(i) they have a common vertex, (ii) they have a common arm and (iii) the other arms of the two angles lie on opposite sides of the common arm.

The given figure below shows a pair of adjacent angles:

(i) they have a common vertex (O). (ii) they have a common arm (OB) and(iii) the other arms OA and OC of the two angles are on opposite sides of the common arm OB.

Explanation:

∠AOB and ∠BOC have the common vertex O. Also they have a common arm OB and their other arms OA and OC lie on the opposite sides of the common arm OB. Therefore, ∠AOB and ∠BOC are adjacent angles.

∠AOC and ∠AOB are not adjacent angles, because their other arms OC and OB are not on the opposite sides of the common arm OA.

Worked-out problems on adjacent angles:

Are the following angles adjacent? Give reasons.

Solution:

(a) ∠1 and ∠2 are not adjacent because they do not have a common arm.

(b) ∠1 and ∠2 are not adjacent because their interiors overlap.

(c) ∠1 and ∠2 are not adjacent because they do not have a common vertex.

(d) ∠1 and ∠2 are adjacent because they have a common arm, a common vertex and interiors do not overlap.