A shorter definition of a rectangle

The Carpenter's Problem has handed us a second definition: a rectangle is a quadrilateral whose diagonals are equal and bisect each other. It says nothing about angles or sides, yet it picks out exactly the same shapes as the first one.

The first definition can be trimmed too. It asked for two things: all angles 90°, and opposite sides equal. Would it be enough to ask only for the angles?

Try to draw a quadrilateral with four right angles whose opposite sides are not equal. You won't manage it. Here's why.

Deduction 4. Take ABCD with all four angles 90°, and join BD. Compare ∆BAD and ∆DCB. BD is in both, and ∠A = ∠C = 90°. What about ∠1 and ∠2?

Since ∠B = 90°, ∠3 + ∠1 = 90°. In ∆BCD, ∠3 + ∠2 + 90° = 180°, so ∠3 + ∠2 = 90°. Hence ∠1 = ∠2: the same trick as in Deduction 2.

By AAS, ∆BAD ≅ ∆DCB. So AD = CB and DC = BA. The equal opposite sides were never an extra condition: they came free with the right angles.

So the definition shortens to: a rectangle is a quadrilateral in which the angles are all 90°. Everything else follows.

The four properties of a rectangle, then:

• all its angles are 90°;
• its opposite sides are equal;
• its opposite sides are parallel: AB is a transversal to AD and BC, and ∠A + ∠B = 180°; when the interior angles on the same side of a transversal add to 180°, the lines are parallel, so AD ∥ BC;
• its diagonals are equal and bisect each other.