Question

Geometry:
Given:
E is the midpoint of seg BC
G is the midpoint of seg AD
seg EF is perpendicular to seg BD
seg GH is perpendicular to seg BD
seg HG is congruent to seg FE
angle 3 is congruent to angle 4
Prove: ABCD is a parallelogram

I can prove already that one opposite sides are parallel because angles 3 and 4 are alternate interior angles. How can I prove that BC and AD are congruent to each other using the only information of their midpoints?