# Linear Pair of Angles

Linear pair of angles are two angles that form a straight angle (angle measuring 180 degrees). This fact leads to a wide range of properties and applications involving linear pairs of angles. Since linear pairs of angles are special types of adjacent angles that are supplementary, it’s important to know what adjacent angles are and the core components needed to whether two angles are adjacent.

In this article, we’ll review what makes linear pair of angles unique, show you the important conditions before we can confirm a pair of angles are considered linear, and also let you try out different problems involving linear pair of angles. We hope that by the end of our discussion, you’ll be ready and confident when dealing with word problems involving linear pairs of angles as well.

## What Are Linear Pair of Angles?

Linear pair of angles are two adjacent angles that form a straight angle when combined.  By straight angle, we mean an angle that forms a straight line and has an angle measure of 180o. This also means that the linear pairs of angles are two adjacent angles that are supplementary (they add up to 180o).

Let’s take a look at the linear pair of angles shown above. We can see that the angles, \angle AOC and \angle COB, are adjacent angles sharing a common side of \overline{OC} and a vertex of O. The linear pair of angles are also supplementary and form a straight angle, so \angle AOC + \angle COB = 180\degree = \angle AOB.

This is the fundamental definition of linear pair of angles- know this concept by heart to help you understand more complex properties and applications of linear pair of angles.

### Problem 1

Determine whether each pair of angles can form a linear pair of angles when combined.

a.

If we merge the two angles, they will form a larger angle, \angle AOC + \angle COB = \angle AOB. Since the two angles share a common side and a common point (\overline{OC} and O), these two are adjacent angles when combined.

Adding the two angles, we can see that their sum is 180o and consequently, they are also supplementary. Since we’ve shown that when we combine the two angles, they become adjacent and supplementary, we can confirm that they are linear pairs of angles.

b.

Visually, we can see that the angles will not form a straight line but will form an angle larger than 180o

By adding the two angles, we’ll also see that the angle measure of \angle    AOB is greater than 180o.

\begin{aligned}\angle AOB &= \angle AOC + \angle COB \\&= 110\degree + 90\degree\\&= 200\degree\end{aligned}

Since the pair of angles form an angle not measuring 180o, they do not form a straight angle.

c.

Let’s now rearrange the angles and try to visualize how they would form a linear angle. From inspection, we can see that the common vertex shared is O and the common side shared by these angles is \overline{OC}.

We’ve shown that the angles form a linear angle and we can also see that \angle AOB has a measure of 180o. This means that the pair of angles is linear pair of angles.

### Problem 2

Two lines, \overline{AB} and \overline{CD}, intersect at point, O. Write down four linear pairs of angles you can find.

Sketch the figure described by the problem to help you visualize the four pairs that you’re looking for.

Let’s take a look at these two pairs – each pair of angles are adjacent angles forming a straight angle each. This means that both pairs are linear pair of angles. Hence, \angle AOD and \angle AOC as well as \angle AOC and \angle COB are both linear pairs of angles.

Let’s take a look at these two pairs – each pair of angles are adjacent angles forming a straight angle each. This means that both pairs are linear pairs of angles. Hence, \angle AOD and \angle AOC as well as \angle AOC and \angle COB are both linear pairs of angles.

Let’s now take a look at the two remaining pairs of linear angles as shown above. Similar to the previous pairs, since each pair is adjacent angles that form straight angles, we can conclude that these two pairs are also linear pairs. This means that we have the following four pairs of linear angles:

\begin{aligned}\angle AOC  \,\&\, \angle AOD\\\angle AOC  \,\&\, \angle AOB\\\angle BOC  \,\&\, \angle BOD\\\angle AOD  \,\&\, \angle DOB\end{aligned}

## Understanding the Difference Between Linear Pair of Angles and Supplementary Angles

These two types of angles are sometimes get interchanged, but it’s important that we know the difference between these two pairs of angles.

Take a look at these two examples: for a pair of angles to be a linear pair, they must always be adjacent angles and are supplementary angles. Meanwhile, supplementary angles are simply pairs of angles that add up to 180o. This means that the angles do not have to be merged and form a straight line.

Hence, all linear pairs of angles are supplementary but not all supplementary angles are linear pairs of angles. This is why it’s important to keep this difference in mind when working with different problems involving angles adding up to 180o.

## What Are Problems Involving Linear Pair of Angles?

There is a wide range of problems involving linear pairs of angles. More often than not, we’re asked to find missing angles’ measures or solve for the unknown values given linear pair of angles. But before trying out different problems, let us first introduce you to the linear pair postulate. It’s an important postulate to use when solving problems involving linear pairs of angles.

\begin{aligned}\angle AOC + \angle COB = 180\degree\end{aligned}

According to the linear pair postulate, if a ray stands on and divides a line, the two adjacent angles that were formed will always have a sum of 180o. The converse statement remains true as well –when to adjacent angles add up to 180o, they form a line.  In the next section, we’ll show you different problems involving linear pairs of angles.

### Problem 3

Suppose that the line PQ is divided by the ray, OC. The ray divides the linear angle into two equal adjacent angles. Show that the angles formed are right angles.

After sketching the illustration similar to the one we’ve shown above, you’ll see that the ray, OC, divides the linear angle in half and reflects into L-shaped angles similar to right angles. But of course, we want to make sure that the linear pair of angles, \angle POC and \angle COQ do form angles each measuring 90o

\begin{aligned}\angle POC &= \angle COQ \end{aligned}

We know that the two angles are equal and that they are supplementary, so we can set up the equation shown below.

\begin{aligned}\angle POC +\angle COQ &= \angle POQ\\ 2\angle POC &= 180\degree\\\angle POC &= 90 \degree\\\angle COQ &= 90\degree\end{aligned}

Since both angles measure 90o, we can conclude that when a ray divides a line equally, the angles will always be right angles or measuring 90o.

### Problem 4

Two angles form a linear pair of angles and the ratio of their angle measures is 5:7. What are the measures of the two angles?

Constructing a sketch illustrating the problem helps us figure out what needs to be done. Since two angles form a linear pair of angles, we construct a ray dividing the line into a pair of angles. The ratio of these angles is 5:7, so if we let x be the common factor shared by the angles, the two angles can be represented as 5x and 7x.

Recall that linear pairs of angles are always supplementary, so equate the sum of the angles to 180o then solve for x.

\begin{aligned}(5x)\degree + (7x)\degree &= 180\degree\\ 12x &= 180\\x &= \dfrac{180}{12}\\&= 15\end{aligned}

To find the measures of the angles, simply multiply 15 to each of the values from the given ratio.

\begin{aligned}5:7&= 15 \times 5 : 15 \times 7\\&= 75\degree: 105\degree\end{aligned}

This means that the angle measures are 75\degree and 105\degree.

### Problem 5

Suppose that \angle POC and \angle COQ are linear pair of angles sharing a common point O and a line segment of \overline{OC}. If the difference between the two angles is 40o, what are the two angles’ measures if \angle POC is the larger angle?

Sketch an illustration to help you when solving the problem. We can let x represent the smaller angle, \angle COQ. Since the difference between the two angles is 40o, the \angle POC is 40o larger. This means that \angle POC = (x + 40)\degree.

Always remember that linear pair of angles will always add up to 180o. Add the two angles measures’ expression in terms of x then equate the sum to 180o

\begin{aligned}\angle POC + \angle COQ &= 180\degree\\(x +40)\degree + x\degree &= 180\degree\\2x+ 40&= 180\\2x&= 140\\x&= 70\end{aligned}

Use the value of x to find measures of \angle POC and \angle COQ.

\begin{aligned}\angle COQ &= x\degree\\&= 70\degree\\\\\angle POC &=  (x + 40)\degree\\&=110\degree \end{aligned}

Hence, the linear pairs of angles’ measures are: \angle POC = 110\degree and \angle COQ = 70\degree.

These examples showcase just three of the many problems that are opened to us once we learn about linear pairs of angles. The problems may be simpler or may even be more complex than the ones we’ve shown, but what’s important is that you feel confident in applying the definition of linear pair of angles to solve these problems!