Use Properties of Trapezoids and Kites

Goal: Use properties of trapezoids and kites.

Vocabulary

Trapezoid: A trapezoid is a quadrilateral with exactly one pair of parallel sides.

Bases of a trapezoid: The parallel sides of a trapezoid are the bases.

Base angles of a trapezoid: A trapezoid has two pairs of base angles. Each pair shares a base as a side.

Vocabulary (Cont.)Legs of a trapezoid: The nonparallel sides of

a trapezoid are the legs.

Isoceles trapezoid: An isosceles trapezoid is a trapezoid in which the legs are congruent.

Midsegment of a trapezoid: The midsegment of a trapezoid is the segment that connects the midpoints of its legs.Kite: A kite is a quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent.

Example 1: Use a coordinate plane

Show that CDEF is a trapezoid.

Solution

Compare the slopes of oppositesides.

4-3 1Slope of DE

4-1 3

2- 0 2 1Slope of CF

6-0 6 3

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

Example 1 (Cont.)

2-4 -2Slope of EF -1

6-4 2

3-0 3Slope of CD 3

1-0 1

The slopes of and are not EF CD Ethe same, so F is

to .not CD

Because quadrilateral CDEF has exactly one pair ofparallel sides, it is a trapezoid.

Theorem 6.14

If a trapezoid is isoceles, then each pair of base angles is congruent.

If trapezoid ABCD is isosceles, then

and .A D B C

Theorem 6.15

If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid.

If (or if ), then trapezoid ABCD is isosceles.A D B C

Theorem 6.16

A trapezoid is isosceles iff its diagonals are congruent.

Trapezoid ABCD is isosceles iff .AC BD

Example 2: Use properties of isosceles trapezoids

Kitchen A shelf fitting into a cupboard in the corner of a kitchen is an isosceles trapezoid.

. and ,, Find MmLmNm

Solution

Step 1 Find . KLMN is an , so

and are congruent base angles, and

isosceles t

50

rapezoid

.

m N

N K

m N m K

Example 2 (Cont.)Step 2 Find . Because and are consecutive

interior angles formed by KL are intersecting two

parallel lines, they are . So,

supplementa

180 50 130 .

ry

m L K L

m L

.130 and ,130,50 So,

.130

and congruent, are they angles, base of

pair a are and Because . Find 3 Step

MmLmNm

LmMm

LMMm

Checkpoint 1

1. In Example 1, suppose the coordinates of point E are (7,5). What type of quadrilateral is CDEF? Explain.

Parallelogram; opposite pairs of sides areparallel.

Checkpoint 2

shown. trapezoidin the and ,, Find 2. DmAmCm

45,45,135 DmAmCm

Theorem 6.17: Midsegment Theorem for Trapezoids

The midsegment of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases.

).(2

1 and

,DC toparallel is MN,AB toparallel is MN

thenABCD, trapezoidof midsegment theis MN If

CDABMN

Example 3: Use the midsegment of a trapezoid

MN. Find PQRS. trapezoidof

midsegment theis MN diagram, In the

Solution

Use Theorem 6.17 to find MN.

Simplify. 12.5

SR.for 9 and PQfor 16 Substitute )916(2

1

6.17. TheoremApply )(2

1

SRPQMN

The length MN is 12.5 inches.

Checkpoint 3

3. Find MN in the trapezoid at the right.

MN = 21 ft

Theorem 6.18

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

.ACthen

kite, a is ralquadrilate If

BD

ABCD

Theorem 6.19

If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent.

D. not is B and C A then

,BABC and kite a is ABCD ralquadrilate If

Example 4:Apply Theorem 6.19right. at theshown kite in the Find Tm

Solution

. find to

equationan solve and Write.

. bemust T and R S, not is Q

Because angles. opposite congruent ofpair

oneexactly has QRST 6.19, TheoremBy

Tm

TmRm

Example 4 (Cont.)

Corollary to Theorem 8.1

Substitut

70 88 360

70 88 360

2 15

e m T for m R.

( ) Combine li8 360 ke terms.

101 Solve for m T.

m T m R

m T m T

m T

m T

Checkpoint 4

below. at theshown kite in theG m Find

100Gm