Adjacent Angles with Examples and FAQs

Adjacent Angles

Let us first understand what an angle is.

An angle is formed by the intersection of two rays at a point. The point where the rays meet is the vertex of that angle and the two rays are called the arms of the angle.

Adjacent Angles are the angles that share a common vertex and a common arm. 

Adjacent Angles

The above diagram depicts angles that are adjacent.

∠COB and ∠BOA are adjacent to each other, their common vertex is O and their common arm is OB.

Properties

Some Properties of two adjacent angles are:

• They share a common vertex.
• They share a common arm and the other two arms, one of each, are uncommon.
• They do not overlap each other.
• No common interior point.
• They can be either complementary or supplementary, depending on the sum of the measurements of the angles. 

Complementary Adjacent Angles

Two angles are complementary if the sum of their angle measurement is 90°.

Complementary Adjacent Angles

In the figure above ∠AOB and ∠BOC are adjacent and complementary angles.

Supplementary or Linear Adjacent Angles 

Two angles are supplementary if the sum of their angle measurement is 180°.

Supplementary Adjacent Angles

In the figure above ∠CAD and ∠DAE are adjacent and supplementary(linear) angles.

Example: From the figure determine the angles that are adjacent. What is the angle measurement of ∠BOC if ∠AOD = 140°?

Solution: 

The two pairs of angles that are adjacent are:

∠AOB and ∠BOC; ∠BOC and ∠COD.

Given ∠AOD = 140°, ∠AOC = 45° , ∠COD = 45°.

∠AOD = ∠AOC + ∠BOC + ∠COD

⇒ 140° = 45° + ∠BOC  + 45°

∴ ∠BOC = 140° – ( 45° + 45° )

⇒ ∠BOC = 140° – 90° = 50°.

Hence, the measurement of ∠BOC = 50°.