To find the measurement of angle \( \angle AOC \), we use the property of angle bisectors. Since ray \( OC \) is the angle bisector of \( \angle AOB \), it divides the angle \( \angle AOB \) into two equal parts.
Given that:
\[ m\angle AOB = 74^\circ \]
We find \( m\angle AOC \) as follows:
\[ m\angle AOC = \frac{m\angle AOB}{2} = \frac{74^\circ}{2} = 37^\circ \]
Therefore, the measurement of \( \angle AOC \) is \( 37 \) degrees.
Thus, \( m\angle AOC \) is 37 degrees.