Let’s write the measurement of ∠BOD, ∠BOC and ∠AOC.

In the above given figure, let us understand the pairs of vertically opposite angle.

∠AOD and ∠COB are lying opposite to each other [in common language lying face to face to each other].

So, first pair of vertically opposite angles is ∠AOD and ∠COB.

∠AOC and ∠DOB are also lying opposite to each other.

Hence ∠AOC and ∠DOB forms our second pair of vertically opposite angles.

Vertically opposite angles are always equal.

Hence ∠AOD must be equal to ∠COB and ∠AOC must be equal to ∠DOB.

Also, we know that sum all vertically opposite angles is 360°.

∠AOD + ∠COB + ∠AOC + ∠DOB = 360°.

Now in the question it is given that,

∠AOD = 120°

So ∠COB must be equal to 120° [as ∠COB = ∠AOD……… vertically opposite angles]

Let ∠DOB = ∠AOC = x [as ∠AOC = ∠DOB……… vertically opposite angles]

So, 120° + 120° + x + x = 360°

240° + 2x = 360°

2x = 360 – 240

2x = 120

x = 120 / 2

x = 60°

So ∠AOC = ∠DOB = 60°.