2.7.2 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 rectangles.

Use properties of rectangles.

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.

>>

>>

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

m1 + m2 = 180m2 + m3 = 180m3 + m4 = 180m4 + m1 = 180

>>

>>

T U

NE

SSSSTS NS, ES US

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

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