A regular polygon has all its angles of equal measure, and its sides of equal length. In the case of a quadrilateral, that would mean a square.
A concave quadrilateral must have at least one reflex angle and so cannot be a square. So it cannot be regular.
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A trapezoid (UK trapezium) is a quadrilateral has only one pair of parallel sides. Otherwise, there are many irregular quadrilaterals that have no parallel sides, such as a kite or dart shape, and these include both concave and convex quadrilaterals.
Concave is a property of [irregular] polygons. A parallelogram cannot be concave.
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square, rectangle, trapezoid, rhombus, parallelogram, kite, and irregular quadrilaterals
Quadrilaterals- Squares, Rectangles, Rhomboids, Parallelograms, kites, chevrons, trapezoids, and irregular quadrilaterals.