If A, B, C are three points on a circle with centre 'O' such that `angleAOB=90^@` and `angleBOC=120^@`, then `angleABC=`

Options

A. `60^@`
B. `75^@`
C. `90^@`
D. `135^@`

Answer

To solve the problem, we need to find the angle \( \angle ABC \) given the angles \( \angle AOB = 90^\circ \) and \( \angle BOC = 120^\circ \).

### Step-by-Step Solution:

1. **Identify the Angles at the Center**:
- We have \( \angle AOB = 90^\circ \) and \( \angle BOC = 120^\circ \).

2. **Calculate the Angle AOC**:
- The angle \( \angle AOC \) can be found by adding the angles \( \angle AOB \) and \( \angle BOC \):
\[ \angle AOC = \angle AOB + \angle BOC = 90^\circ + 120^\circ = 210^\circ \]

3. **Use the Inscribed Angle Theorem**:
- According to the inscribed angle theorem, the angle \( \angle ABC \) is half of the angle \( \angle AOC \):
\[ \angle ABC = \frac{1}{2} \angle AOC = \frac{1}{2} \times 210^\circ = 105^\circ \]

4. **Final Answer**:
- Therefore, the angle \( \angle ABC \) is:
\[ \angle ABC = 105^\circ \]