The Parallelogram and Its Angles

The Parallelogram and Its Angles — Moosa Academy

A parallelogram is defined by one thing only: both pairs of opposite sides are parallel. From that single condition, three further properties follow automatically — about the sides, the angles, and the diagonals.

Concept The shape and its labels

The corners are named in order around the shape, so

AB

and

DC

are opposite sides, as are

AD

and

BC

. The diagonals

AC

and

BD

cross at

M

Theorem The four properties

Opposite sides are parallel:

AB \parallel DC

and

AD \parallel BC

Opposite sides are equal:

AB = DC

and

AD = BC

Opposite angles are equal:

\angle A = \angle C

and

\angle B = \angle D

The diagonals bisect each other:

AM = MC

and

BM = MD

There is a fifth statement that follows from the parallel sides and is used constantly: any two adjacent angles add to 180°.

Concept Why adjacent angles add to 180°

Look at the side $AB$ . It cuts across the two parallel lines $AD$ and $BC$ , so it is a transversal. The angles $\angle A$ and $\angle B$ sit between the parallels on the same side of it — they are co-interior angles, and co-interior angles are supplementary:

\angle A + \angle B = 180^\circ

The same argument applies to every side, so each pair of neighbouring angles adds to 180°. Combined with the opposite angles being equal, this means one known angle determines all four.

Example One angle gives all four

In a parallelogram $ABCD$ , $\angle A = 72^\circ$ . Find the other three angles.

\angle C = \angle A = 72^\circ

— opposite angles are equal

\angle B = 180^\circ - 72^\circ = 108^\circ

— adjacent to

\angle A

\angle D = \angle B = 108^\circ

— opposite angles are equal

Check the total: $72 + 108 + 72 + 108 = 360^\circ$ , which is what any quadrilateral must give.

Note What "bisect" means here

The diagonals of a parallelogram cut each other exactly in half, so $M$ is the midpoint of both. Be careful about what this does not say:

The two diagonals are generally not equal in length

They generally do not meet at right angles

Those extra conditions belong to the special parallelograms — equal diagonals make a rectangle, perpendicular diagonals make a rhombus, and both together make a square.

Summary

A parallelogram has both pairs of opposite sides parallel.
Opposite sides are also equal in length.
Opposite angles are equal.
Adjacent angles add to 180°, because they are co-interior angles.
One known angle therefore determines all four.
The diagonals bisect each other, but need not be equal or perpendicular.