SOLUTION: Find the number of sides in a regular polygon if: the measure of an interior angle exceeds 6 times the measure of an exterior angle by 12

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Question 250513: Find the number of sides in a regular polygon if:
the measure of an interior angle exceeds 6 times the measure of an exterior angle by 12
Answer by Theo(13342) About Me  (Show Source):
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i = measure of the interior angle of regular polygon.
e = measure of the exterior angle of regular polygon.

i = 6*e + 12

s = number of sides in regular polygon.

exterior angle = 360 divided by number of sides = 360/s

interior angle = (s-2)*180 / s

example:

exterior angle of regular triangle = 360 / 3 = 120

interior angle of regular triangle = 1*180/3 = 60

the interior angle of a polygon and its associated exterior angle are supplementary to each other. the sum of their angles is 180.

we get i + e = 180

we also get:

i = 6*e + 12

substitute in supplementary equation to get:

6*e + 12 + e = 180

combine like terms to get:

7*e + 12 = 180

subtract 12 from both sides of equation to get:

7*e = 180 - 12 = 168

divide both sides of equation by 7 to get:

e = 168/7 = 24.

i = 6*24 + 12 = 156

156 + 24 = 180 so the angles are supplementary as they should be.

we have:

i = 156
e = 24

formula for exterior angle of a regular polygon is:

e = 360/s

this becomes

24 = 360/s

solve for s to get:

s = 15

number of sides in the regular polygon appears to be 15.

substitute in equation for interior angle of polygon to get:

156 = (s-2)*180/s

solve for s.

mulltiply both sides of the equation by s to get:

156*s = (s-2)*180

simplify by removing parentheses to get:

156*s = 180*s - 360

solve for s to get:

24*s = 360

s = 360/24 = 15

everything checks out so the answer is:

number of sides of the regular polygon is 15.