If an interior angle is eight times the exterior angle of a regular polygon, fin...

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PROBLEM:

If an interior angle is eight times the exterior angle of a regular polygon, find:

a. The value of an exterior angle

b. The number of sides of the regular polygon

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ANSWER:

> Let's denote the exterior angle as x and the interior angle as 8x.

> Interior and exterior angles that are adjacent to each other and forms a linear pair are supplementary. Thus, we'll equate the sum of the measures of the exterior angles to 180°.

The value of an exterior angle.

$\rm 8x + x = 180 \degree$

$\rm 9x = 180 \degree$

$\rm x = 180 \degree \div 9$

$\large \boxed{ \: \rm x = 20 \degree}$

> Since x is our exterior angle, thus, it measures 20°. If the exterior angle measures 20°, then the interior angle next to it measures 160°, 8 × 20°.

The number of sides of the regular polygon.

> Given the measure of one of the sides of a regular polygon, we can easily solve for the number of sides using the formula for the sum of the interior angles divided by the number of terms which is equal to 160°.

$\rm 160 \degree = \frac{(n - 2) \times 180 \degree}{n} \\$