The Shape Hierarchy

Every square is a rectangle. Every rectangle is a parallelogram. The groups sit inside each other.

Properties pass down

A rectangle has opposite sides parallel, so every square does too, without needing to be told.

It only works one way

Every square is a rectangle, but not every rectangle is a square.

Several correct names

One shape can honestly be called a square, a rectangle, and a parallelogram at once.

Properties are inherited

Every rectangle is a parallelogram, so every property of parallelograms belongs to rectangles too. If parallelograms have opposite sides equal, rectangles do as well, without needing to be told. Categories that nest pass their properties down.

The quadrilateral hierarchy

Quadrilaterals contain parallelograms; parallelograms contain rectangles and rhombuses; both contain squares. A square is a rectangle, a rhombus, a parallelogram and a quadrilateral all at once, and inherits every property of each.

Inheritance runs one way only

Every square is a rectangle, but not every rectangle is a square. Properties flow downwards, not upwards. Reversing the direction is the classic error, and it is worth testing every such claim with a counterexample before accepting it.

Why this is real reasoning

Deducing that squares have equal diagonals because rectangles do — without measuring a single square — is deductive reasoning. It is the same move that runs through all of geometry, and this lesson is where it starts being used deliberately.