Geometry Help - Prove Diagonals of an Isosceles Trapezoid Congruent?

Question

I need to prove the diagonals of an isosceles trapezoid congruent using ONLY the definition.

Def.=a quadrilateral with the two bases parallel and the two legs congruent

I CANNOT use any properties of an isosceles trapezoid. I CANNOT use the base angles congruent.

I don't want the answer, just some help on where to start. Thanks!

Answer 1

Drop perpendiculars from the endpoints of the shorter parallel side. Now it's easy enough to show that it forms a rectangle, because all four angles are 90 degrees. We also know that the opposite sides of a rectangle are equal. And with the equal non-parallel sides of the trapezoid, we have 2 right triangles with hypotenuses and a pair of legs equal. That makes them congruent. Now look at the 2 triangles formed by an altitude that you dropped and a diagonal of the trapezoid. The third side of these 2 triangles are congruent by equals added to equals. And now you have 2 right triangles congruent by both pairs of legs being equal. So the diagonals of the trapezoid are congruent by corresponding parts.

Answer 2

FH = GE because all diagonals in a trapezoid are congruent.