Question 202991
Let I = the measure of an interior angle and E = the measure of the corresponding exterior angle.
Recall that the interior angle and the exterior angle are supplementary so:  I+E = 180 degrees.
You are told in the problem that:
I = 8E "The measure of each interior angle of a regular polygon is 8 times that of an exterior angle."
Substitute I = 8E into the first equation and solve for E.
8E+E = 180
9E = 180
E = 20 Degrees. This is the measure of an exterior angle.
The interior angle is:
I = 8E
I = 8(20)
I = 160 degrees.
Now the measure of an interior angle of a regular polygon of n sides is given by:
{{{I = (n-2)180/n}}} But we just found the measure of an interior angle: I = 160, so we substitute and solve for n.
{{{160 = (n-2)(180)/n}}} Multiply both sides by n.
{{{160n = 180n-360}}} Subtract 160n from both sides.
{{{0 = 20n-360}}} Add 360 to both sides.
{{{360 = 20n}}} Finally, divide both sides by 20.
{{{n = 18}}}
The regular polygon has 18 sides.