The quadrilateral is any 4-sided polygon. But it is the parent of a whole family of figures, each with their own distinct names and characteristics. See also Tetragon.
Quadrangle ("four angles") is another name for a quadrilateral. However, it is a word often used for an open space where people gather, say on a school or university campus. In this case the quadrangle (or "quad") is usually a rectangle or square.
||All sides equal, all angles 90°. See Definition of a square|
||Opposite sides equal, all angles 90°. See Definition of a rectangle|
||Opposite sides parallel. See Definition of a parallelogram|
||Two sides parallel. See Definition of a trapezoid|
||Opposite sides parallel and equal. See Definition of a rhombus|
||Adjacent pairs of sides equal. See Definition of a kite|