The interior angles of a regular polygon measure `150^@` each. The number of diagonals of the polygon is

Options

A. `35`
B. `44`
C. `54`
D. `78`

Explanation

To solve the problem of finding the number of diagonals in a regular polygon where each interior angle measures \(150^\circ\), we can follow these steps:

### Step 1: Use the formula for the interior angle of a polygon
The formula for the interior angle \(A\) of a regular polygon with \(n\) sides is given by:
\[ A = \frac{(n-2) \times 180}{n} \]
Given that \(A = 150^\circ\), we can set up the equation:
\[ 150 = \frac{(n-2) \times 180}{n} \]

### Step 2: Clear the fraction
To eliminate the fraction, multiply both sides by \(n\):
\[ 150n = (n-2) \times 180 \]

### Step 3: Expand and rearrange the equation
Expanding the right side gives:
\[ 150n = 180n - 360 \]
Now, rearranging the equation to isolate \(n\):
\[ 150n - 180n = -360 \]
\[ -30n = -360 \]

### Step 4: Solve for \(n\)
Dividing both sides by \(-30\):
\[ n = \frac{360}{30} = 12 \]
Thus, the polygon has \(12\) sides.

### Step 5: Use the formula for the number of diagonals
The formula for the number of diagonals \(D\) in a polygon with \(n\) sides is:
\[ D = \frac{n(n-3)}{2} \]
Substituting \(n = 12\):
\[ D = \frac{12(12-3)}{2} = \frac{12 \times 9}{2} = \frac{108}{2} = 54 \]

### Conclusion
The number of diagonals in the polygon is \(54\).