Two Column proofs

Question

Given:?XYZ ≅ ZWX

 

Prove WXYZ is a parallelogram

Answer 1

We have angle XYZ = angle ZWX.

It is easy to prove that the quadrilateral XYZW does not have to be a parallelogram:

For example, one could take two equal triangles XWZ and XYZ and put them together along the XZ side to prove that. The resulting quadrilateral XYZW will have two equal angles (angle XYZ = angle ZWX), but the angle WXY could be different from the angle YZW.

Of course, for the particular situation when those two equal triangles XWZ and XYZ are also isosceles triangles, then the quadrilateral XYZW becomes a parallelogram.