Classifying Quadrilaterals: The Shape Hierarchy

Sorting by properties

A quadrilateral is any closed shape with four straight sides. They are sorted by what is true of their sides and angles.

Shape	Defining properties
quadrilateral	four straight sides
trapezoid	at least one pair of parallel sides
parallelogram	two pairs of parallel sides
rectangle	a parallelogram with four right angles
rhombus	a parallelogram with four equal sides
square	four right angles and four equal sides

Each row down the table adds a condition. Adding conditions narrows the group, so there are fewer squares than rectangles and fewer rectangles than parallelograms.

The categories nest

Each shape is defined by having more properties. So the groups sit inside one another:

$$\text{squares} \subset \text{rectangles} \subset \text{parallelograms} \subset \text{trapezoids} \subset \text{quadrilaterals}$$

Read it outward. Every square is a rectangle. Every rectangle is a parallelogram. Every parallelogram is a trapezoid. All of them are quadrilaterals.

Why every square is a rectangle

This is the claim that feels wrong. Reading the definition settles it.

A rectangle is a parallelogram with four right angles. That is the whole requirement.

Does a square have four right angles? Yes. Is it a parallelogram? Yes. So it is a rectangle.

Having four equal sides as well does not disqualify it. A definition lists what a shape must have, not what it must lack.

The comparison that makes it click: a square is to a rectangle what a terrier is to a dog. Every terrier is a dog. Not every dog is a terrier. Nobody finds that confusing, and it is the identical relationship.

$$\text{every square is a rectangle} \qquad \text{not every rectangle is a square}$$

A square is also a rhombus, for the same reason — it is a parallelogram with four equal sides. A square is exactly the shape that is both.

Inheriting properties

This is what the hierarchy is for.

A shape has every property of every category it belongs to. So knowing a shape is a square tells you, without any checking:

From	It inherits
quadrilateral	four sides, angles totaling $360°$
parallelogram	opposite sides parallel and equal
rectangle	four right angles, equal diagonals
rhombus	four equal sides, perpendicular diagonals

That is a lot of facts from one word. Classify a shape before working on it. The category carries information the picture might not show.

The inheritance runs one way only. A parallelogram need not have right angles. That property belongs to a narrower category.

Angles in a quadrilateral

Every quadrilateral’s angles total $360°$.

The reason is short. A diagonal cuts any quadrilateral into two triangles. Each triangle’s angles add to $180°$.

$$2 \times 180° = 360°$$

So a rectangle’s four right angles are no accident: $4 \times 90° = 360°$, which is exactly the budget available.

Worked examples

Common mistakes

Practice problems