Parallelograms & Rectangles

The student is able to (I can):

Prove and apply properties of parallelograms.

Use properties of parallelograms to solve problems.

Prove and apply properties of special parallelograms.

Use properties of special parallelograms to solve problems.

parallelogram

Properties of Parallelograms

A quadrilateral with two pairs of parallel sides.

Therefore, if a quadrilateral is a parallelogram, then it has two pairs of parallel sides.

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T I

ME

TI ME, TE IM

Properties of Parallelograms

If a quadrilateral is a parallelogram, then opposite sides are congruent.

If a quadrilateral is a parallelogram, then opposite angles are congruent.

KI NG, GK IN

K

NG

I

>>

>>

K

NG

O

K N, O G

Properties of Parallelograms

If a quadrilateral is a parallelogram, then consecutive angles are supplementary.

If a quadrilateral is a parallelogram, then its diagonals bisect each other.

1 2

34

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T U

NE

STS NS, ES US

+ =

+ =

+ =

+ =

m 1 m 2 180

m 2 m 3 180

m 3 m 4 180

m 4 m 1 180

Examples Find the value of the variable:

1. x =

2. x =

3. y =

5x + 3 2x + 15

(3x)

(x + 84)

y

Examples Find the value of the variable:

1. x =

2. x =

3. y =

5x + 3 2x + 15

4

(3x)

(x + 84)

y

5x + 3 = 2x + 153x = 12

3x = x + 842x = 84

42

3(42) = 126y = 180 126

54

rectangle A parallelogram with four right angles.

If a parallelogram is a rectangle, then its diagonals are congruent (checking for square).

F I

SH

FS IH

Because a rectangle is a parallelogram, it also inherits all of the properties of a parallelogram:

Opposite sides parallel

Opposite sides congruent

Opposite angles congruent (actually allallallallangles are congruent)

Consecutive angles supplementary

Diagonals bisect each other

Example Find each length.

1. LW

2. OL

3. OW

F O

WL

30

17

Example Find each length.

1. LW

LW = FO = 30

2. OL

OL = FW = 2(17) = 34

3. OW

OWL is a right triangle, so

OW = 16

F O

WL

30

17

+ =2 2 2OW LW OL

+ =2OW 900 1156

=2OW 256

+ =2 2 2OW 30 34

rhombus A parallelogram with four congruent sides. (Plural is either rhombi or rhombuses.)

If a parallelogram is a rhombus, then its diagonals are perpendicular.

If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles.

1 2

3 4

5 6

7 8

1 234

567

8

Since opposite angles are also congruent:1 2 5 63 4 7 8

Examples 1. What is the perimeter of a rhombus whose side length is 7?

2. Find the value of x

3. Find the value of y

x

8

Perimeter = 40

(3y+11)

(13y9)

10

Examples 1. What is the perimeter of a rhombus whose side length is 7?

4(7) = 28

2. Find the value of x

The side = 10

Pyth. triple: 6, 8, 10

x = 6

3. Find the value of y

13y 9 = 3y + 11

10y = 20

y = 2

x

8

Perimeter = 40

(3y+11)

(13y9)

10

square A quadrilateral with four right angles and four congruent sides.

Note: A square has all of the properties of bothbothbothboth a rectangle andandandand a rhombus:

Diagonals are congruent

Diagonals are perpendicular

Diagonals bisect opposite angles.