Adjacent Angles

Two angles in the same plane and with a common vertex and a common arm are called adjacent angles.

Let us take example of the angles, shown in following figure:

∠ COB and ∠ BOA have a common vertex, i.e. O. Additionally, the ray OB is the common arm between these two angles.

Hence, ∠ COB and ∠ BOA are the pairs of adjacent angles.

Example: Find the pairs of adjacent angles in the following figure.

Solution: ∠ 1 and ∠ 2 are adjacent to each other
∠ 2 and ∠ 3 are adjacent to each other

Example: Find the pairs of adjacent angles in the following figure.

Solution: ∠ APD and ∠ DPC are adjacent to each other
∠ DPC and ∠ CPB are adjacent to each other

Note: ∠ APD and ∠ CPB are not adjacent to each other, because they don’t have a common arm in spite of having a common vertex.

Linear Pair of Angles

Two angles make a linear pair if their non-common arms are two opposite rays. In other words, if the non-common arms of a pair of adjacent angles are in a straight line, these angles make a linear pair.

Note: Two acute angles cannot make a linear pair because their sum will always be less than 180°. On the other hand, two right angles will always make a linear pair as their sum is equal to 180°. It can also be said that angles of the linear pair are always supplementary to each other.

Example: Find if following angles can make a linear pair.

Solution: 130° + 50° = 180°
Since the sum of these angles is equal to two right angles, so they can make a linear pair.

Example: Find if following angles can make a linear pair.

Solution: 110° + 70° = 180°
Since the sum of these angles is equal to two right angles, so they can make a linear pair.

Example: If following angles make a linear pair, find the value of q.

Solution: `(7q – 46)° + (3q + 6)° = 180°`

Or, `10q – 40 = 180°`

Or, `10q = 180°+ 40 = 220°`

Or, `q = 220° ÷ 10 = 22°`