Interior angle =4* exterior angle : (n-2)*180/n=4*360/n : n-2=8 : n=10

Interior angle and Exterior angle of a regular polygon:

In a regular polygon, all the angles are equal and all the sides are of the same length. Let's assume the number of sides of the regular polygon is 'n'.

- The interior angle of a regular polygon is the angle formed inside the polygon at each vertex.
- The exterior angle of a regular polygon is the angle formed outside the polygon at each vertex.

Relation between interior and exterior angles:

The sum of the interior and exterior angles at each vertex of a polygon is always 180 degrees. This is because the interior and exterior angles form a straight line.

Forming the equation:

Let's assume the measure of the exterior angle of the regular polygon is 'x' degrees.

According to the given information, the interior angle is four times the exterior angle. So, the measure of the interior angle is 4x degrees.

Using the relation between interior and exterior angles, we can form the equation:

4x + x = 180

Simplifying the equation, we get:

5x = 180

x = 180/5

x = 36

Finding the number of sides:

We know that the sum of all the exterior angles of a polygon is always 360 degrees. So, we can find the number of sides using the formula:

Number of sides = 360/measure of each exterior angle

Number of sides = 360/36

Number of sides = 10

Therefore, the regular polygon has 10 sides.

Conclusion:

The correct answer is option 'C' - 10 sides.