If A, B, C are three points on a circle with centre 'O' such that ∠AOB = 90° and ∠BOC = 120°, then ∠ABC =

Question

Accepted Answer

To determine the measure of ∠ABC, we need to analyze the given information about the circle and the angles.

1. **Understanding the Given Angles:**
   - We have a circle with center 'O'.
   - ∠AOB = 90° and ∠BOC = 120°.

2. **Calculating ∠AOC:**
   - Since ∠AOB and ∠BOC are angles subtended by arcs AB and BC at the center O, we can find ∠AOC by adding these two angles:
     $$
     ∠AOC = ∠AOB + ∠BOC = 90° + 120° = 210°
     $$

3. **Using the Angle at the Center and the Angle at the Circumference:**
   - The angle subtended by an arc at the center of the circle is twice the angle subtended by the same arc at any point on the remaining part of the circle.
   - Therefore, the angle ∠ABC, which is subtended by arc AC at the circumference, is half of ∠AOC:
     $$
     ∠ABC = \frac{1}{2} × ∠AOC = \frac{1}{2} × 210° = 105°
     $$

4. **Re-evaluating the Calculation:**
   - Upon reviewing the calculation, it seems there was an oversight. The correct calculation should be:
     - Since ∠AOC = 210°, the angle subtended by arc AC at the circumference is:
     $$
     ∠ABC = \frac{1}{2} × (360° - ∠AOC) = \frac{1}{2} × (360° - 210°) = \frac{1}{2} × 150° = 75°
     $$

5. **Conclusion:**
   - Therefore, the correct measure of ∠ABC is 75°, which corresponds to Option 2.

6. **Explanation of Incorrect Options:**
   - **Option 1 (60°):** This would imply a different calculation or misunderstanding of the relationship between the angles at the center and the circumference.
   - **Option 3 (90°):** This would suggest that ∠AOC is 180°, which contradicts the given angles.
   - **Option 4 (135°):** This would imply an incorrect calculation of the angle subtended at the circumference.

Thus, the correct answer is indeed Option 2, 75°, as it accurately reflects the relationship between the angles at the center and the circumference of the circle.