A parallelogram is a quadrilateral whose opposite sides are parallel. In the figure below, is a parallelogram. is parallel to and is parallel to .

In this post, aside from being parallel, we will also prove that the opposite sides of a parallelogram are congruent.

**Theorem**

The opposite sides of a parallelogram are congruent.

**Given**

Parallelogram .

**What To Show**

and .

**Proof**

Draw diagonal .

Since is a transversal to parallel lines and , because they are alternate interior angles. (A)

Now, segment is congruent to itself (reflexive proeprty) so, . (S)

Since is parallel to , because they are alternate interior angles. (A)

By ASA congruence, .

Since corresponding parts of congruent triangles are congruent, and .