5.11 Properties of Trapezoids and Kites

Example 1 Use a coordinate planeUse a coordinate plane

Show that CDEF is a trapezoid.

0 ,0C

3 ,1D 4 ,4E

2 ,6F

SolutionCompare the slopes of the opposite sides.

DE of slope 34 14 3

1

CF of slope 02 06 6

2

3

1

The slopes of DE and CF are the same, so DE ___ CF.

EF of slope 42 46 2

21

CD of m 03

01 1

3 3

The slopes of EF and CD are not the same, so EF is ______________ to CD.not parallelBecause quadrilateral CDEF has exactly one pair of _______________, it is a trapezoid.parallel sides

5.11 Properties of Trapezoids and Kites

Theorem 5.29If a trapezoid is isosceles, then each pair of

base angles is _____________.congruent

A D

B C

C. ___ and ___ A If trapezoid ABCD is isosceles, then

D B

5.11 Properties of Trapezoids and Kites

Theorem 5.30If a trapezoid has a pair of congruent

___________, then it is an isosceles trapezoid.base angles

A D

B C

,C B ifor D A If then trapezoid ABCD is isosceles.

5.11 Properties of Trapezoids and Kites

Theorem 5.31A trapezoid is isosceles if and only if its diagonals

are __________.congruent

A D

B C

____. ____ Trapezoid ABCD is isosceles if and only if

AC BD

5.11 Properties of Trapezoids and Kites

Example 2 Use properties of isosceles trapezoidsUse properties of isosceles trapezoids

Kitchen A shelf fitting into a cupboard in the corner of a kitchen is an isosceles trapezoid. Find m N, m L, and m M.

L

K

M

N

o50

SolutionStep 1 Find m N. KLMN is an ___________________, so N

and ___ are congruent base angles, and

isosceles trapezoid

K.______N mm K o50

Step 2

supplementary.______M mm L o130

Find m L. Because K and L are consecutive interior angles formed by KL intersecting two parallel lines, they are _________________.

5.11 Properties of Trapezoids and Kites

Example 2 Use properties of isosceles trapezoidsUse properties of isosceles trapezoids

Kitchen A shelf fitting into a cupboard in the corner of a kitchen is an isosceles trapezoid. Find m N, m L, and m M.

L

K

M

N

o50

SolutionStep 3 Find m M. Because M and ___ are a pair of base

angles, they are congruent, and L

.______M mm L o130.____M ____,L ____,N So, mmm o50 o130 o130

5.11 Properties of Trapezoids and KitesCheckpoint. Complete the following exercises. Checkpoint. Complete the following exercises. 1. In Example 1, suppose the coordinates

of E are (7, 5). What type of quadrilateral is CDEF? Explain.

0 ,0C

3 ,1D

5 ,7E

2 ,6F

DE of slope 35 17 6

2

CF of slope 02 06 6

2

3

13

1

The slopes of DE and CF are the same, so DE ___ CF.

EF of slope 52 76 1

3

3

CD of m 03 01 1

3 3

The slopes of EF and CD are the same, so EF ___ CD.

Because the slopes of DE and CD are not the opposite reciprocals of each other, they are not perpendicular.

Therefore the opposite sides are parallel and CDEF is a parallelogram.

5.11 Properties of Trapezoids and KitesCheckpoint. Complete the following exercises. Checkpoint. Complete the following exercises. 1. Find m C, m A, and m D

in the trapezoid shown. A

B

D

C

o135

ABCD is an isosceles trapezoid, so base angles are congruent and consecutive interior angles are supplementary.

o135C m oo 135180Am o45

o45D m

5.11 Properties of Trapezoids and Kites

Theorem 5.32 Midsegment Theorem of TrapezoidsThe midsegment of a trapezoid is parallel to each

base and its length is one half the sum of the lengths of the bases.

.________ MN and ____,MN ____,MN If MN is the midsegment of trapezoid ABCD,

A

D

B

C

M N

AB CD 2

1AB CD

5.11 Properties of Trapezoids and Kites

Example 3 Use the midsegment of a trapezoidsUse the midsegment of a trapezoids

Solution

In the diagram, MN is the midsegment of trapezoid PQRS. Find MN

P

S

Q

R

M N

Use Theorem 5.32 to find MN.

______MN ____ 2

1PQ SR

__________ 2

116 9

____12.5

Apply Theorem 5.32Apply Theorem 5.32

Substitute ___ for PQ and Substitute ___ for PQ and ___ for SR.___ for SR.

169

Simplify.Simplify.

inches. _____ is MN 12.5

16 in.

9 in.

5.11 Properties of Trapezoids and KitesCheckpoint. Complete the following exercises. Checkpoint. Complete the following exercises. 2. Find MN in the trapezoid

at the right.P

S

Q

R

M

N

30 ft

12 ft

______MN ____ 2

1PS QR

__________ 2

130 12

____ 21

ft 21 is MN

5.11 Properties of Trapezoids and Kites

Theorem 5.33If a quadrilateral is a kite, then its diagonals are

_______________.

______then If quadrilateral ABCD is a kite,

AC BD

A

DB

C

perpendicular

5.11 Properties of Trapezoids and Kites

Theorem 5.34If a quadrilateral is a kite, then exactly one pair

of opposite angles are congruent.

D.B___ and CA___then

A

DB

C

If quadrilateral ABCD is a kite and BC BA,

5.11 Properties of Trapezoids and Kites

Example 4 Use properties of kitesUse properties of kites

SolutionBy Theorem 5.34, QRST has exactly one pair of __________ opposite angles.

Find m T in the kite shown at the right.

RQ

S

T

o70

o88

T. __ So, congruent. bemust T and __ ,SQ Because mmcongruent

R RT. find oequation tan solve and Write m

Corollary to Corollary to Theorem 5.16Theorem 5.16__________RT mm o70 o88 o360

__________TT mm o70 o88 o360 Substitute.Substitute.

_______T__ m Combine like terms.Combine like terms.2 o158 o360Solve.Solve.____T m o101

5.11 Properties of Trapezoids and KitesCheckpoint. Complete the following exercises. Checkpoint. Complete the following exercises. 4. Find m G in the kite

shown at the right. G

I

H

J

o85

o75

ooo 3605857IG mmooo 3605857GG mm

G2 m o160 o360 G2 m o200

o100Gm

5.11 Properties of Trapezoids and Kites

Pg. 339, 5.11 #2-22 even, 23