Linear Pair of Angles

A linear pair of angles (adjacent supplementary angles) consists of two angles that share the same vertex and one common side, while their non-common sides extend in opposite directions along the same straight line.

Put simply, a linear pair is formed by two adjacent angles whose outer sides create a straight line.

The measures of the two angles in a linear pair always add up to a straight angle:

$$ 180^\circ $$

or, in radians:

$$ \pi $$

Because of this property, every linear pair is also made up of supplementary angles.

Difference between a linear pair and adjacent angles. A linear pair is always made up of adjacent angles because the two angles share a common vertex and one side. However, not all adjacent angles form a linear pair. The key difference is that, in a linear pair, the non-common sides must form a straight line. This also means that the two angles always add up to 180°. Adjacent angles, on the other hand, may have any sum because their outer sides are not necessarily aligned.

A practical example

Consider the following two angles:

$$ \alpha = 60° $$

$$ \beta = 120° $$

These angles share the same vertex and one common side. Their other sides lie on the same straight line and point in opposite directions.

For this reason, the two angles form a linear pair of angles.

If we add the two angles together, we obtain:

$$ \alpha + \beta = 180° $$

This confirms that they are supplementary angles and form a linear pair.