Linear Pair of Angles
- What is a Linear Pair of Angles?
- Some Examples of Linear Pair of Angles
- Properties of Linear Pair of Angles
- Postulate of Linear Pair of Angles
- Difference between Linear Pair of Angles and Supplementary Angles
- Linear Pair Perpendicular Theorem
- Solved Examples
- Practice Problems
- Frequently Asked Questions
What is a Linear Pair of Angles?
A linear pair of angles are a pair of angles that are adjacent to each other and they are formed on a straight line.
In the diagram shown above, it can be seen that there are two angles `∠POA` and `∠POB`. These angles are adjacent to each other and are formed on the straight line `AB`.
So, `∠POA` and `∠POB` form a linear pair of angles.
It can be seen that the linear pair of angles are formed on a straight line. The angle formed on a straight line is called a straight angle and measures `180°`.
In the above diagram, `∠POA` and `∠POB` make a straight angle, meaning that the sum of `∠POA` and `∠POB` is `180°`. Hence, it can be said that linear pair of angles add up to `180°`.
Some Examples of Linear Pair of Angles
In the above diagram, some of the linear pairs of angles are as follows:
`∠1` and `∠2`
`∠3` and `∠4`
`∠5` and `∠7`
`∠6` and `∠8`
There are more linear pairs of angles in the diagram above. Try to identify them.
In the next example(figure given below), it can be seen that `∠AOC` and `∠BOC` form a linear pair of angles. `∠AOC` and `∠BOC` measure `120°` and `60°` respectively. When `∠AOC` and `∠BOC` are added, it becomes equal to `180°`.
`∠AOC + ∠BOC = 180°`
`120° + 60° = 180°`
Properties of Linear Pair of Angles
Some of the properties of the linear pair of angles are as follows:
`1`. The two angles that form a linear pair add up to `180°`(supplementary angles).
`2`. The two angles in a linear pair are adjacent to each other.
`3`. The angles that make a linear pair form a straight line.
`4`. There is a common vertex and a common arm between a linear pair of angles.
`5`. The two angles in a linear pair together form a straight angle.
Postulate of Linear Pair of Angles
According to the linear pair of angles postulate the two angles that form a linear pair of angles are supplementary.
But, the opposite is not always true which means that if two angles are supplementary then the two angles may not be a linear pair of angles. In the image below `\angle M` and `\angle N` are supplementary but they do not form a linear pair of angles.
Difference between Linear Pair of Angles and Supplementary Angles
Linear Pair of Angles | Supplementary Angles |
| A linear pair of angles are adjacent to each other. | If two angles are supplementary, they may not be adjacent to each other. |
| If two angles form a linear pair, they add up to `180°`. | Supplementary angles need not necessarily be a linear pair of angles. |
 |  |
Linear Pair Perpendicular Theorem
According to the linear pair perpendicular theorem if two straight lines meet/intersect at a point and a linear pair of equal angles is formed, then the two lines are perpendicular.
In the diagram above `∠1 = ∠2` and lines `L1` and `L2` intersect each other. So, as per the linear pair perpendicular theorem, it can be said that line `L1` is perpendicular to `L2`.
Solved Examples
Example `1`: In the diagram given below, identify the linear pair of angles.
Solution:
There are two lines `AB` and `XY`. On line `XY` there are two angles: `∠XCA` and `∠YCA`. These two form a linear pair of angles, `AC` being the common arm. Similarly, all the linear pairs of angles in the above diagram are
- `∠XCA` and `∠YCA`
- `∠XCA` and `∠XCB`
- `∠XCB` and `∠YCB`
- `∠YCA` and `∠YCB`
Example `2`. Two angles form a linear pair, one of the angles is `65°`. Find the measure of the other angle.
Solution:
Measure of the first angle of the linear pair `= 65°`.
Both the angles form a linear pair, so they add up to `180°`.
Therefore, the measure of the other angle `= 180° - 65° = 115°`
Example `3`. Two angles form a linear pair such that the first angle is twice the other, find the measure of both angles.
Solution:
Let one angle be `x°` and the other angle be `2x°`.
Both angles form a linear pair.
Therefore, `x°+2x°=180°`.
`3x° = 180°`
`x° = 60°`
So, the angles are `60°` and `120°`.
Practice Problems
Q`1`. Do angles `113°` and `37°` form a linear pair? Explain your answer.
- Yes, because their non-common sides form a straight line.
- No, because they do not share a common side.
- Yes, because they share a common side and their non-common sides form a straight line.
- No, because the sum of their measures is not \(180^\circ\).
Answer: d
Q`2`. One of the angles of a linear pair of angles is 101°, find the other angle.
- 79°
- 101°
- 179°
- 280°
Answer: a
Q`3`. Identify the linear pair of angles in the diagram given below:
- `\angle 1` and `\angle 2`
- `\angle 2` and `\angle 3`
- `\angle 3` and `\angle 4`
- `\angle 4` and `\angle 5`
Answer: d
Frequently Asked Questions
Q`1`. What is a linear pair of angles?
Answer: A linear pair of angles consists of two adjacent angles whose non-common sides are opposite rays, forming a straight line.
Q`2`. How do you identify a linear pair of angles?
Answer: To identify a linear pair, check if two angles are adjacent (share a common side) and if their non-common sides together form a straight line.
Q`3`: What is the sum of the measures of angles in a linear pair?
Answer: The sum of the measures of angles in a linear pair is always \(180^\circ\).
Q`4`. If one angle in a linear pair is given, how do you find the measure of the other angle?
Answer: Subtract the given angle's measure from \(180^\circ\) to find the measure of the other angle in the linear pair.
Q`5`: Are all adjacent angles linear pairs?
Answer: No, for adjacent angles to form a linear pair, their non-common sides must be opposite rays, creating a straight line.
Q`6`: If the sum of two angles is \(180^\circ\), are they always a linear pair?
Answer: No, the sum of \(180^\circ\) only guarantees supplementary angles. For a linear pair, they must be adjacent and have non-common sides forming a straight line.