Given quadrilateral ABCD, where the diagonals AC and BD intersect at point E. AE≅EC and BE≅DE. Can you prove can you prove that the figure is a parallelogram? Explain.

Answer 1

This problem can be understood much better if I can draw the quadrilateral and diagonals.

Given AE≅EC and DE≅EB

Angle AEB≅DEC. ( vertical angles are congruent)

Therefore triangles AEB≅triangleDEC by side, angle, side congruence.

Similarly AE≅EC, DE≅EB and vertical angles AED≅vertical angle BEC Therefore triangle AED≅BEC triangle

That makes AD≅BC, when triangles are congruent, all their corresponding sides are also congruent.

And AB≅DC

When both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram