# Linear Pair

‘Linear’ means ‘arranged in a straight line.’ A linear pair of angles comprises a pair of angles formed by the intersection of two straight. Thus, two angles are said to form a linear pair if they are adjacent (next to each other) and supplementary (measures add up to 180°.)

In the below figure, ∠ABC and ∠CBD form a linear pair of angles.

In the figure below, ∠MOP and ∠PON, ∠PON and ∠NOQ, ∠POM and ∠MOQ, and ∠MOQ and ∠QON are the linear pairs of angles.

Thus, the linear pairs share a common arm and a common vertex, and their non-common arms are on opposite sides.

However, all adjacent angles do not form linear pairs.

Here, ∠WYX and ∠WYZ are adjacent angles, but they are not a linear pair.

## Postulate

It states that if two angles form a linear pair, they are supplementary.

However, the converse of the above postulate is not true, which means if two angles are supplementary, they are not always a linear pair of angles.

From (a) and (b), ∠XOZ and ∠PQR are supplementary angles but not linear pairs.

Thus, all non-adjacent supplementary angles are not linear pairs.

## Axioms

If a ray stands on a line, the adjacent angles are supplementary.

The converse of the above axiom is also true, which states that if two angles form a linear pair, the non-common arms of both the adjacent angles form a straight line.

## Perpendicular Theorem

The linear pair perpendicular theorem states that if two angles of a linear pair are congruent, the lines are perpendicular.

Let us verify this with the following figure, as shown:

Here, ∠XOZ and ∠YOZ are congruent angles (m∠XOZ = m∠YOZ).

Since they form a linear pair, we have ∠XOZ + ∠YOZ = 180°

⇒ ∠XOZ + ∠XOZ = 180°

⇒ ∠XOZ = ∠YOZ = 90°

Thus, the lines are perpendicular (OZ ⊥ XY).

## Solved Examples

Find the value of each angle.

Solution:

As we observe, ∠MON and ∠MOP form a linear pair.
∠MON + ∠MOP = 180°
⇒ (x + 1)° + (2x – 70)° = 180°
⇒ (x + 2x)° = 180° + 70° – 1°
⇒ 3x° = 249°
⇒ x° = 83°
Thus, ∠MON = (x + 1)° = (83 + 1)° = 84° and ∠MOP = (2x – 70)° = (2 × 83° – 70°) = 96°

Find the value of an angle that forms a linear pair with ∠ABC = 150°.

Solution:

∠ABC + ∠ABD = 180°
⇒ 150° + ∠ABD = 180°
⇒ ∠ABD = 180° – 150° = 30°

Which angles are linear pairs? Check all that apply.
a) ∠AOC and ∠AOB
b) ∠AOE and ∠EOD
c) ∠COD and ∠DOE
d) ∠COD and ∠AOB

Solution:

Here, option c) forms a pair of linear.