Parallelograms

Rectangles have parallel opposite sides. Are there quadrilaterals with parallel opposite sides that aren't rectangles? Easily: draw two pairs of parallel lines that don't meet at right angles, and the shape between them is one.

Quadrilaterals whose opposite sides are parallel are called parallelograms. Is a rectangle one? Its opposite sides are parallel, so yes: a rectangle is a special parallelogram, the one whose angles are all 90°. In a Venn diagram, squares sit inside rectangles, which sit inside parallelograms.

Deduction 6: the angles. In parallelogram ABCD, AB ∥ CD and AD crosses both, so ∠A + ∠D = 180° (interior angles on the same side of a transversal). Likewise ∠A + ∠B = 180° and ∠C + ∠D = 180°. In the book's example with a 30° angle this gives ∠D = 150°, ∠B = 150°, ∠C = 30°.

Make it general with x. If ∠P = x, then the next angle ∠R = 180 − x, and the one after that is 180 − (180 − x) = x. So the opposite angles of a parallelogram are always equal, and neighbouring angles always add to 180°.

Deduction 7: the sides. Draw diagonal BD and compare ∆ABD and ∆CDB:

• one pair of angles is equal, being opposite angles of the parallelogram;
• another pair is equal as alternate angles (AD ∥ BC, with BD as the transversal);
• BD is in both.

By AAS, ∆ABD ≅ ∆CDB. So AD = CB and AB = CD: the opposite sides of a parallelogram are equal. We never assumed that, only that the opposite sides are parallel.

Deduction 8: the diagonals. In parallelogram EASY with diagonals meeting at O, AE = YS (opposite sides) and two pairs of alternate angles are equal. By ASA, ∆AOE ≅ ∆YOS, so OA = OY and OE = OS. O is the midpoint of both diagonals.

The four properties of a parallelogram:

• opposite sides equal;
• opposite sides parallel;
• neighbouring angles add to 180°, and opposite angles are equal;
• the diagonals bisect each other.

One caution (check it with your own drawing): the diagonals of a parallelogram need not be equal. That's what separates it from a rectangle.