Answer: \(\angle BOC = 110^\circ\)
Explanation:
This problem involves the properties of circles, inscribed angles, and central angles. The key concept here is that the measure of an inscribed angle is half the measure of the intercepted arc, and the measure of a central angle is equal to the measure of the intercepted arc. The given angle of 55° at point A is an inscribed angle, which intercepts an arc of the circle. Using this, we can find the measure of the arc, and then determine \(\angle BOC\), which is a central angle.
Steps:
- Identify the given information:
- \(\angle BAC = 55^\circ\) (since the angle at A is marked as 55°)
- \(O\) is the center of the circle
- \(\angle BOC\) is a central angle subtending the same arc as \(\angle BAC\)
- Recall the inscribed angle theorem:
- An inscribed angle measures half the measure of its intercepted arc.
- Therefore, if \(\angle BAC = 55^\circ\), then the intercepted arc \(BC\) measures:
\[
\text{Arc } BC = 2 \times 55^\circ = 110^\circ
\]
- Determine the measure of \(\angle BOC\):
- \(\angle BOC\) is a central angle that intercepts the same arc \(BC\).
- The measure of a central angle is equal to the measure of its intercepted arc.
- Hence:
\[
\angle BOC = \text{Arc } BC = 110^\circ
\]
Therefore, the measure of \(\angle BOC\) is \(\boxed{110^\circ}\).