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.
Draw diagonal .
Now, (S) and (S).
Also, a segment is congruent to itself (Relfexive Property), so (S).
Therefore, by the SSS Triangle Congruence, .
Since corresponding angles of congruent triangles are congruent,
But these pairs of angles are corresponding angles, so by the Parallel Line Postulate,
So, quadrilateral is a parallelogram.