Rectangles, Rhombuses, and Squares

By the end of this lesson, you will be able to state and apply the special diagonal properties of rectangles, rhombuses, and squares; classify quadrilaterals using the hierarchy; and solve algebraic problems using diagonal and angle properties.

The Big Idea

Rectangles, rhombuses, and squares are special parallelograms — they inherit all five parallelogram properties and add more. The additional properties come from the extra constraint: right angles (rectangle), equal sides (rhombus), or both (square). Their diagonal properties are especially useful for proofs and applications.

The Quadrilateral Hierarchy

Quadrilateral
  └─ Parallelogram (2 pairs parallel sides)
       ├─ Rectangle (4 right angles)
       │    └─ Square (4 right angles AND 4 equal sides)
       └─ Rhombus (4 equal sides)
            └─ Square (4 equal sides AND 4 right angles)

A square is both a rectangle and a rhombus.

Rectangle

Definition: a parallelogram with four right angles.

All parallelogram properties plus:

Special diagonal property: Diagonals of a rectangle are congruent.

$AC = BD$

Theorem: A parallelogram is a rectangle if and only if its diagonals are congruent.

Rhombus

Definition: a parallelogram with four congruent sides.

All parallelogram properties plus:

Special diagonal properties:

1. Diagonals of a rhombus are perpendicular ($AC \perp BD$)
2. Each diagonal bisects a pair of opposite angles

Theorem: A parallelogram is a rhombus if and only if its diagonals are perpendicular.

Square

Definition: a parallelogram with four right angles AND four congruent sides.

Equivalently: a square is both a rectangle and a rhombus.

Diagonal properties (combined):

• Congruent (from rectangle)
• Perpendicular (from rhombus)
• Bisect opposite angles (from rhombus) — each at 45°

Example 1 — Rectangle Diagonal