### Linear Pairs

Two angles are **adjacent** if they have the same vertex, share a side, and do not overlap. \begin{align*}\angle PSQ\end{align*} and \begin{align*}\angle QSR\end{align*} are adjacent.

A **linear pair** is two angles that are adjacent and whose non-common sides form a straight line. If two angles are a linear pair, then they are supplementary (add up to \begin{align*}180^\circ\end{align*}). \begin{align*}\angle PSQ\end{align*} and \begin{align*}\angle QSR\end{align*} are a linear pair.

What if you were given two angles of unknown size and were told they form a linear pair? How would you determine their angle measures?

### Examples

For Examples 1 and 2, use the diagram below. Note that \begin{align*}\overline{NK} \perp \overleftrightarrow{IL}\end{align*}.

#### Example 1

Name one linear pair of angles.

\begin{align*} \angle MNL \end{align*} and \begin{align*}\angle LNJ\end{align*}

#### Example 2

What is \begin{align*}m\angle INL\end{align*}?

\begin{align*}180^\circ\end{align*}

#### Example 3

What is the measure of each angle?

These two angles are a linear pair, so they add up to \begin{align*}180^\circ\end{align*}.

\begin{align*}(7q-46)^\circ + (3q+6)^\circ &= 180^\circ\\ 10q - 40^\circ &= 180^\circ\\ 10q & = 220^\circ\\ q & = 22^\circ\end{align*}

Plug in \begin{align*}q\end{align*} to get the measure of each angle. \begin{align*}m\angle ABD = 7(22^\circ) - 46^\circ = 108^\circ \ m\angle DBC = 180^\circ - 108^\circ = 72^\circ\end{align*}

#### Example 4

Are \begin{align*}\angle CDA\end{align*} and \begin{align*}\angle DAB\end{align*} a linear pair? Are they supplementary?

The two angles are not a linear pair because they do not have the same vertex. They are supplementary because they add up to \begin{align*}180^\circ\end{align*}: \begin{align*}120^\circ + 60^\circ = 180^\circ\end{align*}.

#### Example 5

Find the measure of an angle that forms a linear pair with \begin{align*}\angle MRS\end{align*} if \begin{align*} m\angle MRS\end{align*} is \begin{align*} 150^\circ\end{align*}.

Because linear pairs have to add up to \begin{align*}180^\circ\end{align*}, the other angle must be \begin{align*}180^\circ-150^\circ=30^\circ\end{align*}.

### Review

For 1-5, determine if the statement is true or false.

- Linear pairs are congruent.
- Adjacent angles share a vertex.
- Adjacent angles overlap.
- Linear pairs are supplementary.
- Supplementary angles form linear pairs.

For exercise 6, find the value of \begin{align*}x\end{align*}.

Find the measure of an angle that forms a linear pair with \begin{align*}\angle MRS\end{align*} if \begin{align*} m\angle MRS\end{align*} is:

- \begin{align*}61^\circ\end{align*}
- \begin{align*}23^\circ\end{align*}
- \begin{align*}114^\circ\end{align*}
- \begin{align*}7^\circ\end{align*}
- \begin{align*}179^\circ\end{align*}
- \begin{align*}z^\circ\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 1.9.