Find and Use Linear Pairs

A linear pair is a pair of angles that "lie" on a line.

More formally, two angles form a linear pair if and only if all of the following conditions hold:

• the two angles share a side;

• the two angles share a vertex (meaning the lines that form the two angles all meet at the same point);

• the sides that the angles do not share lie on the same line.

The Linear Pair Theorem states that two angles that form a linear pair are supplementary; that is, their measures add up to $180$ degrees.

Suppose that we have the points $P$, $Q$, $R$, and $S$ such that $P$, $Q$, and $R$ are collinear, and $Q$ lies between $P$ and $R$.

Clearly, the angles $\angle PQS$ and $\angle RQS$ form a linear pair.

By the Angle Addition Postulate, $m\angle PQS+m\angle RQS=m\angle PQR$.

However, $\angle PQR$ is a straight angle, so $m\angle PQR=180^\circ$.

Thus, $\angle PQS$ and $\angle RQS$ are supplementary, as desired.

A linear pair of angles are two angles whose measures sum to $180^\circ$

Linear pairs are also supplementary angles. Recall that supplementary angles also sum to $180^\circ$.

Another way of remembering what linear angles are is to recognize that the word 'line' is in 'linear'. $\rightarrow$ $\textcolor{red}{line}ar$ angles form a straight $\color{red}{line}$.