# The Opposite Sides of a Parallelogram are Congruent

A parallelogram is a quadrilateral whose opposite sides are parallel. In this post, we show that asides from being parallel, they are also congruent. In the figure below, $ABCD$ is a parallelogram; $\overline{AB}$ is parallel to $\overline{CD}$ and $\overline{AD}$ is parallel to $\overline{BC}$.

To prove that the opposite sides of $ABCD$ are congruent, we have to show that $\overline{AD} \cong BC$ and $\overline{AB} \cong CD$.

Theorem: The opposite sides of a parallelogram are congruent.

Given: Parallelogram $ABCD$.

Proof: Draw $\overline{BD}$

Notice that $\overline{BD}$ serves as a transversal to the parallel line segments.

Clearly, $\angle 1 \cong \angle 3$ because they are alternate interior angles (A).

Also, $BD \cong BD$ since any segment is congruent to itself (S).

Lastly, $\angle 2 \cong \angle 4$ because they are alternate interior angles (A)

Since the side is included by the two angles, by ASA Congruence, triangle $ABC \cong CDB$

Therefore, $AB \cong CD$ and $AD \cong BC$ since corresponding sides of congruent triangles are congruent. $\blacksquare$

So, the opposite sides of a parallelogram are congruent.