4.4 Parallelograms
A rectangle has parallel opposite sides. Are there quadrilaterals with parallel opposite sides that are not rectangles? Certainly, draw two pairs of parallel lines that avoid meeting at right angles, and a leaning quadrilateral appears.
Definition: Parallelogram A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.
Since a rectangle has parallel opposite sides, every rectangle is a parallelogram, a particular one with all angles 90^\circ. On the Venn diagram, the rectangle nests inside the parallelogram, and the square inside the rectangle.
To probe its angles and sides, build a parallelogram ABCD with adjacent sides AB = 6\ \text{cm}, AD = 4\ \text{cm} and a 35^\circ angle between them; finish it by drawing a line through D parallel to AB and a line through B parallel to AD, meeting at C.
Deduction 6: opposite angles are equal
Because AB \parallel CD with AD as transversal, the co-interior angles give \angle A + \angle D = 180^\circ, so \angle D = 180 - 35 = 145^\circ. Likewise \angle A + \angle B = 180^\circ gives \angle B = 145^\circ, and \angle C = 35^\circ. In general, set \angle P = x in parallelogram PQRS: then \angle S = 180 - x (co-interior with \angle P), and \angle R = 180 - \angle S = 180 - (180 - x) = x. So opposite angles are equal and adjacent angles are supplementary (they total 180^\circ).
Deduction 7: opposite sides are equal
In parallelogram ABCD draw diagonal BD. Compare \triangle ABD and \triangle CDB: the opposite angles \angle A = \angle C (just shown), the alternate angles cut by transversal BD across AD \parallel BC are equal, and BD is shared. By AAS, \triangle ABD \cong \triangle CDB, so AD = CB and AB = CD. Opposite sides are equal.
Deduction 8: the diagonals bisect each other
In parallelogram ABCD, let the diagonals meet at O. Compare \triangle AOB and \triangle COD: AB = CD (opposite sides), and the two pairs of alternate angles (from the parallel sides cut by the diagonals) are equal. By ASA, \triangle AOB \cong \triangle COD, so OA = OC and OB = OD, O is the midpoint of each diagonal. The diagonals bisect each other. (They need not be equal, equal diagonals were special to the rectangle.)
Properties of a Parallelogram 1. Opposite sides are equal. 2. Opposite sides are parallel. 3. Adjacent angles add to 180^\circ; opposite angles are equal. 4. The diagonals bisect each other (but are generally not equal).
Worked Example
In parallelogram ABCD, \angle A = 65^\circ. Find the other three angles.
Adjacent angles are supplementary, so \angle B = 180 - 65 = 115^\circ. Opposite angles are equal, so \angle C = \angle A = 65^\circ and \angle D = \angle B = 115^\circ.
Worked Example
In parallelogram KLMN, side KL = 9\ \text{cm} and side LM = 6\ \text{cm}. The diagonals meet at O with KO = 5\ \text{cm}. Find MN, NK and the full diagonal KM.
Opposite sides are equal, so MN = KL = 9\ \text{cm} and NK = LM = 6\ \text{cm}. The diagonals bisect each other, so O is the midpoint of KM; hence KM = 2 \times KO = 2 \times 5 = 10\ \text{cm}.
Enrichment: A Useful Converse Deduction 8 says a parallelogram’s diagonals bisect each other. The converse is also true and very handy: if the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
Proof. Let the diagonals of ABCD meet at O with OA = OC and OB = OD. In \triangle AOB and \triangle COD: OA = OC, OB = OD, and \angle AOB = \angle COD (vertically opposite). By SAS they are congruent, so AB = CD and the alternate angles \angle OAB = \angle OCD are equal, which makes AB \parallel CD. The same argument on \triangle AOD and \triangle COB gives AD \parallel BC. Both pairs of opposite sides are parallel, so ABCD is a parallelogram.