SOLUTION: In a convex n-gon, the sum of all angles except one angle is 2270 . find the number of sides of n-gon .

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Question 578709: In a convex n-gon, the sum of all angles except one angle is 2270 . find the number of sides of n-gon .
Found 2 solutions by Edwin McCravy, richard1234:
Answer by Edwin McCravy(20054) About Me  (Show Source):
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Let the number of sides be n and the angle not included in the
sum of 2270° have measure k°.

Then:

 (n-1)180° = 2270° + k°  

180n - 180 = 2270 + k

      180n = 2450 + k

         n = %282450+%2B+k%29%2F180

Since n must be a positive integer, 2450 + k must be a multiple of 180.

The smallest multiple of 180 greater than 2450 is 2520 or 14*180 which would 

make k = 2520-2450 or 70.  Thus n=14 (with the omitted angle 70°) is a 
solution.

The next multiple of 180°, 2700 or 15×180 would cause k° to be 250° which
would give us a concave polygon.  But we are given that the polygon is
convex, i.e., all interior angles are less than 180°.

Therefore there is just one solution, a polygon of 14 sides.

Edwin

Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
We have

where k is the unknown angle.

Since k is between 0 and 180 (convex polygon), 180(n-1) must be a little more than 2270. Since 180*12 = 2160 and 180*13 = 2340, we conclude that n-1 = 13, so n = 14.