TODAY> 8.2 Properties of Parallelograms

By definition, a parallelogram is a quadrilateral with 2 pairs of parallel sides

Given: GEOM is a parallelogram.

Prove:

(i.e. opposite sides are )

GE OM GM OE G E

OM

Given: GEOM is a parallelogram. Prove:a) G and E are supplementary. E and O are supplementary. O and M are supplementary. M and G are supplementary. (i.e. consecutive angles are supplementary)

b) G O, M E (i.e. opposite angles are congruent)

G E

OM

Given: GEOM is a parallelogram.

Prove: Diagonals bisect each other.

G E

OM

T

8.2 //ogram Properties

2 pairs of opposite sides are // (by defn.) 2 pairs of consecutive interior s are

supplementary 2 pairs of opposite s are 2 pairs of opposite sides are The diagonals bisect each other.

Exercises: p. 512 #8, 11, 15, 23 – 28, 33, 36

TODAY> 8.3 Proving Parallelograms

Aside from using the definition of a parallelogram (opposite sides are parallel), there are five (5) other ways to prove that a quadrilateral is a parallelogram.

Given:

Quadrilateral GEOM

G and E are supplementary.

E and O are supplementary.

O and M are supplementary.

M and G are supplementary.

Prove: GEOM is a parallelogram.

G E

OM

Given:

Quadrilateral GEOM

M E and G O

Prove:

GEOM is a parallelogram.

G E

OM

b

ba

a

Given:

Quadrilateral GEOM

Diagonals bisect each other at T.

Prove:

GEOM is a parallelogram.G E

OM

T

Given:

Quadrilateral GEOM

Prove:

GEOM is a parallelogram.

GE OM GM OE G E

OM

Given:

Quadrilateral GEOM

Prove:

GEOM is a parallelogram.

GM EO G E

OM

GM EO

A quadrilateral is a parallelogram if:

2 pairs of opposite sides are // (by defn.) 2 pairs of consecutive interior s are

supplementary 2 pairs of opposite s are 2 pairs of opposite sides are The diagonals bisect each other. One pair of opposite sides are // and .

8.3 Proving Parallelograms

1. Given: ABCD is a parallelogram & .

Prove: AECF is a parallelogram.

Warm-up: p. 521 #15 – 18 BF DE

8.3 Proving Parallelograms

3. Given: ABCD is a parallelogram.

E and F are midpoints.

Prove: EFCD is a parallelogram.

A B

CD

E F

8.3 Proving Parallelograms

4. Given: JOHN is a parallelogram.

Prove: JBHD is a parallelogram. JD ON HB ON

J O

HN

BD

8.3 Proving Parallelograms

TODAY> 8.4 Special Parallelograms

Rhombus Properties

The diagonals are bisectors of each other. The diagonals bisect the angles of the rhombus.

Remember your P.T. & Special Right s.

Rectangle Properties

The measure of each of a rectangle is 90o. The diagonals of a rectangle are and bisect

each other.

How many Isosceles

s are there?

Square Properties

The diagonals of a square are , and bisect each other.

Exercises: p. 531

How many isosceles RIGHT s are there?

A-S-N (True)

1. The diagonals of a parallelogram are congruent.

2. The consecutive angles of a rectangle are congruent and supplementary.

3. The diagonals of a rectangle bisect each other.

4. The diagonals of a rectangle bisect the angles.

5. The diagonals of a square are perpendicular bisectors of each other.

6. The diagonals of a square divides it into 4 isosceles right triangles.

7. Opposite angles in a parallelogram are congruent.

8. Consecutive angles in a parallelogram are congruent.

SUMMARYParallelogram Rhombus Rectangle Square

Opp sides are //

Opp sides are

Opp s are

Diagonals bisect each other

Diagonals are

Diagonals are

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y Y

Y Y

Proving

1. Given: MPQS is a rhombus.

G, H, I and K are midpoints.

Prove: GHIK is a rectangle.S

BG

K Q

M P

I

H