the vertex diagonal bisects the vertex angles.

Vertex diagonal bisects the non-vertex diagonal

D

C

A

B

If a quadrilateral is a kite then the vertex diagonal bisects the non-vertex diagonal

Definition-a quadrilateral with exactly one pair of parallel sides.

Leg Leg Leg Leg

BaseBase

BaseBaseAA BB

CC DD››

››

Trapezoid

<A + <C = 180<A + <C = 180

<B + <D = 180<B + <D = 180

AA BB

CC DD››

››

Leg Angles are Supplementary

Property of a Trapezoid

Isosceles Trapezoid

Definition - A trapezoid with congruent legs.

Isosceles Trapezoid - Properties

| | ||

1) Base Angles Are Congruent2) Diagonals Are Congruent

Example

PQRS is an isosceles trapezoid. Find m P, m Q and mR.

50S R

P Q

m R = 50 since base angles are congruent

mP = 130 and mQ = 130 (consecutive angles of parallel lines cut by a transversal are )

Find the measures of the angles in trapezoid

48

m< A = 132m< B = 132m< D = 48

Find BE

AC = 17.5, AE = 9.6

E

Example

Find the side lengths of the kite.

20

12

1212

UW

Z

Y

X

Example Continued

WX = 4 34

likewise WZ = 4 34

XY =12 2

likewise ZY =12 2

20

12

1212

UW

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

Find the lengths of the sides of the kite

W

X

Y

Z

4

55

8

Find the lengths of the sides of kite to the nearest tenth

4

2

2

7

Example 3Find mG and mJ.

60132

J

G

H K

Since GHJK is a kite G J

So 2(mG) + 132 + 60 = 360

2(mG) =168

mG = 84 and mJ = 84

Try This!

RSTU is a kite. Find mR, mS and mT.

x

125

x+30

S

U

R T

x +30 + 125 + 125 + x = 360

2x + 280 = 360

2x = 80

x = 40

So mR = 70, mT = 40 and mS = 125

Try These

base

base

legleg

A B

D C base

base

legleg

A B

D C

1. If <A = 134, find m<D2. m<C = x +12 and m<B = 3x – 2, find x and the measures of the 2 angles

m<D = 46

x = 42.5m<C = 54.5m<B = 125.5

Using Properties of Trapezoids

Find the area of this trapezoid.

When working with a trapezoid, the height may be measured anywhere between the two bases.  Also, beware of "extra" information.  The 35 and 28 are not needed to compute this area.

Area of trapezoid = 212

1bbh

A = ½ * 26 * (20 + 42)

A = 806

Using Properties of Trapezoids

Find the area of a trapezoid with bases of 10 in and 14 in, and a height of 5 in.

Example 2

Using Properties of Kites

D

A

B

C

Area Kite = one-half product of diagonals

212

1ddA

BDACArea 2

1

Using Properties of Kites

D

A

B

C

Example 6

E

24 4

4

ABCD is a Kite.

a) Find the lengths of all the sides.

b) Find the area of the Kite.

Venn Diagram:

http://teachers2.wcs.edu/high/rhs/staceyh/Geometry/Chapter%206%20Notes.ppt#435,22,6.2 – Properties of Parallelograms

Flow Chart:

Homework