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