Question

solve for the missing angles in this rectangle. int and ent is abbreviations for interior and exterior. start solving for nonagon and below. don't solve the 2 top sections

Answer 1

In general, in the case of a regular polygon, the size of each interior angle is

$\frac{180\degree(n-2)}{n}$

And the measure of each exterior angle is

$\frac{360\degree}{n}$

1) A nonagon has 9 sides and its interior/exterior angles are

$\begin{gathered} \text{ interior:}180\degree\frac{(9-2)}{9}=140\degree \\ \text{sum interior:}140\degree\cdot9=1260\degree \end{gathered}$

and

$\begin{gathered} \text{ exterior:}360\degree\frac{1}{9}=40\degree \\ \text{ sum exterior:}360\degree \end{gathered}$

2) One of the interior angles of the polygon is 150°; thus,

$\begin{gathered} 150=\frac{180(n-2)}{n} \\ \Rightarrow150n=180n-360 \\ \Rightarrow360=30n \\ \Rightarrow n=12 \end{gathered}$

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,

$\begin{gathered} \text{ exterior:}360\degree\frac{1}{12}=30\degree \\ \text{ sum exterior:}360\degree \end{gathered}$

3) Since the regular polygon has 15 sides, it is called a pentadecagon.

$\begin{gathered} \text{ interior:}180\degree\frac{(15-2)}{15}=156\degree \\ \text{sum interior:}156\degree\cdot15=2340\degree \end{gathered}$

As for its exterior angles

$\begin{gathered} 1\text{ exterior:}\frac{360\degree}{15}=24\degree \\ \text{sum of exteriors:360}\degree \end{gathered}$