Quadrilateral is Parallelogram iff One Pair of Opposite Sides is Equal and Parallel

Theorem
Let $ABCD$ be a quadrilateral.

Then:
 * $ABCD$ is a parallelogram


 * $AB = CD$ and $AB \parallel CD$
 * $AB = CD$ and $AB \parallel CD$

where $AB \parallel CD$ denotes that $AB$ is parallel to $CD$.

Sufficient Condition
Let $ABCD$ be a parallelogram.

Then $AB \parallel CD$ by definition.

From Opposite Sides and Angles of Parallelogram are Equal it follows that $AB = CD$.

Necessary Condition
Let $AB = CD$ and $AB \parallel CD$.

Then from Lines Joining Equal and Parallel Straight Lines are Parallel, $AD \parallel BC$.

Thus we have $AB \parallel CD$ and $AD \parallel BC$, and so by definition $ABCD$ is a parallelogram.