**Linear pair of angles** are formed when two lines intersect each other at a single point. The angles are said to be linear if they are adjacent to each other after the intersection of the two lines. The sum of angles of a linear pair is always equal to 180Â°. Such angles are also known as supplementary angles. The adjacent angles are the angles which have a common vertex. Hence, here as well the linear angles have a common vertex. Also, there will be a common arm which represents both the angles. A real-life example of a linear pair is a ladder which is placed against a wall, forms linear angles at the ground.

Linearity represents one which is straight. So here also, linear angles are the one which are formed into a straight line. The pair of adjacent angles here are constructed on a line segment,Â but not all adjacent angles are linear. Hence, we can also say, that linear pair of angles are the adjacent angles whose non-common arms are basically opposite rays.

## Explanation for Linear Pair of Angles

When the angle between two lines is 180Â°, they form a straight angle. A straight angle is just another way to represent a straight line. A straight line can be visualized as a circle with an infinite radius. A line segment is any portion of a line which has two endpoints. Also, a portion of any line with only one endpoint is called a ray. A line segment with A and B as two endpoints is represented as \(\overline{AB}\). The figure shown below represents a line segment AB and the two arrows at the end indicate a line.

If a point *O* is taken anywhere on the line segment AB as shown, then the angle between the two line segments *AO* and *OB* is a straight angle i.e. 180Â°.

Consider a ray \(\overrightarrow{OP}\) stand on the line segment \overline{AB} as shown:

The angles which are formed at *O* are âˆ *POB* and âˆ *POA*. It is known that the angle between the two line segments AO and OB is 180Â°. therefore, the angles âˆ *POB* and âˆ *POA* add up to 180Â°.

Thus, âˆ *POB* + âˆ *POA* = âˆ *AOB* = 180Â°

âˆ POB and âˆ POA are adjacent to each other and when the sum of adjacent angles is 180Â° then such angles form linear pair of angles.

The above discussion can be stated as an axiom.

**Also, read:**

### Axioms

**Axiom 1:** If a ray stands on a line then the adjacent angles form a linear pair of angles.

In the figure above, all the line segments pass through the point O as shown. As the ray OAÂ lies on the line segment CD, angles âˆ *AOD* and âˆ *AOC* form a linear pair. Similarly, âˆ *QOD* and âˆ *POD* form a linear pair and so on.

The converse of the stated axiom is also true, which can also be stated as the following axiom.

**Axiom 2:** If two angles form a linear pair, then uncommon arms of both the angles form a straight line.

Figure 3 Adjacent angles with different measures

In the figure shown above, only the last one represents a linear pair, as the sum of the adjacent angles is 180Â°. Therefore, *AB* represents a line. The other two pairs of angles are adjacent but they do not form a linear pair. They do not form a straight line.

The two axioms mentioned above form the Linear Pair Axioms and are very helpful in solving various mathematical problems.

### Example

**Suppose two anglesÂ âˆ AOC and âˆ BOC form a linear pair at point O in a line segment AB. If the difference between the two angles is 60. Then find both the angles.**

Solution: Given,Â âˆ AOC and âˆ BOC form a linear pair

So,Â âˆ AOC + âˆ BOC =180 ………(1)

Also given,

âˆ AOC – âˆ BOC = 60 ………(2)

Adding eq. 1 and 2, we get;

2âˆ AOC = 180 + 60 = 240

âˆ AOC = 240/2 = 120

Now putting the value ofÂ âˆ AOC in equation 1, we get;

âˆ BOC = 180 –Â âˆ AOC = 180 – 120

âˆ BOC = 60

There is a lot more to learn about lines and angles. To know more about properties of pair of angles, download BYJUâ€™s-The Learning App from Google Play Store.