How to prove that this is a parellelogram?

Question

Prove that ABED is a parallelogram

Given:

1. ABCD is a trapezium

2. F and G are the midpoints of AB and DC respectively

3. FHG is a straight line

4. AD is equal to and parallel to BE

My attempts have included trying to show that $\angle BAD + \angle ADE = 180^{\circ}$ and trying to show that the opposite angles are equal. But these have not led me anywhere as at some point I am required to assume that AB and DE are parallel, which is what has to be proven. I'd like a hint; any help is much appreciated.

Answer 1

Steps 1,2,3 need not be given.

AD = BE and parallel to it, ABED is a parallelogram

So AB = DE and parallel to it, by parallelogram definition and property.

Answer 2

HINT: draw $AE$ and show that triangles $ABE$ and $ADE$ are congruent.