How about a trapeziod
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Squares, rectangles, and all parallelograms all fit this description.
A 4 sided arrowhead quadrilateral would fit the description.
Triangle, quadrilateral or hexagon. Triangle, quadrilateral or hexagon. Triangle, quadrilateral or hexagon. Triangle, quadrilateral or hexagon.
A 4 sided quadrilateral kite would fit the description given.
A cyclic quadrilateral is one that has concyclic vertices (its corners all fit on the same circle) and, for a simple cyclic quadrilateral, opposite angles are supplementary.