In a polygon with n sides, the sum of the measures of the interior angles, S, is
[tex] S = (n - 2)180 [/tex]
In a regular polygon with n sides, the measure of each interior angle, a, is the sum of the measures of the interior angles divided by the number of sides, or
[tex] a = \dfrac{(n - 2)180}{n} [/tex]
In this case, we know the measure of one angle, a = 157.5°, so we plug it into the second formula and solve for n.
[tex] a = \dfrac{(n - 2)180}{n} [/tex]
[tex] 157.5 = \dfrac{(n - 2)180}{n} [/tex]
[tex] 157.5n = (n - 2)180 [/tex]
[tex] 157.5n = 180n - 360 [/tex]
[tex] -22.5n = -360 [/tex]
[tex] n = 16 [/tex]
Answer: The polygon has 16 sides.
