The exterior and interior angles of a polygon are supplementary angles (they add up to 180 degrees). Since the exterior angle of the polygon has a measure of 10 degrees, then the interior angles must have a measure of 170 degrees.
On the other hand, the measure of the internal angle of an n-sided polygon is:
[tex]180\times\frac{n-2}{n}[/tex]Since the internal angle must have a measure of 170 degrees, then:
[tex]180\times\frac{n-2}{n}=170[/tex]Solve for n:
[tex]\begin{gathered} \Rightarrow180(n-2)=170n \\ \Rightarrow180n-360=170n \\ \Rightarrow180n-170n=360 \\ \Rightarrow10n=360 \\ \Rightarrow n=\frac{360}{10} \\ \Rightarrow n=36 \end{gathered}[/tex]Therefore, the polygon has 36 sides.
