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Polygons – Rhombuses and Trapezoids Rhombus - four congruent sides

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Polygons – Rhombuses and Trapezoids

Rhombus - four congruent sides

Polygons – Rhombuses and Trapezoids

Rhombus - four congruent sides

- opposite angles are congruent

Polygons – Rhombuses and Trapezoids

Rhombus - four congruent sides

- opposite angles are congruent

Polygons – Rhombuses and Trapezoids

Rhombus - four congruent sides

- opposite angles are congruent

- diagonals bisect the angles at the vertex

B

A

C

D

Polygons – Rhombuses and Trapezoids

Rhombus - four congruent sides

- opposite angles are congruent

- diagonals bisect the angles at the vertex

- diagonals bisect each other and are perpendicular

B

A

E

C

D

Polygons – Rhombuses and Trapezoids

Rhombus - four congruent sides

- opposite angles are congruent

- diagonals bisect the angles at the vertex

- diagonals bisect each other and are perpendicular

B

A

E

14

C

60°

D

EXAMPLE : If AD = 14, what is the measure of EB ?

Polygons – Rhombuses and Trapezoids

Rhombus - four congruent sides

- opposite angles are congruent

- diagonals bisect the angles at the vertex

- diagonals bisect each other and are perpendicular

B

A

E

14

C

60°

D

EXAMPLE : If AD = 14, what is the measure of EB ?

SOLUTION : With angle ADE = 60 degrees we have a 30 – 60 – 90 triangle.

So segment EB = Segment ED which is half of AD.

Polygons – Rhombuses and Trapezoids

Rhombus - four congruent sides

- opposite angles are congruent

- diagonals bisect the angles at the vertex

- diagonals bisect each other and are perpendicular

B

A

E

14

C

60°

D

EXAMPLE : If AD = 14, what is the measure of EB ?

SOLUTION : With angle ADE = 60 degrees we have a 30 – 60 – 90 triangle.

So segment EB = Segment ED which is half of AD. ED = 7

Polygons – Rhombuses and Trapezoids

Rhombus - four congruent sides

- opposite angles are congruent

- diagonals bisect the angles at the vertex

- diagonals bisect each other and are perpendicular

B

A

E

14

C

60°

D

EXAMPLE : What is the measure of angle ECD ?

Polygons – Rhombuses and Trapezoids

Rhombus - four congruent sides

- opposite angles are congruent

- diagonals bisect the angles at the vertex

- diagonals bisect each other and are perpendicular

B

A

E

14

C

60°

D

EXAMPLE : What is the measure of angle ECD ?

SOLUTION : Again we have a 30 – 60 – 90 triangle. So angle DAC = 30 degrees.

Polygons – Rhombuses and Trapezoids

Rhombus - four congruent sides

- opposite angles are congruent

- diagonals bisect the angles at the vertex

- diagonals bisect each other and are perpendicular

B

A

E

14

C

60°

D

EXAMPLE : What is the measure of angle ECD ?

SOLUTION : Again we have a 30 – 60 – 90 triangle. So angle DAC = 30 degrees.

So angle ECD would also be 30 degrees.

Polygons – Rhombuses and Trapezoids

Trapezoid - two parallel sides that are not congruent

║

B

A

D

C

Polygons – Rhombuses and Trapezoids

Trapezoid - two parallel sides that are not congruent

║

- these parallel sides are called bases
- - non-parallel sides are calledlegs

base 1

B

A

leg

leg

D

C

base 2

Polygons – Rhombuses and Trapezoids

Trapezoid - two parallel sides that are not congruent

║

- these parallel sides are called bases
- - non-parallel sides are calledlegs

base 1

B

A

leg

leg

D

C

base 2

- there are two pairs of base angles

Polygons – Rhombuses and Trapezoids

Trapezoid - two parallel sides that are not congruent

║

- these parallel sides are called bases
- - non-parallel sides are calledlegs

base 1

B

A

leg

leg

D

C

base 2

- there are two pairs of base angles
- diagonal base angles are supplementary

Polygons – Rhombuses and Trapezoids

Trapezoid - two parallel sides that are not congruent

║

- these parallel sides are called bases
- - non-parallel sides are calledlegs

base 1

B

A

leg

leg

D

C

base 2

- there are two pairs of base angles
- diagonal base angles are supplementary
- base angles that share a leg are also supplementary

Polygons – Rhombuses and Trapezoids

Isosceles Trapezoid - has all the properties of a trapezoid

- legs are congruent

- base angles are congruent

A

B

D

C

Polygons – Rhombuses and Trapezoids

Isosceles Trapezoid - has all the properties of a trapezoid

- legs are congruent

- base angles are congruent

- diagonals have the same length

A

B

D

C

Polygons – Rhombuses and Trapezoids

Median of a Trapezoid

- parallel with both bases

- equal to half the sum of the bases

- joins the midpoints of the legs

A

B

X

Y

D

C

Polygons – Rhombuses and Trapezoids

Let’s try some problems…

EXAMPLE : What is the median length ?

A

20

B

D

C

28

Polygons – Rhombuses and Trapezoids

Let’s try some problems…

EXAMPLE : What is the median length ?

A

20

B

24

D

C

28

Polygons – Rhombuses and Trapezoids

Let’s try some problems…

EXAMPLE : If AD = 18, what is the measure of AX ?

A

B

18

X

Y

D

C

Polygons – Rhombuses and Trapezoids

Let’s try some problems…

EXAMPLE : If AD = 18, what is the measure of AX ?

The median joins the midpoints of the legs

A

B

18

X

Y

D

C

Polygons – Rhombuses and Trapezoids

Let’s try some problems…

EXAMPLE : ABCD is an isosceles trapezoid. If angle DAB = 110°, what is the measure of angle ABC ?

A

B

D

C

Polygons – Rhombuses and Trapezoids

Let’s try some problems…

EXAMPLE : ABCD is an isosceles trapezoid. If angle DAB = 110°, what is the measure of angle ABC ?

110°-

base angles are congruent in an isosceles trapezoid

A

B

D

C

Polygons – Rhombuses and Trapezoids

Let’s try some problems…

EXAMPLE : What is the length of side AB?

?

A

B

40

X

Y

D

C

50

Polygons – Rhombuses and Trapezoids

Let’s try some problems…

EXAMPLE : What is the length of side AB?

?

A

B

40

X

Y

D

C

50

Polygons – Rhombuses and Trapezoids

Let’s try some problems…

EXAMPLE : What is the length of side AB?

?

A

B

40

X

Y

D

C

50

Polygons – Rhombuses and Trapezoids

Let’s try some problems…

EXAMPLE : What is the length of side AB?

?

A

B

40

X

Y

D

C

50