6.5 Trapezoids and Kites€¦ · 8.5 Trapezoids and Kites . Using properties of trapezoids ... The slope of AD = 4 – 0 = 4 = 2 7 – 5 2

8.5 Trapezoids

and Kites

Using properties of trapezoids A trapezoid is a

quadrilateral with exactly one pair of parallel sides.

The parallel sides are the bases.

A trapezoid has two pairs of base angles.

For instance in trapezoid ABCD D and C are one pair of base angles. The other pair is A and B. The nonparallel sides are the legs of the trapezoid.

base

base

legleg

A B

D C

Using properties of trapezoids

If the legs of a

trapezoid are

congruent, then

the trapezoid is

an isosceles

trapezoid.

Trapezoid Theorems

Theorem 8.14

If a trapezoid is

isosceles, then

each pair of base

angles is congruent.

A ≅ B, C ≅ D

A B

D C

Trapezoid Theorems

Theorem 8.15

If a trapezoid has a

pair of congruent

base angles, then it

is an isosceles

trapezoid.

ABCD is an isosceles

trapezoid

A B

D C

Trapezoid Theorems

Theorem 8.16

A trapezoid is isosceles

if and only if its

diagonals are

congruent.

ABCD is isosceles if and

only if AC ≅ BD.

A B

D C

Ex. 1: Using properties of Isosceles

Trapezoids

PQRS is an isosceles trapezoid. Find mP, mQ, mR.

PQRS is an isosceles trapezoid, so mR = mS = 50°.

Because S and P are consecutive interior angles formed by parallel lines, they are supplementary.

So mP = 180°- 50° = 130°, and mQ = mP = 130°

m PS = 2.16 cm

m RQ = 2.16 cm

S R

P Q

50°

You could also add 50 and 50,

get 100 and subtract it from

360°. This would leave you

260/2 or 130°.

Ex. 2: Using properties of trapezoids Show that ABCD is a trapezoid.

Compare the slopes of opposite sides.

The slope of AB = 5 – 0 = 5 = - 1

0 – 5 -5

The slope of CD = 4 – 7 = -3 = - 1

7 – 4 3

The slopes of AB and CD are equal,

so AB ║ CD.

The slope of BC = 7 – 5 = 2 = 1

4 – 0 4 2

The slope of AD = 4 – 0 = 4 = 2

7 – 5 2

The slopes of BC and AD are not equal, so BC is not parallel to AD.

So, because AB ║ CD and BC is not parallel to AD, ABCD is a trapezoid.

8

6

4

2

5 10 15A(5, 0)

D(7, 4)

C(4, 7)

B(0, 5)

Midsegment of a trapezoid

The midsegment of a

trapezoid:

the segment that

connects the

midpoints of its legs.

Theorem 8.17 is

similar to the

Midsegment Theorem

for triangles.

midsegment

B C

DA

Theorem 8.17: Midsegment of a

trapezoid

The midsegment of

a trapezoid is

parallel to each

base and its length

is one half the sums

of the lengths of the

bases.

MN║AD, MN║BC

MN = ½ (AD + BC)

NM

A D

CB

Ex. 3: Finding Midsegment lengths of

trapezoids A baker is making a

cake like the one at the right. The top layer has a diameter of 8 inches and the bottom layer has a diameter of 20 inches. How big should the middle layer be?

Ex. 3: Finding Midsegment lengths of

trapezoids

Use the midsegment theorem for trapezoids.

DG = ½(EF + CH)=

½ (8 + 20) = 14”

C

D

E

D

G

F

Using properties of kites

A kite is a quadrilateral

that has two pairs of

consecutive congruent

sides, but opposite

sides are not

congruent.

Kite Theorems

Theorem 8.18

If a quadrilateral is

a kite, then its

diagonals are

perpendicular.

AC BD

B

C

A

D

Kite Theorems

Theorem 8.19

If a quadrilateral is a

kite, then exactly

one pair of opposite

angles is congruent.

A ≅ C, B ≅ D

B

C

A

D

Ex. 4: Using the diagonals of a kite

WXYZ is a kite so the diagonals are perpendicular.

You can use the Pythagorean Theorem to find the side lengths.

WX = √202 + 122 ≈ 23.32

XY = √122 + 122 ≈ 16.97

Because WXYZ is a kite, WZ = WX ≈ 23.32, and ZY = XY ≈ 16.97

12

12

20

12

U

X

Z

W Y

Ex. 5: Angles of a kite

Find mG and mJ in the

diagram.

SOLUTION:

GHJK is a kite, so G ≅ J and mG = mJ.

2(mG) + 132° + 60° = 360° Sum of measures of int. s

of a quad. is 360°

2(mG) = 168° Simplify

mG = 84° Divide each side by 2.

So, mJ = mG = 84°

J

G

H K

132° 60°

Homework

Chapter 8.5

3, 7, 9, 12, 14, 16, 19, 26,

34, 37