Prove ABCD is a parallelogram?

E must be defined to be the point of intersection of the diagonals. AB and DC are parallel because EDC and EBA are equal alternate interior angles. Segments DE and EB are equal, as are segments AE and EC. The angles DEA and BEC are equal because of [I forget the actual name; lines intersect in alternating pairs of angles...]. From SAS, triangles AED and CEB are congruent, so angles DAE and BCE are equal. Since these are equal alternate interior angles, DA and BC are parallel. So, it's a parallelogram.