If the sum and difference of two angles are `135^(@)and(pi)/(12)` respectively , then the value of the angles in degree measure are

Options

A. `70^(@),65^(@)`
B. `75^(@),60^(@)`
C. `45^(@),90^(@)`
D. `80^(@),55^(@)`

Answer

To solve the problem, we need to find two angles A and B given their sum and difference.

1. **Set up the equations**:
- The sum of the angles is given as:
\[ A + B = 135^\circ \quad \text{(1)} \]
- The difference of the angles is given as:
\[ A - B = \frac{\pi}{12} \quad \text{(2)} \]

2. **Convert \(\frac{\pi}{12}\) to degrees**:
- We know that \(\pi\) radians is equal to \(180^\circ\). Thus:
\[ \frac{\pi}{12} = \frac{180^\circ}{12} = 15^\circ \]
- Now we can rewrite equation (2):
\[ A - B = 15^\circ \quad \text{(3)} \]

3. **Add equations (1) and (3)**:
- Adding equations (1) and (3):
\[ (A + B) + (A - B) = 135^\circ + 15^\circ \]
This simplifies to:
\[ 2A = 150^\circ \]
- Now, divide both sides by 2:
\[ A = \frac{150^\circ}{2} = 75^\circ \]

4. **Substitute A back into equation (1)**:
- Now that we have \(A\), we can find \(B\) by substituting \(A\) back into equation (1):
\[ 75^\circ + B = 135^\circ \]
- Rearranging gives:
\[ B = 135^\circ - 75^\circ = 60^\circ \]

5. **Final values of A and B**:
- Therefore, the values of the angles are:
\[ A = 75^\circ, \quad B = 60^\circ \]

### Summary of the Solution:
- The two angles are \(A = 75^\circ\) and \(B = 60^\circ\).