Rectangles and squares

We look closely at squares and rectangles. Take rectangle ABCD.

• The points A, B, C and D are its corners.
• The lines AB, BC, CD and DA are its sides.
• Its angles are ∠A, ∠B, ∠C and ∠D.

AB and CD lie opposite each other, so they are called opposite sides; AD and BC are the other pair of opposite sides.

What makes a rectangle a rectangle? Two properties, and it needs both:

• R1: the opposite sides are equal in length.
• R2: all the angles are 90°.

A slanted four-sided figure can have its opposite sides equal and still not be a rectangle, because its angles are not 90°. A square has corners and sides in the same way, and two properties of its own:

• S1: all the sides are equal.
• S2: all the angles are 90°.

Naming matters too. Going round rectangle ABCD, you meet A, then B, then C, then D. So it can also be named BCDA, CDAB or DABC (starting at a different corner), or ADCB, DCBA, CBAD and BADC (going round the other way). But it cannot be named ABDC or ACBD. In ABDC, the step from B to D jumps across the middle of the rectangle instead of going along a side. In a valid name, the corners come in an order of travel around the shape, starting from any corner.

The book asks which is not a name for square SPQR. In that square, S is the top-left corner, P the top-right, Q the bottom-right and R the bottom-left, so going round you meet S, P, Q, R. SPQR, RSPQ and QRSP all follow this order. PQSR does not: after Q it jumps across to S. So PQSR is not a name for this square.