Tests for a parallelogram

Grade 8 facts about a parallelogram:

• (a) opposite sides are equal;
• (b) opposite angles are equal;
• (c) the diagonals bisect each other.

Their converses are true too, so each one is a test for a parallelogram. When proving them, forget the properties: a parallelogram means only 'both pairs of opposite sides parallel'. You may not use equal sides to prove equal sides.

(a) Opposite sides equal ⇒ parallelogram. Draw diagonal AC: SSS gives ΔACD ≅ ΔCAB, so the alternate angles are equal, and the sides are parallel.

(b) Opposite angles equal ⇒ parallelogram. The four angles make 360°, so 2(∠A + ∠B) = 360°: neighbouring angles add to 180°, and the sides are parallel.

(c) Diagonals bisect each other ⇒ parallelogram. SAS gives ΔAED ≅ ΔCEB, so the alternate angles are equal.

Theorem 5: one pair of opposite sides both equal and parallel ⇒ parallelogram.

A converse is never automatic. 'If x = y then x² = y²' is true, but cannot be turned round: 3² = (–3)². Each converse above needed its own proof.