One angle of a triangle is `45^(@)` and the other two are in the ratio 2:3. find these angles.

Options

A. `50^(@),85^(@)`
B. `54^(@),81^(@)`
C. `54^(@),70^(@)`
D. `44^(@),81^(@)`

Answer

To solve the problem, we need to find the angles of a triangle where one angle is given as \(45^\circ\) and the other two angles are in the ratio \(2:3\).

### Step-by-Step Solution:

1. **Identify the Given Information:**
- One angle of the triangle (let's call it angle B) is \(45^\circ\).
- The other two angles (angle A and angle C) are in the ratio \(2:3\).

2. **Express the Angles in Terms of a Variable:**
- Let angle A = \(2x\) (since it corresponds to the ratio of 2).
- Let angle C = \(3x\) (since it corresponds to the ratio of 3).

3. **Use the Triangle Angle Sum Property:**
- The sum of the angles in a triangle is always \(180^\circ\).
- Therefore, we can write the equation:
\[ \text{Angle A} + \text{Angle B} + \text{Angle C} = 180^\circ \]
- Substituting the known values:
\[ 2x + 45^\circ + 3x = 180^\circ \]

4. **Combine Like Terms:**
- Combine \(2x\) and \(3x\):
\[ 5x + 45^\circ = 180^\circ \]

5. **Isolate the Variable:**
- Subtract \(45^\circ\) from both sides:
\[ 5x = 180^\circ - 45^\circ \]
\[ 5x = 135^\circ \]

6. **Solve for \(x\):**
- Divide both sides by 5:
\[ x = \frac{135^\circ}{5} \]
\[ x = 27^\circ \]

7. **Find the Values of Angle A and Angle C:**
- Substitute \(x\) back into the expressions for angle A and angle C:
- Angle A:
\[ \text{Angle A} = 2x = 2 \times 27^\circ = 54^\circ \]
- Angle C:
\[ \text{Angle C} = 3x = 3 \times 27^\circ = 81^\circ \]

8. **Final Angles of the Triangle:**
- Angle A = \(54^\circ\)
- Angle B = \(45^\circ\)
- Angle C = \(81^\circ\)

9. **Verification:**
- To ensure the angles are correct, check if they sum to \(180^\circ\):
\[ 54^\circ + 45^\circ + 81^\circ = 180^\circ \]

### Conclusion:
The angles of the triangle are:
- Angle A = \(54^\circ\)
- Angle B = \(45^\circ\)
- Angle C = \(81^\circ\)