SOLUTION: at each vertex of a regular n-gon, the ratio of the interior angle to the exterior angle is 19:1. find the value of n.

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Question 1210728: at each vertex of a regular n-gon, the ratio of the interior angle to the exterior angle is 19:1. find the value of n.
Answer by ikleyn(54062) About Me  (Show Source):
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at each vertex of a regular n-gon, the ratio of the interior angle to the exterior angle is 19:1.
find the value of n.
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Since the n-gon is regular, all the exterior angles are congruent 
(just as all interior angles are congruent).

Let the measure of each exterior angle be x. 
Then each interior angle measures 19x, according to the problem.


    x + 19x = 180  degrees.

      20x   = 180  degrees

        x   = 180/20 = 9 degrees.


If n is the number of vertices, then  nx = 360 degrees,  or

    9n = 360  degrees,

     n = 360/9 = 40.


ANSWER.  The value of n is 40.  n = 40.

Solved.