Question

ABCD is a quadrilateral, AC and BD meet at E. It is given that angle A is equal to angle C, and BE=ED. Prove that ABCD is a parallelogram.

Answer 1

Since they are vertical angles, you know that \(\angle BEC\cong\angle AED\) and \(\angle AEB\cong\angle CED\).
Notice that \(\angle BAE\cong\angle DEC\). This means you have congruent triangles, \(\Delta ABE\cong\Delta CDE\) and \(\Delta ACE\cong\Delta ADE\). (You could use the angle-angle side postulate to expand on this).

I think that's all you need.