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## Search For a Convex Example

This is one of request or problem sent by a former student (G.H.), now an experienced mathematics teacher.

Recently in my Geometry classes, we were studying quadrilaterals where we had to prove that they were parallelograms. There are six established ways, both sets of opposite sides congruent, both sets of opposite angles congruent, both sets of sides parallel, the diagonals bisect each other and consecutive angles that are supplementary and finally when one set of sides that are both congruent and parallel then the quadrilateral is a parallelogram.

However, we had a question given that one set of sides were congruent and one pair of opposite angles are congruent could that be enough information to prove that it is a parallelogram. I was able to draw a concave quadrilateral to show that this couldn’t happen—the counterexample.

I was wondering if there exists a CONVEX quadrilateral that shows that it isn’t a parallelogram. I have been drawing convex quads and it seems that in every scenario the quadrilateral is a parallelogram. I know I cannot prove that the triangles are congruent (Side-Side-Angle—incorrect way of proving triangles congruent) when I draw a diagonal so I know that there is a problem.

If you could help me out on this, I would appreciate it. This would be a great discussion question ....

Send me your responses and I will forward them to G.H.

Hint: Enough information is supplied in the posed question...