Sure, let's solve the problem step-by-step.
### Problem:
The size of each interior angle of a regular polygon is five times the size of the exterior angle. Find the number of sides of the polygon.
### Solution:
1. Understanding Interior and Exterior Angles:
- Let [tex]\( x \)[/tex] be the measure of the exterior angle.
- The measure of the interior angle is given as [tex]\( 5x \)[/tex].
2. Relationship Between Interior and Exterior Angles:
- The sum of an interior angle and its corresponding exterior angle is always [tex]\( 180 \)[/tex] degrees (since they are supplementary angles).
- Therefore, we can write the equation:
[tex]\[
x + 5x = 180
\][/tex]
3. Solving for [tex]\( x \)[/tex]:
- Simplify the equation:
[tex]\[
6x = 180
\][/tex]
- Divide both sides by 6:
[tex]\[
x = 30
\][/tex]
- Hence, the measure of the exterior angle is [tex]\( 30 \)[/tex] degrees.
4. Finding the Number of Sides of the Polygon:
- The formula for the measure of an exterior angle of a regular polygon with [tex]\( n \)[/tex] sides is:
[tex]\[
\text{Exterior Angle} = \frac{360}{n}
\][/tex]
- Substitute the measure of the exterior angle [tex]\( x = 30 \)[/tex] degrees into the formula:
[tex]\[
30 = \frac{360}{n}
\][/tex]
- Solving for [tex]\( n \)[/tex]:
[tex]\[
n = \frac{360}{30}
\][/tex]
[tex]\[
n = 12
\][/tex]
- Therefore, the number of sides of the polygon is 12.
### Answer:
The number of sides of the polygon is [tex]\( 12 \)[/tex].
