UNIT 7 LESSON 4B PROVING PARALLELOGRAMSCCSS

G-CO 11: Prove theorems about parallelograms.

LESSON GOALS

Use properties of parallelograms to prove that a quadrilateral is a parallelogram in a two-column or coordinate plane proof.

ESLRs: Becoming Effective Communicators, Competent Learners and Complex Thinkers

COORDINATE GEOMETRY

When a figure is in the coordinate plane you can use:

the Distance formula to prove that sides are congruent

the slope formula to prove that sides are parallel

the midpoint formula to prove the diagonals bisect each other.

FORMULAS

DistanceFormula

Slope Formula

Midpoint Formula

Show that quadrilateral ABCD is a parallelogram

(–1, 2) A

C (1, – 2)

x

y

B ( 3, 2)

(– 3, – 2) D

Not on the coordinate

plane.

(definition)Opposite sides are ||

Opposite sides are

Opposite angles are

Consecutive 's are supplementary

Diagonals bisect each other

One pair of opposite sides are both and

EXAMPLE

Show that quadrilateral ABCD is a parallelogram

(–1, 2) A

C (1, – 2)

x

y

B ( 3, 2)

(– 3, – 2) D

(definition)Opposite sides are ||

METHOD 1

ABm

DCm

ADm

BCm

0

0

2

2

AB DC AD BC

Show that quadrilateral ABCD is a parallelogram

(–1, 2) A

C (1, – 2)

x

y

B ( 3, 2)

(– 3, – 2) D

Opposite sides are

METHOD 2

4

4DC

AB AB DC

2 52 22 4 223 1 2 2

202 24 2 2 22 2 1 3

20

2 5AD BC

AD

BC

Show that quadrilateral ABCD is a parallelogram

(–1, 2) A

C (1, – 2)

x

y

B ( 3, 2)

(– 3, – 2) D

METHOD 3

4

4DC

AB

AB DC

One pair of opposite sides are both and

ABm 0

0DCm

AB DC

Show that quadrilateral ABCD is a parallelogram

(–1, 2) A

C (1, – 2)

x

y

B ( 3, 2)

(– 3, – 2) D

Diagonals bisect each other

METHOD 4

2 21 1,

2 2

0,0

midpoint BD

midpoint AC

3 3 2 2,

2 2

0,0

TODAY’S ASSIGNMENT

p. 342: 8, 21, 24, 25, 26