Circle Geometry: Calculating Inscribed Angle $\angle ABC$

This problem involves finding an angle within a circle given other angles related to the center and points on the circumference. We are given a circle with center O and three points A, B, and C on its circumference. We know the central angles subtended by the arcs AB and BC: $\angle AOB = 90^\circ$ and $\angle BOC = 120^\circ$. We need to find the measure of the inscribed angle $\angle ABC$.

Understanding Key Concepts

To solve this, we use a fundamental theorem in circle geometry:

• The angle subtended by an arc at the center of the circle is double the angle subtended by the same arc at any point on the remaining part of the circle.
• Mathematically, if O is the center and A, B, C are points on the circle, the angle subtended by arc AC at the center is $\angle AOC$. The angle subtended by the same arc AC at a point B on the circumference is $\angle ABC$. The relationship is: $\angle ABC = \frac{1}{2} \angle AOC$ (assuming B is on the major arc AC).

Step-by-Step Calculation

1. Identify the arcs: We are given angles related to arcs AB and BC.

• The central angle $\angle AOB = 90^\circ$ corresponds to arc AB.
• The central angle $\angle BOC = 120^\circ$ corresponds to arc BC.

2. Determine the arrangement of points: Assuming the points A, B, and C are arranged consecutively around the circle, we can find the measure of the arc AC that passes through B.

• Measure of arc ABC = Measure of arc AB + Measure of arc BC
• Measure of arc ABC = $\angle AOB + \angle BOC = 90^\circ + 120^\circ = 210^\circ$.

3. Find the measure of the arc AC not containing B: The angle $\angle ABC$ subtends the arc AC that *does not* contain the point B. The measure of the entire circle is $360^\circ$.

• Measure of arc AC (minor) = $360^\circ$ - Measure of arc ABC
• Measure of arc AC (minor) = $360^\circ - 210^\circ = 150^\circ$.

4. Calculate the central angle $\angle AOC$: The central angle subtending the minor arc AC is $\angle AOC$. Its measure is equal to the measure of the minor arc AC.

• $\angle AOC = 150^\circ$.

5. Calculate the inscribed angle $\angle ABC$: Now, we apply the theorem relating the central angle and the inscribed angle subtending the same arc.

• $\angle ABC = \frac{1}{2} \times \angle AOC$
• $\angle ABC = \frac{1}{2} \times 150^\circ$
• $\angle ABC = 75^\circ$.

Conclusion

Therefore, the measure of angle $\angle ABC$ is $75^\circ$. This corresponds to the second option provided.