Given: ABCD is a quadrilateral, Segment AB is congruent to Segment CD, <1 is congruent to <2 Prove: ABCD is a parallelogram

Answer 1

In a nutshell...

If angle 1 and angle 2 are adjacent then their adjacent sides are parallel. The diagonal through 1 will get the other two angles congruent per alternate interior.

If angle 1 and angle 2 are across from each other then the diagonal will produce two congruent triangles by ASA.