Definition: Parallelogram A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.

A

D

B

C

A

D

B

C

If both pairs of opposite sides of a quadrilateral are parallel, the quadrilateral is a parallelogram. (reverse of the definition)

A

D

B

C

Here is a quadrilateral with both pairs of opposite sides congruent.

A

D

B

C

A

D

B

C

Triangle ADC is congruent to triangle CBD by SSS.

A

D

B

C

The marked pairs of angles are congruent by CPCTC.

A

D

B

C

The pairs of opposite sides are parallel by alternate interior angles.

A

D

B

C

If both pairs of opposite sides are congruent, a quadrilateral is a parallelogram.

A

D

B

C

This quadrilateral has one pair of opposite sides parallel and congruent.

A

D

B

C

A

D

B

C

The two angles marked in green are congruent by alternate interior angles.

A

D

B

C

Triangle ADC is congruent to triangle CBD by SAS.

A

D

B

C

The remaining pairs of corresponding angles and sides are congruent by CPCTC.

A

D

B

C

AD and CB are parallel by alternate interior angles.

A

D

B

C

If a quadrilateral has one pair of opposite sides both congruent and parallel, it is a parallelogram.

A

B

M

D

C

Here is a quadrilateral in which the diagonals bisect each other.

A

B

M

D

C

The angles marked in green are congruent, because they are vertical angles.

A

B

M

D

C

Triangle AMB is congruent to triangle CMD by SAS.

A

B

M

D

C

The pair of sides marked in red and the pair of angles marked in red are each congruent by CPCTC.

A

B

M

D

C

AB is parallel to CD by alternate interior angles.

A

B

M

D

C

The quadrilateral is parallelogram, because one pair of opposite sides are both congruent and parallel.

A

B

M

D

C

A quadrilateral is a parallelogram if the diagonals bisect each other.

A

D

B

C

This quadrilateral has both pairs of opposite angles congruent.

A

D

B

C

The triangle sum theorem tells us that the sum of the measures of the angles of each triangle is 180º.

A

D

B

C

Since the angles of the two triangles form the four angles of the quadrilateral, the sum of the measures of the four angles of the quadrilateral must be 360º.

A

yº

xº yº D

B

xº C

Let each red angle have a measure, xº, and each green angle have a measure, yº.

A

yº

xº yº

xº

D

C

2x + 2y = 360

B

A

yº

xº yº

xº

D

C

2x + 2y = 360 x + y = 180

B

A

D

B

C

Each red angle is supplementary to each green angle.

A

D

B

C

Each pair of opposite sides is parallel, because supplementary interior angles on the same side of a transversal give parallel lines. The quadrilateral, therefore, is a parallelogram.

A

D

B

C

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

Ways to prove a quadrilateral is a parallelogram.

Ways to prove a quadrilateral is a parallelogram. If both pairs of opposite sides of a quadrilateral are parallel, the quadrilateral is a parallelogram.

Ways to prove a quadrilateral is a parallelogram. If both pairs of opposite sides of a quadrilateral are parallel, the quadrilateral is a parallelogram.

If both pairs of opposite sides are congruent, a quadrilateral is a parallelogram.

Ways to prove a quadrilateral is a parallelogram. If both pairs of opposite sides of a quadrilateral are parallel, the quadrilateral is a parallelogram. If both pairs of opposite sides are congruent, a quadrilateral is a parallelogram. If a quadrilateral has one pair of opposite sides both congruent and parallel, it is a parallelogram.

Ways to prove a quadrilateral is a parallelogram. If both pairs of opposite sides of a quadrilateral are parallel, the quadrilateral is a parallelogram. If both pairs of opposite sides are congruent, a quadrilateral is a parallelogram. If a quadrilateral has one pair of opposite sides both congruent and parallel, it is a parallelogram. A quadrilateral is a parallelogram if the diagonals bisect each other.

Ways to prove a quadrilateral is a parallelogram. If both pairs of opposite sides of a quadrilateral are parallel, the quadrilateral is a parallelogram. If both pairs of opposite sides are congruent, a quadrilateral is a parallelogram. If a quadrilateral has one pair of opposite sides both congruent and parallel, it is a parallelogram. A quadrilateral is a parallelogram if the diagonals bisect each other. If the opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.