8.5 T

RAPEZOID

S AND

KITES

QUADRILATE

RALS

OBJECTIVES:

Use properties of trapezoids.

Use properties of kites.

ASSIGNMENT:

pp. 541 through 549

H.W problems # 4, 8, 10, 14, 16, 18, 20, 34, 36, 45 on pages 546-549 of the textbook

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.

USING PROPERTIES OF TRAPEZOIDS

base

base

legleg

A B

D C

If the legs of a trapezoid are congruent, then the trapezoid is an isosceles trapezoid.

USING PROPERTIES OF TRAPEZOIDS

Theorem 8.14

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

A ≅ B, C ≅ D

TRAPEZOID THEOREMS

A B

D C

Theorem 8.15

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

ABCD is an isosceles trapezoid

TRAPEZOID THEOREMS

A B

D C

Theorem 8.16

A trapezoid is isosceles if and only if its diagonals are congruent.

ABCD is isosceles if and only if AC ≅ BD.

TRAPEZOID THEOREMS

A B

D C

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°

EX. 1: USING PROPERTIES OF ISOSCELES TRAPEZOIDS

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°.

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.

EX. 2: USING PROPERTIES OF TRAPEZOIDS

8

6

4

2

5 10 15A(5, 0)

D(7, 4)

C(4, 7)

B(0, 5)

The midsegment of a trapezoid is the segment that connects the midpoints of its legs. Theorem 6.17 is similar to the Midsegment Theorem for triangles.

MIDSEGMENT OF A TRAPEZOID

midsegment

B C

DA

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║BCMN = ½ (AD + BC)

THEOREM 8.17: MIDSEGMENT OF A TRAPEZOID

NM

A D

CB

LAYER CAKE 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”

EX. 3: FINDING MIDSEGMENT LENGTHS OF TRAPEZOIDS

C

D

E

D

G

F

A kite is a quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent.

USING PROPERTIES OF KITES

Theorem 8.18

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

AC BD

KITE THEOREMS

B

C

A

D

Theorem 8.19

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

A ≅ C, B ≅ D

KITE THEOREMS

B

C

A

D

WXYZ is a kite so the diagonals are perpendicular. You can use the Pythagorean Theorem to find the side lengths.

WX = √202 + 122 ≈ 23.32XY = √122 + 122 ≈ 16.97Because WXYZ is a kite,

WZ = WX ≈ 23.32, and ZY = XY ≈ 16.97

EX. 4: USING THE DIAGONALS OF A KITE

12

1220

12

U

X

Z

W Y

EX. 5: ANGLES OF A KITE

Find mG and mJ in the diagram at the right.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 K132° 60°