Quadrangle that is not a parallelogram, but has congruent opposites and 2 parallel sides Does there exist a quadrangle $ABCD$ that is not a parallelogram, but opposite sides are congruent and $AB$ is parallel to $CD$? My initial thought was something of a bowtie shape or a trapezoid, but neither of these work.

Yes.

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Sorry the letters are wrong, based on your question, but you get the idea.

NOTE: One definition of a parallelogram is a quadrilateral whose opposite sides are congruent and parallel, so the answer cannot be a quadrilateral.