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Trapezoids and Kites Chapter 8, Section 5 (8.5)

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Page 1: Trapezoids and Kites Chapter 8, Section 5 (8.5) Essential Questions How do I use properties of trapezoids? How do I use properties of kites?

Trapezoids and Kites

Chapter 8, Section 5

(8.5)

Page 2: Trapezoids and Kites Chapter 8, Section 5 (8.5) Essential Questions How do I use properties of trapezoids? How do I use properties of kites?

Essential Questions

How do I use properties of trapezoids?

How do I use properties of kites?

Page 3: Trapezoids and Kites Chapter 8, Section 5 (8.5) Essential Questions How do I use properties of trapezoids? How do I use properties of kites?

VocabularyTrapezoid – a quadrilateral with exactly one pair of parallel sides.

A trapezoid has two pairs of base angles. In this example the base angles are A & B and C & D

A

D C

B

leg leg

base

base

Page 4: Trapezoids and Kites Chapter 8, Section 5 (8.5) Essential Questions How do I use properties of trapezoids? How do I use properties of kites?

8.14 Base Angles Trapezoid Theorem

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

A

D C

B

A B, C D

Page 5: Trapezoids and Kites Chapter 8, Section 5 (8.5) Essential Questions How do I use properties of trapezoids? How do I use properties of kites?

8.15 Base Angles Trapezoid Converse

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

D C

B

ABCD is an isosceles trapezoid

Page 6: Trapezoids and Kites Chapter 8, Section 5 (8.5) Essential Questions How do I use properties of trapezoids? How do I use properties of kites?

8.16 Diagonals of a Trapezoid Theorem

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

A

D C

B

BDAC ifonly and if isosceles is ABCD

Page 7: Trapezoids and Kites Chapter 8, Section 5 (8.5) Essential Questions How do I use properties of trapezoids? How do I use properties of kites?

Example 1PQRS is an isosceles trapezoid. Find m P, m Q and mR.

50S R

P Q

m R = 50 since base angles are congruent

mP = 130 and mQ = 130 (consecutive angles of parallel lines cut by a transversal are )

Page 8: Trapezoids and Kites Chapter 8, Section 5 (8.5) Essential Questions How do I use properties of trapezoids? How do I use properties of kites?

Definition

Midsegment of a trapezoid – the segment that connects the midpoints of the legs.

midsegment

Page 9: Trapezoids and Kites Chapter 8, Section 5 (8.5) Essential Questions How do I use properties of trapezoids? How do I use properties of kites?

8.17 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.

NM

A D

B C

BC) AD2

1 MN ,BC ll MNAD llMN ( ,

Page 10: Trapezoids and Kites Chapter 8, Section 5 (8.5) Essential Questions How do I use properties of trapezoids? How do I use properties of kites?

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

Page 11: Trapezoids and Kites Chapter 8, Section 5 (8.5) Essential Questions How do I use properties of trapezoids? How do I use properties of kites?

8.18 Theorem: Perpendicular Diagonals of a Kite

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

D

C

A

B

BDAC

Page 12: Trapezoids and Kites Chapter 8, Section 5 (8.5) Essential Questions How do I use properties of trapezoids? How do I use properties of kites?

8.19 Theorem: Opposite Angles of a Kite

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

D

C

A

B

A C, B D

Page 13: Trapezoids and Kites Chapter 8, Section 5 (8.5) Essential Questions How do I use properties of trapezoids? How do I use properties of kites?

Example 2Find the side lengths of the kite.

20

12

1212

UW

Z

Y

X

Page 14: Trapezoids and Kites Chapter 8, Section 5 (8.5) Essential Questions How do I use properties of trapezoids? How do I use properties of kites?

Example 2 Continued

WX = 4 34

likewise WZ = 4 34

20

12

1212

UW

Z

Y

X

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

122 + 202 = (WX)2

144 + 400 = (WX)2

544 = (WX)2

122 + 122 = (XY)2

144 + 144 = (XY)2

288 = (XY)2

XY =12 2

likewise ZY =12 2

Page 15: Trapezoids and Kites Chapter 8, Section 5 (8.5) Essential Questions How do I use properties of trapezoids? How do I use properties of kites?

Example 3

Find mG and mJ.

60132

J

G

H K

Since GHJK is a kite G J

So 2(mG) + 132 + 60 = 360

2(mG) =168

mG = 84 and mJ = 84

Page 16: Trapezoids and Kites Chapter 8, Section 5 (8.5) Essential Questions How do I use properties of trapezoids? How do I use properties of kites?

Try This!RSTU is a kite. Find mR, mS and mT.

x

125

x+30

S

U

R T

x +30 + 125 + 125 + x = 360

2x + 280 = 360

2x = 80

x = 40

So mR = 70, mT = 40 and mS = 125

Page 17: Trapezoids and Kites Chapter 8, Section 5 (8.5) Essential Questions How do I use properties of trapezoids? How do I use properties of kites?

Homework

Pages 546 (7-15)