In general, in the case of a regular polygon, the size of each interior angle is
[tex]\frac{180\degree(n-2)}{n}[/tex]
And the measure of each exterior angle is
[tex]\frac{360\degree}{n}[/tex]
1) A nonagon has 9 sides and its interior/exterior angles are
[tex]\begin{gathered} \text{ interior:}180\degree\frac{(9-2)}{9}=140\degree \\ \text{sum interior:}140\degree\cdot9=1260\degree \end{gathered}[/tex]
and
[tex]\begin{gathered} \text{ exterior:}360\degree\frac{1}{9}=40\degree \\ \text{ sum exterior:}360\degree \end{gathered}[/tex]
2) One of the interior angles of the polygon is 150°; thus,
[tex]\begin{gathered} 150=\frac{180(n-2)}{n} \\ \Rightarrow150n=180n-360 \\ \Rightarrow360=30n \\ \Rightarrow n=12 \end{gathered}[/tex]
The number of sides of the fourth figure is 12 (dodecagon). The sum of its inner triangles is 12*150°=1800°. As for its exterior angles,
[tex]\begin{gathered} \text{ exterior:}360\degree\frac{1}{12}=30\degree \\ \text{ sum exterior:}360\degree \end{gathered}[/tex]
3) Since the regular polygon has 15 sides, it is called a pentadecagon.
[tex]\begin{gathered} \text{ interior:}180\degree\frac{(15-2)}{15}=156\degree \\ \text{sum interior:}156\degree\cdot15=2340\degree \end{gathered}[/tex]
As for its exterior angles
[tex]\begin{gathered} 1\text{ exterior:}\frac{360\degree}{15}=24\degree \\ \text{sum of exteriors:360}\degree \end{gathered}[/tex]
