Find the measure of the angles of a triangle which has one of its exterior angles of `120^(@)` and the two interior opposite angles in the ratio 1:2.

Answer

To find the measure of the angles of a triangle with one of its exterior angles measuring 120 degrees and the two interior opposite angles in the ratio 1:2, follow these steps:

### Step-by-Step Solution:

1. **Identify the Exterior Angle**:
- Given that one of the exterior angles is 120 degrees.

2. **Set Up the Interior Angles**:
- Let the two interior angles be \(X\) and \(2X\) (since they are in the ratio 1:2).

3. **Use the Exterior Angle Property**:
- According to the property of triangles, the exterior angle is equal to the sum of the two opposite interior angles.
- Therefore, we can write the equation:
\[ X + 2X = 120^\circ \]

4. **Combine Like Terms**:
- This simplifies to:
\[ 3X = 120^\circ \]

5. **Solve for \(X\)**:
- Divide both sides by 3:
\[ X = \frac{120^\circ}{3} = 40^\circ \]

6. **Find the Second Interior Angle**:
- Since the second angle is \(2X\):
\[ 2X = 2 \times 40^\circ = 80^\circ \]

7. **Calculate the Third Angle**:
- The sum of all interior angles in a triangle is 180 degrees. Let the third angle be \(Y\):
\[ X + 2X + Y = 180^\circ \]
- Substitute \(X\) and \(2X\):
\[ 40^\circ + 80^\circ + Y = 180^\circ \]

8. **Solve for \(Y\)**:
- Combine the known angles:
\[ 120^\circ + Y = 180^\circ \]
- Subtract 120 degrees from both sides:
\[ Y = 180^\circ - 120^\circ = 60^\circ \]

9. **Conclusion**:
- The measures of the angles of the triangle are:
- \(40^\circ\)
- \(80^\circ\)
- \(60^\circ\)

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