Rectangles, Rhombuses, and Squares

All the properties of parallelograms apply to rectangles, rhombuses, and squares. But each of these figures has additional properties that make them more specialized.

Rectangle

A rectangle is a parallelogram with four right angles. Its special properties:

  1. All rules of a parallelogram apply
  2. All four angles are 90°
  3. The diagonals are congruent (equal in length)
  4. The diagonals bisect each other

Rectangle ABCD

Example: In rectangle ABCD, diagonals AC and BD intersect at point R. AR = \(2x - 6\) and CR = \(x + 10\). Find BD.

Since the diagonals bisect each other, AR = CR:

$$2x - 6 = x + 10$$ $$x = 16$$

So AR = \(2(16) - 6 = 26\). Since R is the midpoint of both diagonals, CR = 26 as well — meaning the full diagonal AC = 52. And since the diagonals of a rectangle are equal in length, BD = 52.

Rhombus

A rhombus is a parallelogram with four congruent sides. Its special properties:

  1. All rules of a parallelogram apply
  2. All four sides are equal in length
  3. The diagonals intersect at right angles (they are perpendicular)
  4. Each diagonal bisects the vertex angles it passes through

Rhombus

Example: ABCD is a rhombus and ∠D = 60°. Find ∠A and ∠B.

In a rhombus, opposite angles are equal, so ∠B = ∠D = 60°.

Consecutive angles are supplementary:

$$\angle A = 180 - 60 = 120°$$

Rhombus with diagonal drawn

You can also verify with the triangle formed by the diagonal. In triangle AEB (where E is the intersection of the diagonals), the diagonals are perpendicular so ∠AEB = 90°. Since the diagonal bisects ∠B, we have ∠ABE = 30°. Then:

$$\angle A \text{ (half)} = 180 - 90 - 30 = 60°$$

So the full ∠A = 2 × 60 = 120°. ✓

Square

A square is a parallelogram that is both a rectangle and a rhombus — it has four right angles and four equal sides.

Square ABCD

Example: In square ABCD, AB = \(x + 4\). Find the perimeter.

All four sides are equal, so:

$$P = 4s = 4(x + 4) = 4x + 16$$

How They Relate

Every square is a rectangle and a rhombus. Every rectangle and rhombus is a parallelogram. But a rectangle is not necessarily a rhombus (unless it's also a square), and vice versa.

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