Properties of Parallelograms

Take a rectangle drawn on a stiff card frame, pin the corners so they can swing, and give the top edge a gentle push sideways. The rectangle slumps into a leaning shape — but look closely: the top and bottom edges are still parallel, and so are the two slanted sides. You have just made a parallelogram.

A parallelogram is a quadrilateral with both pairs of opposite sides parallel. That is the only thing you have to promise. The astonishing part is what comes free: from that one property, a whole cascade of others is forced to be true — the opposite sides must be equal, the opposite angles must match, and the diagonals must slice each other exactly in half. You never have to measure a single one of them.

The four properties

Every fact below is a consequence of "both pairs of opposite sides are parallel" — none of them is a separate rule to memorise, they all fall out of that one idea.

In any parallelogram:

opposite sides are equal in length;
opposite angles are equal;
consecutive angles add to 180^\circ — each pair of sides is parallel, so two angles next to each other are exactly a co-interior pair;
the diagonals bisect each other — each diagonal cuts the other exactly in half.

Why the angles work

Nothing here needs measuring — it all comes straight from "both pairs of sides are parallel". Step through a parallelogram ABCD and watch the angle facts appear.

Because AD \parallel BC with AB crossing them, the two angles at A and B are co-interior, so they add to 180^\circ. Slide round the shape the same way and you find the opposite angles at A and C must be equal.

Worked example 1 — the angles

A parallelogram ABCD has one angle \angle A = 70^\circ. Find all the other angles.

\angle C is opposite to \angle A, so \angle C = 70^\circ.
\angle B is next to \angle A (consecutive), so \angle B = 180^\circ - 70^\circ = 110^\circ.
\angle D is opposite \angle B, so \angle D = 110^\circ.

Check: the four angles are 70,\;110,\;70,\;110, and they add to 360^\circ — exactly what the angles of any quadrilateral must total.

Worked example 2 — the sides

In parallelogram PQRS, side PQ = 8\text{ cm} and side QR = 5\text{ cm}. What are the other two sides, and the perimeter?

Opposite sides are equal, so RS = PQ = 8\text{ cm} and SP = QR = 5\text{ cm}. A parallelogram is really only two side lengths, each used twice, so the perimeter is

P = 2(8 + 5) = 26\text{ cm}.

Worked example 3 — the diagonals

The two diagonals of a parallelogram always cross at their shared midpoint M — that is what "bisect each other" means. So if a diagonal AC is 12\text{ cm} long, the centre splits it into AM = MC = 6\text{ cm}. And if you are told the half-length BM = 5\text{ cm}, the whole diagonal is BD = 2 \times 5 = 10\text{ cm}.

Worked example 4 — a pinch of algebra

Opposite angles are equal, and that lets us solve for an unknown. Suppose one angle of a parallelogram is (2x + 10)^\circ and the angle opposite it is (3x - 20)^\circ. Because they must be equal, set the two expressions equal:

2x + 10 = 3x - 20.

Subtract 2x from both sides and add 20: 30 = x. So x = 30, and each of those angles is 2(30) + 10 = 70^\circ. (The other two angles are then 110^\circ each.)

The family: rectangle, rhombus, square

A parallelogram is the parent of some shapes you already know. Add one extra promise and you climb into a more special member of the family — but every one of them is still a parallelogram, so everything above stays true.

A rectangle is a parallelogram whose angles are all 90^\circ.
A rhombus is a parallelogram whose four sides are all equal.
A square is the overachiever — a parallelogram that is both a rectangle and a rhombus: all sides equal and all angles 90^\circ.

The property is "opposite angles are equal" — the two angles across the diagonal from each other. It is very tempting to slide that into "all four angles are equal", but that is false for a leaning parallelogram. The two adjacent angles (the ones next to each other) are supplementary: they add to 180^\circ, not equal. All four angles are equal only when the shape is a rectangle (all 90^\circ).

The diagonals trip people up too. They do bisect each other — but in general they are not the same length (they are equal only in a rectangle), and they do not cross at right angles (that happens only in a rhombus). "Bisect" means "cut in half", not "equal" and not "perpendicular".

Everything! The "push a rectangle sideways and the opposite sides stay parallel" trick is a superpower for engineers. Pin four rods into a parallelogram with swinging corners and you get a parallelogram linkage: as it flexes, the opposite bars are forced to stay parallel. That is exactly the mechanism inside a folding clothes-drying rack, a scissor lift, the arm of a desk lamp that keeps its shade pointing the same way, and a pantograph (the diamond linkage that copies a drawing at a bigger scale, and the springy arm that presses a train's power cable). The geometry does the guaranteeing — the parts cannot tilt out of parallel — so machines can move in controlled, predictable ways.