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# Linear Pairs

## Two adjacent angles that form a straight line.

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Linear Pairs

What if you notice two angles in a picture that make a straight line? What information does this give you about the angles? After completing this Concept, you'll be able to apply the properties of linear pairs to help you solve problems.

### Guidance

Adjacent angles are two angles that have the same vertex, share a side, and do not overlap. In the picture below, \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.

\begin{align*}\angle PSQ\end{align*} and \begin{align*}\angle QSR\end{align*} are a linear pair.

\begin{align*}m \angle PSR & = 180^\circ\\ m \angle PSQ + m \angle QSR & = m \angle PSR\\ m \angle PSQ + m \angle QSR & = 180^\circ\end{align*}

#### Example A

What is the value of each angle?

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

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

So, 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 \quad m \angle DBC = 180^\circ - 108^\circ = 72^\circ\end{align*}

#### Example B

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. However, they are supplementary, \begin{align*}120^\circ + 60^\circ = 180^\circ\end{align*}.

#### Example C

Name one linear pair in the diagram below.

One example is \begin{align*} \angle INM\end{align*} and \begin{align*} \angle MNL\end{align*}.

Watch this video for help with the Examples above.

### Vocabulary

Adjacent angles are two angles that have the same vertex, share a side, and do not overlap. 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.

### Guided Practice

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

2. What is \begin{align*}m\angle LNK\end{align*}?

3. If \begin{align*}m\angle INJ = 63^\circ\end{align*}, find \begin{align*}m\angle MNI\end{align*}.

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

2. \begin{align*}90^\circ\end{align*}

3. \begin{align*}180^\circ - 63^\circ=117^\circ\end{align*}

### Practice

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

1. Linear pairs are congruent.
2. Adjacent angles share a vertex.
4. Linear pairs are supplementary.
5. Supplementary angles form linear pairs.

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:

1. \begin{align*}54^\circ\end{align*}
2. \begin{align*}32^\circ\end{align*}
3. \begin{align*}104^\circ\end{align*}
4. \begin{align*}71^\circ\end{align*}
5. \begin{align*}149^\circ\end{align*}
6. \begin{align*}x^\circ\end{align*}

For 12-16, find the value of \begin{align*}x\end{align*}.

### Vocabulary Language: English

Two angles are adjacent if they share a side and vertex. The word 'adjacent' means 'beside' or 'next-to'.
Diagram

Diagram

A diagram is a drawing used to represent a mathematical problem.
linear pair

linear pair

Two angles form a linear pair if they are supplementary and adjacent.