The side AE is the sum of AD and DE and the side DE is the sum of EH and DE.

Since AD = EH, the sides AE and DE are equal, as well.

Thus triangles ABE and DCH are congruent.

Begin with the two parallelograms ABCD and EBCF between two parallels.

The first step of the proof is to establish that the triangles ABE and DCH are congruent.

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Because AB and DC are parallel with the transversal AD, the angles BAE and CDH, the exterior to the interior, are equal.

Since ABCD is a parallelogram, the sides AB and DC are equal.

 

Now label the point of intersection of DC and BE as G.

The parallelogram ABCD is equal to the triangle ABE plus the triangle BGC minus the triangle DGE.

The parallelogram EBCF is equal to the triangle DCH plus the triangle BGC minus the triangle DGE.

Since triangles ABE and DCH are congruent and hence equal, the parallelograms ABCE and EBCH must be equal.

QED