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 find the angle \( \angle ABC \) given the angles \( \angle AOB = 90^\circ \) and \( \angle BOC = 120^\circ \), we can follow these steps:

### Step 1: Understand the relationship between angles at the center and angles at the circumference
In a circle, the angle subtended at the center by an arc is twice the angle subtended at any point on the circumference by the same arc. This means if we have \( \angle AOC \) at the center, then \( \angle ABC \) at the circumference is given by the formula:
\[ \angle ABC = \frac{1}{2} \angle AOC \]

### Step 2: Calculate \( \angle AOC \)
We know that the sum of angles around point \( O \) is \( 360^\circ \). Therefore, we can find \( \angle AOC \) using the following equation:
\[ \angle AOB + \angle BOC + \angle AOC = 360^\circ \]
Substituting the known values:
\[ 90^\circ + 120^\circ + \angle AOC = 360^\circ \]
This simplifies to:
\[ 210^\circ + \angle AOC = 360^\circ \]
Now, isolate \( \angle AOC \):
\[ \angle AOC = 360^\circ - 210^\circ = 150^\circ \]

### Step 3: Find \( \angle ABC \)
Now that we have \( \angle AOC = 150^\circ \), we can find \( \angle ABC \):
\[ \angle ABC = \frac{1}{2} \angle AOC = \frac{1}{2} \times 150^\circ = 75^\circ \]

### Conclusion
Thus, the value of \( \angle ABC \) is \( 75^\circ \).