5-minute checkIn a brief paragraph explain all the properties

you know about parallelograms.Explain the properties of the following:RhombusRectangleSquare

Objectives:Students will analyze and apply properties of

trapezoids and kites. Why? So you can measure part of a building,

as in Example 2.Mastery is 80% or better on 5-minute checks

and practice problems.

Using properties of trapezoidsA 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 trapezoidsIf the legs of a

trapezoid are congruent, then the trapezoid is an isosceles trapezoid.

Trapezoid TheoremsTheorem 8.14If a trapezoid is

isosceles, then each pair of base angles is congruent.

A ≅ B, C ≅ D

A B

D C

Trapezoid TheoremsTheorem 8.15If 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 TheoremsTheorem 8.16A trapezoid is

isosceles if and only if its diagonals are congruent.

ABCD is isosceles if and only if AC BD.≅

A B

D C

Skill DevelopEx. 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°.

Skill DevelopEx. 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 trapezoidThe midsegment of a

trapezoid is 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 trapezoidThe 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)

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.

HG = ½(EF + CD)=½ (8 + 20) = 14”

C

H

E

D

G

F

Using properties of kitesA kite is a

quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent.

Kite theoremsTheorem 8.18If a quadrilateral is a

kite, then its diagonals are perpendicular.

AC BD

B

C

A

D

Kite theoremsTheorem 8.19If a quadrilateral is a

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

A ≅ C, B ≅ D

B

C

A

D

Think…Ink…ShareEx. 4: Using the diagonals of a kiteWXYZ 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

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°

What was the objective for today?Students will analyze and apply properties of

trapezoids and kites. Why? So you can measure part of a building,

as in Example 2.Mastery is 80% or better on 5-minute checks

and practice problems.

HomeworkPage 546-547# 1-34 allQuiz tomorrow 8.4-8.5Test Coming Soon