To determine the number of sides of a regular polygon where each exterior angle is [tex]\( \frac{1}{3} \)[/tex] of its supplement (interior angle), follow these steps:
1. Let the exterior angle be [tex]\( E \)[/tex] degrees and the interior angle be [tex]\( I \)[/tex] degrees.
2. Establish the relationship between the exterior and interior angles:
Since the problem states that [tex]\( E \)[/tex] is [tex]\( \frac{1}{3} \)[/tex] of its supplement, we know the interior angle [tex]\( I \)[/tex] is the supplement of [tex]\( E \)[/tex]. Therefore,
[tex]\[
E = \frac{1}{3} I
\][/tex]
Also, since they are supplementary,
[tex]\[
E + I = 180^\circ
\][/tex]
3. Substitute [tex]\( I \)[/tex] with [tex]\( 3E \)[/tex] in the supplementary equation:
From the relationship given,
[tex]\[
I = 3E
\][/tex]
Substituting [tex]\( I \)[/tex] in the supplementary equation, we get:
[tex]\[
E + 3E = 180^\circ
\][/tex]
Simplify this equation:
[tex]\[
4E = 180^\circ
\][/tex]
[tex]\[
E = 45^\circ
\][/tex]
4. Determine the number of sides [tex]\( n \)[/tex] using the exterior angle:
For a regular polygon, each exterior angle [tex]\( E \)[/tex] is given by:
[tex]\[
\frac{360^\circ}{n} = E
\][/tex]
Substitute the known value of [tex]\( E \)[/tex]:
[tex]\[
\frac{360^\circ}{n} = 45^\circ
\][/tex]
Solve for [tex]\( n \)[/tex]:
[tex]\[
n = \frac{360^\circ}{45^\circ}
\][/tex]
[tex]\[
n = 8
\][/tex]
Therefore, the regular polygon has [tex]\(\boxed{8}\)[/tex] sides.
