In a `triangle`ABC, if 2 `/_A` = 3 `/_`B = 6`/_`C, then `/_B` = ? `

Options

A. `30^(@)`
B. `90^(@)`
C. `60^(@)`
D. `45^(@)`

Answer

To solve the problem, we need to find the value of angle B in triangle ABC given the relationships between the angles.

### Step-by-Step Solution:

1. **Set Up the Relationships**:
We are given that:
\[ 2 \angle A = 3 \angle B = 6 \angle C \]
Let's denote \( \angle B \) as \( B \). Then we can express \( \angle A \) and \( \angle C \) in terms of \( B \):
\[ \angle A = \frac{3}{2}B \]
\[ \angle C = \frac{1}{2}B \]

2. **Use the Triangle Sum Property**:
The sum of the angles in a triangle is always \( 180^\circ \):
\[ \angle A + \angle B + \angle C = 180^\circ \]
Substituting the expressions for \( \angle A \) and \( \angle C \):
\[ \frac{3}{2}B + B + \frac{1}{2}B = 180^\circ \]

3. **Combine Like Terms**:
Combine the terms on the left side:
\[ \left(\frac{3}{2}B + B + \frac{1}{2}B\right) = \left(\frac{3}{2}B + \frac{2}{2}B + \frac{1}{2}B\right) = \frac{6}{2}B = 3B \]
So we have:
\[ 3B = 180^\circ \]

4. **Solve for Angle B**:
Divide both sides by 3:
\[ B = \frac{180^\circ}{3} = 60^\circ \]

5. **Conclusion**:
Therefore, the value of angle B is:
\[ \angle B = 60^\circ \]