If ray $\overrightarrow{\mathrm{AB}}$ is rotated counterclockwise about the origin to coincide with ray $\overrightarrow{\mathrm{AC}}$ in standard position, and the measure of angle $\angle BAC$ is $120$ degrees, how many degrees has $\mathrm{ABC}$ been rotated counterclockwise about the origin?

• A. $240$ degrees
• B. $60$ degrees
• C. $120$ degrees
• D. $90$ degrees

Answer: (C)

Verified step by step guidance

1. Identify the given information: ray \( \overrightarrow{AB} \) is rotated counterclockwise about the origin to coincide with ray \( \overrightarrow{AC} \), and the measure of angle \( \angle BAC \) is 120 degrees.
2. Recall that the angle \( \angle BAC \) represents the amount of rotation from ray \( \overrightarrow{AB} \) to ray \( \overrightarrow{AC} \) when rotating about point A (the origin in this case).
3. Since the rotation is counterclockwise and the angle between the two rays is given as 120 degrees, the measure of the rotation is exactly the measure of \( \angle BAC \).
4. Therefore, the number of degrees that ray \( \overrightarrow{AB} \) has been rotated counterclockwise about the origin to coincide with ray \( \overrightarrow{AC} \) is 120 degrees.
5. Conclude that the rotation angle \( \angle BAC = 120^\circ \) is the answer to the problem.