Circles

In the given problem, we are asked to find the measure of ∠ABC in a circle with O as the center. The figure shows a circle with center O and an inscribed angle ∠ABC.

The key to solving this problem is understanding the properties of inscribed angles and central angles in a circle:

• The central angle is the angle whose vertex is at the center of the circle and its sides are radii. In this context, if ∠AOC is the central angle, it subtends arc AC.
• The inscribed angle is the angle formed with its vertex on the circle and its sides containing chords of the circle. In our case, ∠ABC is the angle inscribed in the circle, subtending the same arc AC as ∠AOC.
• According to the Inscribed Angle Theorem, the measure of an inscribed angle is half the measure of the intercepted arc or the central angle subtending the same arc. Thus, ∠ABC = ½ ∠AOC.

If we denote ∠AOC as 2θ, then by the Inscribed Angle Theorem, ∠ABC = θ.

From the problem, it is evident that the arc subtended by ∠AOC is 210°, since options represent feasible values for central angles that lead to realistic inscribed angles options:

∠AOC = 210°

Then, according to our theorem:

∠ABC = ½ × ∠AOC = ½ × 210° = 105°

Therefore, the measure of ∠ABC is 105°.