An exterior angle of a triangle measure `110^(@)` and its interior oposite angles are in the rato `2:3` .Find the angles of the triangle.

Options

A. `40^@` and `70^@`
B. `44^@` and `66^@`
C. `50^@` and `60^@`
D. none of these

Answer

To solve the problem, we need to find the angles of the triangle given that an exterior angle measures 110 degrees and the two interior opposite angles are in the ratio of 2:3.

### Step-by-Step Solution:

1. **Understand the relationship between exterior and interior angles**:
The exterior angle of a triangle is equal to the sum of the two opposite interior angles. In this case, we know the exterior angle is 110 degrees.

\[ \text{Exterior Angle} = \text{Interior Angle 1} + \text{Interior Angle 2} \]

Let the two interior angles be \(2x\) and \(3x\) (since they are in the ratio of 2:3).

2. **Set up the equation**:
According to the relationship mentioned above, we can write:

\[ 110 = 2x + 3x \]

Simplifying this gives:

\[ 110 = 5x \]

3. **Solve for \(x\)**:
To find \(x\), divide both sides of the equation by 5:

\[ x = \frac{110}{5} = 22 \]

4. **Find the interior angles**:
Now that we have \(x\), we can find the two interior angles:

- First angle:
\[ 2x = 2 \times 22 = 44^\circ \]

- Second angle:
\[ 3x = 3 \times 22 = 66^\circ \]

5. **Find the third angle of the triangle**:
The sum of the interior angles of a triangle is always \(180^\circ\). Therefore, we can find the third angle by subtracting the sum of the two known angles from \(180^\circ\):

\[ \text{Third angle} = 180 - (44 + 66) = 180 - 110 = 70^\circ \]

6. **Conclusion**:
The angles of the triangle are \(44^\circ\), \(66^\circ\), and \(70^\circ\).

### Final Answer:
The angles of the triangle are \(44^\circ\), \(66^\circ\), and \(70^\circ\).

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