To determine the measure of angle \( \angle BAC \), we first note that if ray \( AC \) bisects angle \( \angle BAD \), it divides \( \angle BAD \) into two equal parts. The measure of angle \( \angle BAD \) is given as 46 degrees.
Since \( AC \) bisects \( \angle BAD \), we can find the measure of \( \angle BAC \) as follows:
\[ m\angle BAC = \frac{m\angle BAD}{2} = \frac{46^\circ}{2} = 23^\circ \]
Thus, \( m\angle BAC = 23^\circ \).
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