In the following figure, O is the center of the circle, and the line segments AB and AC intersect the circle at D and E, respectively. If m( ACD = 30 degrees and angle BEC = 50 degrees, then which one

Here are the solutions to the questions from the exam:

9. In the following figure, O is the center of the circle, and the line segments AB and AC intersect the circle at D and E, respectively. If $m(\angle ACD) = 30^\circ$ and $m(\angle BEC) = 50^\circ$, then which one of the following is equal to $m(\angle BAC)$?

Step 1: Identify the given angles and their relationship to arcs.
The angle $m(\angle ACD) = 30^\circ$ is an inscribed angle subtending arc AD. Therefore, the measure of arc AD is twice the angle:

$$m(\text{arc AD}) = 2 \times m(\angle ACD) = 2 \times 30^\circ = 60^\circ$$

The angle $m(\angle BEC) = 50^\circ$ is an inscribed angle subtending arc BC. Therefore