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.
Practise: chase the angles
A parallelogram: opposite angles are equal, consecutive angles add to
180^\circ. Fill in every angle you can, ending with the highlighted
one. Refresh for a new parallelogram; Check names the property.