Quadrilateral with Congruent Opposite Sides are Parallelogram

This is the first of a series of post about the conditions of a quadrilateral to be a parallelogram.  In this post, I will be discussing the proof that if the opposite sides of a quadrilateral are congruent, then it is a parallelogram. Given

Quadrilateral $ABCD$ $\overline{AB} \cong \overline{CD}$ $\overline {BC} \cong \overline{AD}$

Proof

Draw diagonal $\overline{BD}$. Now, $\overline{AB} \cong \overline{CD}$ (S) and $\overline {BC} \cong \overline{AD}$ (S).

Also, a segment is congruent to itself (Relfexive Property), so $\overline{BD} \cong \overline{BD}$ (S).

Therefore, by the SSS Triangle Congruence, $\triangle ABD \cong \triangle BDA$.

Since corresponding angles of congruent triangles are congruent, $\angle ADB \cong \angle CBD$ and $\angle ABD \cong \angle CDB$.

But these pairs of angles are corresponding angles, so by the Parallel Line Postulate, $AB \parallel DC$ and $AD \parallel BD$.

So, quadrilateral $ABCD$ is a parallelogram.