Rectangles, Rhombuses & Squares

All three are parallelograms with something extra.

Rectangle: four right angles. Its diagonals are congruent.
Rhombus: four congruent sides. Its diagonals are perpendicular and bisect the angles.
Square: both at once. It gets every property of both.

The diagonals are the fastest way to tell them apart:

• congruent diagonals → rectangle
• perpendicular diagonals → rhombus
• both → square

Think of the family tree. Every square is a rhombus and a rectangle; every rectangle and rhombus is a parallelogram. It does not run the other way — most rectangles are not squares.

A rhombus's diagonals split it into four congruent right triangles, which is why its area can be found as half the product of the diagonals.

Each is a parallelogram with an added constraint, and each constraint has an equivalent diagonal characterisation.

A parallelogram is a rectangle if and only if its diagonals are congruent. It is a rhombus if and only if its diagonals are perpendicular, equivalently if and only if each diagonal bisects a pair of opposite angles. A square satisfies both, so its diagonals are congruent, perpendicular and angle-bisecting.

The hierarchy is strict: square ⊂ rhombus ⊂ parallelogram and square ⊂ rectangle ⊂ parallelogram. Inclusion statements must be read in the right direction — every square is a rectangle, but the converse fails.

A rhombus's perpendicular diagonals give the area formula A = ½d₁d₂, which holds for any quadrilateral with perpendicular diagonals, kites included.

In coordinate geometry the tests are computational: congruent diagonals via the distance formula, perpendicular diagonals via slopes multiplying to −1, and the parallelogram base established first via the midpoint test.