In the given figure, straight lines AB and CD intersect at O. If `angleAOC=phi`, `angleBOC=theta` and `theta=3 phi`, then `phi=?`

Options

A. `30^(@)`
B. `40^(@)`
C. `45^(@)`
D. `60^(@)`

Answer

To solve the problem, we need to find the value of \(\phi\) given that \(\theta = 3\phi\) and the angles formed by the intersection of the lines.

### Step-by-Step Solution:

1. **Identify the Angles**:
- We have two angles formed by the intersection of lines AB and CD at point O:
- \( \angle AOC = \phi \)
- \( \angle BOC = \theta \)
- According to the problem, we know that \( \theta = 3\phi \).

2. **Use the Straight Line Property**:
- Since AB is a straight line, the sum of the angles on a straight line is \(180^\circ\).
- Therefore, we can write the equation:
\[ \angle AOC + \angle BOC = 180^\circ \]
- Substituting the values of the angles:
\[ \phi + \theta = 180^\circ \]

3. **Substitute \(\theta\)**:
- We know from the problem that \(\theta = 3\phi\).
- Substitute this into the equation:
\[ \phi + 3\phi = 180^\circ \]

4. **Combine Like Terms**:
- Combine the terms on the left side:
\[ 4\phi = 180^\circ \]

5. **Solve for \(\phi\)**:
- Divide both sides by 4 to isolate \(\phi\):
\[ \phi = \frac{180^\circ}{4} = 45^\circ \]

### Final Answer:
\[ \phi = 45^\circ \]