Question

in a regular polygon, the ratio of the measure of the exterior angle to the measure of its adjacent interior angle is 1 to 4. how many sides does the polygon have?

Answer 1

A polygon's exterior angle is 180 degrees.
So if you have a regular n-gon where one of the interior angles is 4 times larger than the adjacent exterior angle, then the sum of the interior angles must be 4 times the sum of the exterior angles, the equation should look like this,
180(n-2) = 4 (180)

then solve for n,
180n-360=720
180n = 1080
then divide both sides by 180 and the answer will be n=6