To find the measure of angle AOC, we need to understand the property of an angle bisector. An angle bisector divides an angle into two equal parts.
Given that the measure of angle AOB is 84 degrees and ray OC is the angle bisector, we can conclude that it divides angle AOB into two equal angles:
\[ \text{Measure of angle AOC} = \frac{\text{Measure of angle AOB}}{2} = \frac{84 \text{ degrees}}{2} = 42 \text{ degrees} \]
Thus, the measure of angle AOC is 42 degrees, which corresponds to option a. 42.
This is the correct answer because an angle bisector by definition divides the angle into two equal halves, and since we know the total angle (84 degrees), dividing it by two gives us the measure of each of the angles formed by the bisector.