Question

Given: The measure of each interior angle of a regular n-gon is x times that of an exterior angle.
a.Express x in terms of n
b.For what values of n will x be an integer

Answer 1

The sum on an int. angle and ext. angle is 180 deg;
The sum of all int. angles in a regular n-gon is 180 * (n-2);
( = 180n - 360)
Therefore one int. angle in a n-gon is (180n - 360)/ n. (because a regular n-gon has n angles)
( = 180 - 360/n)
One ext. angle of the same n-gon is 180 - (180 - 360/ n)
( = 360/n)

x = (180 -360/n)/(360/n)
x = n/2 -1

for x to be an integer, n/2 -1 must be an integer; Thus following the logic n/2 is an integer too.
Therefore n is any integer times 2; n is even.

Answer 2

Forgot to add that n must be bigger than 2 as 2-sided polygon does not exist in euclidean geometry.

Answer 3

Thank you so much! For some reason I got (n-2)/2=x

Answer 4

I did (180n-360)n = x (360/n) and then I must of messed things up