If the measure of each interior angle of a regular polygon be 144 degree the number of sides of the polygon is:

Options

A. a.30
B. b.20
C. c.10
D. d.none of (a) ,(B ) ,(c )

Answer

To find the number of sides of a regular polygon where each interior angle measures 144 degrees, we can follow these steps:

### Step 1: Use the formula for the sum of interior angles
The sum of the interior angles \( S \) of a polygon with \( n \) sides is given by the formula:
\[ S = (n - 2) \times 180 \]

### Step 2: Relate the sum of interior angles to the measure of each angle
Since the polygon is regular, each interior angle can also be expressed as:
\[ \text{Each interior angle} = \frac{S}{n} \]
Given that each interior angle is 144 degrees, we can set up the equation:
\[ 144 = \frac{S}{n} \]

### Step 3: Substitute the sum of interior angles into the equation
Substituting the expression for \( S \) into the equation gives:
\[ 144 = \frac{(n - 2) \times 180}{n} \]

### Step 4: Multiply both sides by \( n \) to eliminate the fraction
Multiplying both sides by \( n \) results in:
\[ 144n = (n - 2) \times 180 \]

### Step 5: Expand the right-hand side
Expanding the right-hand side gives:
\[ 144n = 180n - 360 \]

### Step 6: Rearrange the equation
To isolate \( n \), we can move all terms involving \( n \) to one side:
\[ 360 = 180n - 144n \]
This simplifies to:
\[ 360 = 36n \]

### Step 7: Solve for \( n \)
Dividing both sides by 36 gives:
\[ n = \frac{360}{36} = 10 \]

### Conclusion
The number of sides of the polygon is \( n = 10 \).