polygons – rhombuses and trapezoids rhombus - four congruent sides

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

A

C

B

D

2

2A

CAD

ACAB

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

A

C

B

D

E ACBD

ECAE

EDBE

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

A

C

B

D

E14

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

60°

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

A

C

B

D

E14

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.

60°

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

A

C

B

D

E14

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

60°

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

A

C

B

D

E14

EXAMPLE : What is the measure of angle ECD ?

60°

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

A

C

B

D

E14

EXAMPLE : What is the measure of angle ECD ?

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

60°

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

A

C

B

D

E14

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.

60°

Polygons – Rhombuses and Trapezoids

Trapezoid - two parallel sides that are not congruent

D

A B

C

CDAB ║

CDAB

Polygons – Rhombuses and Trapezoids

Trapezoid - two parallel sides that are not congruent

D

A B

C

CDAB ║

CDAB - these parallel sides are called bases

- non-parallel sides are called legs

base 1

base 2

leg leg

Polygons – Rhombuses and Trapezoids

Trapezoid - two parallel sides that are not congruent

D

A B

C

CDAB ║

CDAB - these parallel sides are called bases

- non-parallel sides are called legs

base 1

base 2

leg leg

- there are two pairs of base angles

Polygons – Rhombuses and Trapezoids

Trapezoid - two parallel sides that are not congruent

D

A B

C

CDAB ║

CDAB - these parallel sides are called bases

- non-parallel sides are called legs

base 1

base 2

leg leg

- there are two pairs of base angles

- diagonal base angles are supplementary

Polygons – Rhombuses and Trapezoids

Trapezoid - two parallel sides that are not congruent

D

A B

C

CDAB ║

CDAB - these parallel sides are called bases

- non-parallel sides are called legs

base 1

base 2

leg leg

- 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

D

A B

C

DC

BA

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

D

A B

C

DC

BA

BDAC

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

D

A B

C

2

21 basebase

X Y

Polygons – Rhombuses and Trapezoids

Let’s try some problems…

D

A B

C

EXAMPLE : What is the median length ?

20

28

Polygons – Rhombuses and Trapezoids

Let’s try some problems…

D

A B

C

EXAMPLE : What is the median length ?

20

28

242

48

2

2820

2

21

basebase

24

Polygons – Rhombuses and Trapezoids

Let’s try some problems…

D

A B

C

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

18 X Y

Polygons – Rhombuses and Trapezoids

Let’s try some problems…

D

A B

C

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

18 X Y

92

18 The median joins the midpoints of the legs

Polygons – Rhombuses and Trapezoids

Let’s try some problems…

D

A B

C

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

Polygons – Rhombuses and Trapezoids

Let’s try some problems…

D

A B

C

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

Polygons – Rhombuses and Trapezoids

Let’s try some problems…

D

A B

C

EXAMPLE : What is the length of side AB?

?

50

YX 40

Polygons – Rhombuses and Trapezoids

Let’s try some problems…

D

A B

C

EXAMPLE : What is the length of side AB?

?

50

YX 40

2

25040

base

Polygons – Rhombuses and Trapezoids

Let’s try some problems…

D

A B

C

EXAMPLE : What is the length of side AB?

?

50

YX 40

22

250240

base2

25040

base

Polygons – Rhombuses and Trapezoids

Let’s try some problems…

D

A B

C

EXAMPLE : What is the length of side AB?

?

50

YX 40

230

25080

22

250240

base

base

base

2

25040

base