6.3

What If Both Sides Are Parallel?

Pg. 13Properties of Trapezoids

6.3 – What If Both Sides Aren't Parallel?___Properties of Trapezoids

In the previous lesson, you learned that parallelograms have both pairs of opposite sides parallel. Today you will study a shape that has only one pair of opposite sides parallel.

Trapezoid:

Quadrilateral with one pair of parallel sides

a. Saundra noticed that two identical trapezoids can be arranged to form a parallelogram. Trace the trapezoid shown below onto a piece of tracing paper. Be sure to label its bases and height as shown in the diagram. Then determine a formula to find the area of the original trapezoid.

b1

b2

b1

b2

A(parallelogram) = h(b1 + b2)

A(trapezoid)

1

2h b

1 b

2

Trapezoid 1 2

1

2A h b b

6.15 –AREA OF A TRAPEZOIDCalculate the exact areas of the trapezoids below. Don't forget units.

32

32

A h b b1 2

1

2

A

1

216

A un2416

20 32

A h b b1 2

1

2

A

1

28

A un2108

6 21

A h b b1 2

1

2

A

1

28

2120A unx2 + 62 = 102

x2 + 36 = 100

x2 = 64

x = 8

8

6 24

14 23

O

A

tan 42 23

x

20.71x

20.71 A h b b1 2

1

2

A

1

220.71

2528.11A un

14 37

6.16 –RIGHT TRAPEZOIDSA quadrilateral with two consecutive right angles is called a right trapezoid. 

a. If two consecutive angles are 90°, does it have to be a trapezoid? How do you know? Explain using the proof below.

givenaddition

Consecutive int. are supp.

One pair opp. sides //

b. What do you know about and now that you know that ABCD is a trapezoid?

C B

B C 180

c. What if the angles were not consecutive? Does it still have to be a trapezoid? Draw a picture to support your answer.

No. The angles have to be consecutive

6.17 –ISOSCELES TRAPEZOIDSA trapezoid with its legs congruent is called isosceles.

 a. Using reflection symmetry, what can you say about the angles in the picture? Complete the statements.

A _______B

 a. Using reflection symmetry, what can you say about the angles in the picture? Complete the statements.

_______D C

 a. Using reflection symmetry, what can you say about the angles in the picture? Complete the statements.

A _______ 180D

 a. Using reflection symmetry, what can you say about the angles in the picture? Complete the statements.

B _______ 180C

b. Complete the two new properties of isosceles trapezoids.

Isosceles Trapezoid:

• Base angles are congruent

• Diagonals are congruent

6.18 –MISSING ANGLESFind the measure of the missing angles in the isosceles trapezoids.

122122

58

10971

71

6.19 –TRAPEZOIDS ON THE GRIDProve the following shape is a trapezoid. Then prove it isn’t isosceles.

Slopes of: AB = BC =

CD = DA =

2

3

.undef

1

3.undef

Lengths of: AB = BC =

CD = DA =

22 32 x2

4 9 x2

13 2

12 32 x2

1 9 x2

10 5

trapezoid

Midsegment of a Trapezoid:• Connects the midpoints of the legs of a

trapezoid

midsegment

base

base midsegment = base + base 2

x =7 + 13 2

=20 2

= 10

6.20 –TRAPEZOIDS ON THE GRIDFind x.

x

M =b + b 2

20 =7x +12 2

1

7x + 12 = 40

x = 4

7x = 28

12

Parallelogram

Rectangle

Rhombus

Square

Trapezoid

IsoscelesTrapezoid

Kite

Triangle

Trapezoid

• One pair of parallel sides

• Consecutive angles supplementary

A h b b1 2

1

2

Isosceles Trapezoid

• Properties listed above

• Legs are congruent

• Base angles congruent

• Diagonals are congruent

A h b b1 2

1

2