If the sum of two angles is `135^(@)` and their difference is `(pi)/(12)` , then the circular measure of the greater angle is

Options

A. `(2pi)/(3)`
B. `(3pi)/(5)`
C. `(5pi)/(12)`
D. `(pi)/(3)`

Answer

To solve the problem, we need to find the greater angle given the sum and difference of two angles. Let's denote the two angles as A and B.

### Step 1: Set up the equations
We know from the problem that:
1. \( A + B = 135^\circ \)
2. \( A - B = \frac{\pi}{12} \)

### Step 2: Convert degrees to radians
To work with radians, we need to convert \( 135^\circ \) to radians. The conversion formula is:
\[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \]
So,
\[ 135^\circ = 135 \times \frac{\pi}{180} = \frac{135\pi}{180} = \frac{3\pi}{4} \]
Thus, we can rewrite our first equation:
\[ A + B = \frac{3\pi}{4} \]

### Step 3: Solve the system of equations
Now we have the system of equations:
1. \( A + B = \frac{3\pi}{4} \)
2. \( A - B = \frac{\pi}{12} \)

We can add these two equations to eliminate B:
\[ (A + B) + (A - B) = \frac{3\pi}{4} + \frac{\pi}{12} \]
This simplifies to:
\[ 2A = \frac{3\pi}{4} + \frac{\pi}{12} \]

### Step 4: Find a common denominator
To add the fractions, we need a common denominator. The least common multiple of 4 and 12 is 12. Therefore:
\[ \frac{3\pi}{4} = \frac{3\pi \times 3}{4 \times 3} = \frac{9\pi}{12} \]
Now we can add:
\[ 2A = \frac{9\pi}{12} + \frac{\pi}{12} = \frac{10\pi}{12} \]

### Step 5: Solve for A
Now, divide both sides by 2:
\[ A = \frac{10\pi}{12} \times \frac{1}{2} = \frac{10\pi}{24} = \frac{5\pi}{12} \]

### Step 6: Find B
Now that we have A, we can find B using the first equation:
\[ B = \frac{3\pi}{4} - A = \frac{3\pi}{4} - \frac{5\pi}{12} \]
Again, we need a common denominator to subtract. The least common multiple of 4 and 12 is 12:
\[ \frac{3\pi}{4} = \frac{3\pi \times 3}{4 \times 3} = \frac{9\pi}{12} \]
Thus,
\[ B = \frac{9\pi}{12} - \frac{5\pi}{12} = \frac{4\pi}{12} = \frac{\pi}{3} \]

### Conclusion
The greater angle is \( A \):
\[ A = \frac{5\pi}{12} \]

### Final Answer
The circular measure of the greater angle is \( \frac{5\pi}{12} \).

---