Question

1. assume the following are regular polygons. for both polygons, find the measure of one interior angle and one exterior angle: a. measure of 1 interior angle:__ measure of 1 exterior angle: b. measure of 1 interior angle: measure of 1 exterior angle:__

Explanation:

Step1: Recall the formula for exterior angle of a regular polygon

The measure of an exterior angle $\theta_{e}$ of a regular polygon with $n$ sides is given by $\theta_{e}=\frac{360^{\circ}}{n}$.

Step2: Recall the formula for interior angle of a regular polygon

The measure of an interior angle $\theta_{i}$ of a regular polygon with $n$ sides is given by $\theta_{i}=\frac{(n - 2)\times180^{\circ}}{n}=180^{\circ}-\frac{360^{\circ}}{n}$.

For the pentagon ($n = 5$):

Step3: Calculate the exterior - angle

Using the exterior - angle formula $\theta_{e}=\frac{360^{\circ}}{n}$, substituting $n = 5$, we get $\theta_{e}=\frac{360^{\circ}}{5}=72^{\circ}$.

Step4: Calculate the interior - angle

Using the interior - angle formula $\theta_{i}=\frac{(n - 2)\times180^{\circ}}{n}$, substituting $n = 5$, we have $\theta_{i}=\frac{(5 - 2)\times180^{\circ}}{5}=\frac{3\times180^{\circ}}{5}=108^{\circ}$.

For the octagon ($n = 8$):

Step5: Calculate the exterior - angle

Using the exterior - angle formula $\theta_{e}=\frac{360^{\circ}}{n}$, substituting $n = 8$, we get $\theta_{e}=\frac{360^{\circ}}{8}=45^{\circ}$.

Step6: Calculate the interior - angle

Using the interior - angle formula $\theta_{i}=\frac{(n - 2)\times180^{\circ}}{n}$, substituting $n = 8$, we have $\theta_{i}=\frac{(8 - 2)\times180^{\circ}}{8}=\frac{6\times180^{\circ}}{8}=135^{\circ}$.

Answer:

a. For the pentagon: Measure of 1 exterior angle: $72^{\circ}$, Measure of 1 interior angle: $108^{\circ}$
b. For the octagon: Measure of 1 exterior angle: $45^{\circ}$, Measure of 1 interior angle: $135^{\circ}$