If the angles of a triangle are in the ratio `2:3:4` . determine three angles.

Answer

To determine the three angles of a triangle given that they 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 \( k \).
- Let angle A = \( 2k \)
- Let angle B = \( 3k \)
- Let angle C = \( 4k \)

2. **Use the Triangle Angle Sum Property**:
The sum of the angles in any triangle is always 180 degrees. Therefore, we can set up the equation:
\[ 2k + 3k + 4k = 180^\circ \]

3. **Combine Like Terms**:
Combine the terms on the left side of the equation:
\[ (2 + 3 + 4)k = 180^\circ \]
This simplifies to:
\[ 9k = 180^\circ \]

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

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

6. **Conclusion**:
The three 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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