Theorem

Opposite sides of a parallelogram are equal

Geometry • Quadrilateral • Parallelogram

Formula

A B = D C and A D = B C

A B C D

is a parallelogram, that is

A B ∥ D C

and

A D ∥ B C

, then

A B = D C

and

A D = B C

. Conversely, a quadrilateral in which both pairs of opposite sides are equal is a parallelogram, and so is one in which a single pair of opposite sides is both equal and parallel.

How to use it

1Transfer a known length across the figure: once one side is given, the side opposite it is the same length without further work.
2Use the converse to prove a quadrilateral is a parallelogram from one pair of sides that is both equal and parallel — this is the usual route when a midpoint or a congruence has just produced that pair.
3Combine with the triangle made by a diagonal: the diagonal splits the parallelogram into two congruent triangles, so corresponding parts of those triangles are equal too.

Worked example

In a parallelogram

P QR S

P Q = 9

cm and the perimeter is

28

cm. Find

QR

Step-by-step

1
$S R = P Q = 9$ cm
Why: $S R$ is the side opposite $P Q$ .
2
$P Q + QR + R S + S P = 28 \Rightarrow 9 + QR + 9 + QR = 28$
Why: $S P = QR$ for the same reason, so only one unknown is left.
3
$2 QR = 10 \Rightarrow QR = 5$ cm

Answer

5

Watch out

Equality of opposite sides alone makes a quadrilateral a parallelogram only if the two equal sides are the opposite ones; a single pair of equal sides that are adjacent, or one pair equal and a different pair parallel, proves nothing — an isosceles trapezium has a pair of equal sides and is not a parallelogram.

Builds on

Congruence criteria for triangles Angles made by a transversal of two parallel lines

Also known as

opposite sides of a parallelogram are equal, opposite sides equal and parallel, sides of a parallelogram, one pair of sides equal and parallel, properties of a parallelogram

Appears in past papers

2020 (I)

Q79