Linear Pair of Angles

When two straight lines intersect at a point, then the two adjascent angles formed are called a Linear Pair of Angles. Two angles of a linear pair always add to 180°

Linear pair of angles

In diagram P,
∠1 and ∠2 are a linear pair of angles, so ∠1 + ∠2 = 180°
∠2 and ∠3 are a linear pair of angles, so ∠2 + ∠3 = 180°
∠3 and ∠4 are a linear pair of angles, so ∠3 + ∠4 = 180°
∠4 and ∠1 are a linear pair of angles, so ∠4 + ∠1 = 180°

In diagram Q, line OC stands on line AB.
∠AOC and ∠BOC are linear pair of angles, so ∠AOC + ∠BOC = 180°

The two angles in a linear pair are also called supplementary angles.

Opposite angles

∠1 and ∠3 are opposite angles.
∠2 and ∠4 are opposite angles.
Opposite angles are always equal.
∴ ∠1 = ∠3 and ∠2 = ∠4

If one of a linear pair of angles is given, we can find the other angle by deducting it from 180°.

Given 60° as one of a linear pair of angles, the other angle is (180 - 60)° = 120°
Given 90° as one of a linear pair of angles, the other angle is (180 - 90)° = 90°
Given θ° as one of a linear pair of angles, the other angle is (180 - θ)°

When a straight line stands on another straight line, then the adjacent angles form a linear pair of angles.

If two adjascent angles form a linear pair, then the uncommon sides of both the angles form a straight line. In diagram Q, the common arm is OC, uncommon arms are AO and OB. ∴ AO and OB form a straight line.

In the following diagram, with the help of theory of linear pair of angles prove that
∠AOB + ∠BOC + ∠COD + ∠DOE + ∠EOF + ∠FOA = 360°

--- x ---