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

`(180\degree(n-2))/(n)`

And the measure of each exterior angle is

`(360\degree)/(n)`

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

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

and

`\begin{gathered} \text{ exterior:}360\degree(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=(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(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((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:}(360\degree)/(15)=24\degree \\ \text{sum of exteriors:360}\degree \end{gathered}`