Linear Pair

â€˜Linearâ€™ means â€˜arranged in a straight line.â€™ 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 180Â°.)

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

In the figure below, âˆ MOP and âˆ PON, âˆ PON and âˆ NOQ, âˆ POM and âˆ MOQ, and âˆ MOQ and âˆ 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, âˆ WYX and âˆ 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), âˆ XOZ and âˆ 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, âˆ XOZ and âˆ YOZ are congruent angles (mâˆ XOZ = mâˆ YOZ).

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

â‡’ âˆ XOZ + âˆ XOZ = 180Â°

â‡’ âˆ XOZ = âˆ YOZ = 90Â°

Thus, the lines are perpendicular (OZ âŠ¥ XY).

Solved Examples

Find the value of each angle.

Solution:

As we observe, âˆ MON and âˆ MOP form a linear pair.
âˆ MON + âˆ MOP = 180Â°
â‡’ (x + 1)Â° + (2x – 70)Â° = 180Â°
â‡’ (x + 2x)Â° = 180Â° + 70Â° – 1Â°
â‡’ 3xÂ° = 249Â°
â‡’ xÂ° = 83Â°
Thus, âˆ MON = (x + 1)Â° = (83 + 1)Â° = 84Â° and âˆ MOP = (2x – 70)Â° = (2 Ã— 83Â° – 70Â°) = 96Â°

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

Solution:

âˆ ABC + âˆ ABD = 180Â°
â‡’ 150Â° + âˆ ABD = 180Â°
â‡’ âˆ ABD = 180Â° – 150Â° = 30Â°

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

Solution:

Here, option c) forms a pair of linear.