the quadrilateral hierarchy

All four-sided figures are quadrilaterals, but some are more special than others, and the special ones nest inside the general ones like a family tree or a set of Russian dolls. The quadrilateral hierarchy is the map of which four-sided shapes contain which — telling you, for instance, that every square is also a rectangle, a rhombus, a parallelogram, and a trapezoid, but not the reverse.

Reading from most general to most special: a quadrilateral is any four-sided polygon. Add 'at least one pair of parallel sides' and you reach a trapezoid. Add the second pair of parallel sides and you reach a parallelogram, which then splits into two branches — the rectangle (add right angles) and the rhombus (add equal sides) — that meet again at the square, the figure with both. Off to the side, the kite (two pairs of adjacent equal sides) branches separately, overlapping the parallelogram line only at the rhombus. Each downward step adds a condition, so each lower figure has every property of those above it plus its own.

The hierarchy is really a chain of conditional statements, and its whole point is that conditionals do not automatically reverse (see converse). 'If a shape is a square, then it is a rhombus' is true; 'if a rhombus, then a square' is false. This is the single most common source of error with quadrilaterals — assuming a property of the special case holds for the general one, or treating 'is a' as symmetric. A clean way to test a claim is to look for a counterexample: a non-square rhombus instantly refutes 'every rhombus is a square'. The inclusive-versus-exclusive trapezoid definition also shifts where parallelograms attach, so the bottom of the tree depends on that convention.