To find the measure of $\angle ACB$, we can use the Inscribed Angle Theorem. This theorem states that an inscribed angle is half of the central angle that subtends the same arc.

Given: $\angle AOB = 150^\circ$

Using the Inscribed Angle Theorem: $\angle ACB = \frac{1}{2} \times \angle AOB = \frac{1}{2} \times 150^\circ = 75^\circ$

So, the answer is $75^\circ$.