Chapter 3.5, Problem 9CE

To determine

ToComplete:The table for regular polygon for the measure of exterior and interior angles.

Answer to Problem 9CE

  $\begin{array}{cccccccc}\text{Number of sides}& 6& 10& 20& 36& 18& 360& 4\\ \text{Measure of each ext}\text{.}\angle & 60°& 36°& 18°& 10°& 20°& 1°& 90°\\ \text{Measure of each int}\text{.}\angle & 120°& 144& 162°& 170°& 160°& 179°& 90°\end{array}$

Explanation of Solution

Given Information:Consider the table given below.

  

Calculations:

For a polygon having n sides.

The measure of an exterior angle is:

  $=\frac{360°}{n}$

The measure of an interior angle is:

  $=\frac{\left(n-2\right)}{n}\times 180°$

In the first column, a polygon has 6 sides.

Here, $n=6$ .

  $\begin{array}{l}\text{ext}\angle =\frac{360°}{n}\\ \text{ext}\angle =\frac{360°}{6}\\ \text{ext}\angle =60°\end{array}$

So, the measure of the exterior angle is $60°$ .

  $\begin{array}{l}\mathrm{int}\angle =\frac{\left(n-2\right)}{n}\times 180°\\ \mathrm{int}\angle =\frac{\left(6-2\right)}{6}\times 180°\\ \mathrm{int}\angle =120°\end{array}$

So, the measure of the interior angle is $120°$ .

Calculating the number of sides from given interior angle.

  $\begin{array}{l}\mathrm{int}\angle =90°\\ 90°=\frac{\left(n-2\right)}{n}\times 180°\\ 90n=180n-360°\\ 90n=360°\\ n=4\end{array}$

So, a polygon whose interior angle is $90°$ , has 4 sides.

Calculating the number of sides from given exterior angle.

  $\begin{array}{l}\text{ext}\angle =10°\\ 10°=\frac{360°}{n}\\ 10°\times n=360°\\ n=\frac{360°}{10°}\\ n=36\end{array}$

So, a polygon whose exterior angle is $10°$ , has 36 sides.

Similarly, completing the tables.

  $\begin{array}{cccccccc}\text{Number of sides}& 6& 10& 20& 36& 18& 360& 4\\ \text{Measure of each ext}\text{.}\angle & 60°& 36°& 18°& 10°& 20°& 1°& 90°\\ \text{Measure of each int}\text{.}\angle & 120°& 144& 162°& 170°& 160°& 179°& 90°\end{array}$