In a regular polygon, if the number of diagonals is k times the number of sides and each interior angle is `theta`, then k is:

Options

A. `(3theta-2pi)/(2pi-2theta)`
B. `(4theta-pi)/(2pi-2theta)`
C. `(3theta-pi)/(2pi-2theta)`
D. `(3(theta-pi))/(2(pi-theta)`

Answer

To solve the problem, we need to find the value of \( k \) in a regular polygon where the number of diagonals is \( k \) times the number of sides, and each interior angle is \( \theta \).

### Step-by-Step Solution:

1. **Understand the relationship between interior angle and sides**:
The interior angle \( \theta \) of a regular polygon with \( n \) sides can be expressed as:
\[ \theta = 180^\circ - \text{exterior angle} \]
The exterior angle can be calculated as:
\[ \text{exterior angle} = \frac{360^\circ}{n} \]
Therefore, we can write:
\[ \theta = 180^\circ - \frac{360^\circ}{n} \]

2. **Convert the angle to radians**:
Since the problem uses \( \pi \), we convert the degrees to radians:
\[ \theta = \pi - \frac{2\pi}{n} \]

3. **Rearranging to find \( n \)**:
Rearranging the equation gives:
\[ \frac{2\pi}{n} = \pi - \theta \]
From this, we can solve for \( n \):
\[ n = \frac{2\pi}{\pi - \theta} \]

4. **Formula for the number of diagonals**:
The formula for the number of diagonals \( D \) in a polygon with \( n \) sides is:
\[ D = \frac{n(n - 3)}{2} \]
According to the problem, the number of diagonals is \( k \) times the number of sides:
\[ D = k \cdot n \]

5. **Setting up the equation**:
Substituting the formula for \( D \) into the equation gives:
\[ \frac{n(n - 3)}{2} = k \cdot n \]
We can cancel \( n \) (assuming \( n \neq 0 \)):
\[ \frac{n - 3}{2} = k \]

6. **Substituting the value of \( n \)**:
Now substitute \( n = \frac{2\pi}{\pi - \theta} \) into the equation for \( k \):
\[ k = \frac{\frac{2\pi}{\pi - \theta} - 3}{2} \]

7. **Simplifying the expression**:
Simplifying the expression for \( k \):
\[ k = \frac{2\pi - 3(\pi - \theta)}{2(\pi - \theta)} \]
This simplifies to:
\[ k = \frac{2\pi - 3\pi + 3\theta}{2(\pi - \theta)} = \frac{3\theta - \pi}{2(\pi - \theta)} \]

8. **Final expression for \( k \)**:
Thus, we arrive at the final expression:
\[ k = \frac{3\theta - \pi}{2\pi - 2\theta} \]

### Conclusion:
The value of \( k \) is:
\[ k = \frac{3\theta - \pi}{2\pi - 2\theta} \]