Find the angles of a triangle which are in the ratio 2 :3 : 4.

Options

A. `20^(@), 60^(@)` and `90^(@)`
B. `40^(@), 60^(@)` and `80^(@)`
C. `30^(@), 50^(@)` and `80^(@)`
D. `10^(@), 60^(@)` and `90^(@)`

Answer

To find the angles of a triangle that are in the ratio 2:3:4, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Ratio**:
The angles of the triangle are in the ratio 2:3:4. This means we can express the angles in terms of a variable \( x \). Let:
- Angle A = \( 2x \)
- Angle B = \( 3x \)
- Angle C = \( 4x \)

2. **Use the Triangle Angle Sum Property**:
The sum of the angles in any triangle is always 180 degrees. Therefore, we can write the equation:
\[ \text{Angle A} + \text{Angle B} + \text{Angle C} = 180^\circ \]
Substituting the expressions for the angles:
\[ 2x + 3x + 4x = 180^\circ \]

3. **Combine Like Terms**:
Combine the terms on the left side:
\[ (2x + 3x + 4x) = 9x \]
So, we have:
\[ 9x = 180^\circ \]

4. **Solve for \( x \)**:
To find \( x \), divide both sides of the equation by 9:
\[ x = \frac{180^\circ}{9} = 20^\circ \]

5. **Calculate Each Angle**:
Now that we have the value of \( x \), we can find each angle:
- Angle A = \( 2x = 2 \times 20^\circ = 40^\circ \)
- Angle B = \( 3x = 3 \times 20^\circ = 60^\circ \)
- Angle C = \( 4x = 4 \times 20^\circ = 80^\circ \)

6. **Conclusion**:
The angles of the triangle are:
- Angle A = \( 40^\circ \)
- Angle B = \( 60^\circ \)
- Angle C = \( 80^\circ \)

### Final Answer:
The angles of the triangle are \( 40^\circ, 60^\circ, \) and \( 80^\circ \).

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