Each interior angle of a regular polygon is `18^(@)` more than eight times an exterior angle. The number of sides of the polygon is

Options

A. A)10
B. B)15
C. C)20
D. D)25

Answer

To solve the problem, we need to find the number of sides of a regular polygon given that each interior angle is 18 degrees more than eight times an exterior angle. Let's break this down step by step.

### Step 1: Define the variables
Let the exterior angle of the polygon be denoted as \( E \). According to the problem, the interior angle \( I \) can be expressed in terms of \( E \):
\[ I = 8E + 18 \]

### Step 2: Use the relationship between interior and exterior angles
We know that the sum of the interior angle and the exterior angle of a polygon is always 180 degrees:
\[ I + E = 180 \]

### Step 3: Substitute the expression for \( I \)
Now, we can substitute the expression for \( I \) from Step 1 into the equation from Step 2:
\[ (8E + 18) + E = 180 \]

### Step 4: Simplify the equation
Combine like terms:
\[ 9E + 18 = 180 \]

### Step 5: Solve for \( E \)
Subtract 18 from both sides:
\[ 9E = 162 \]
Now, divide both sides by 9:
\[ E = 18 \]

### Step 6: Calculate the number of sides of the polygon
The number of sides \( n \) of a regular polygon can be found using the formula:
\[ n = \frac{360}{E} \]
Substituting the value of \( E \):
\[ n = \frac{360}{18} = 20 \]

### Conclusion
Thus, the number of sides of the polygon is \( \boxed{20} \).

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