SOLUTION: The sum of all but one of the angles of a particular convex polygon is 1578 degrees. What is the measure, in degrees, of the remaining interior angle?

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Question 949307: The sum of all but one of the angles of a particular convex polygon is 1578 degrees. What is the measure, in degrees, of the remaining interior angle?
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
The measure of each (interior) angle in a convex polygon is less than 180%5Eo .
The sum of the measures of all the (interior) angles of a convex polygon with n sides is
%28n-2%29%2A180%5Eo
Since 8%2A180%5Eo=1440%5Eo%3C1578%5Eo and 9%2A180%5Eo=1620%5Eo%3E1578%5Eo ,
we know that the polygon must have at least
n-2=9<--->n=9%2B2=11 sides.

Could it have n%3E=12<--->n-2%3E=10 ?
That would make the sum of the measures of all its angles at least
10%2A180%5E8=1800%5Eo , and then, the measure of the remaining interior angle would be at least
1800%5Eo-1578%5Eo=222%5Eo%3E180%5Eo .

Then, n-2=9<--->n=9%2B2=11 ,
and the sum of the measures of all the (interior) angles is
9%2A180%5Eo=1620%5Eo%3E1578%5Eo .
The measure of the remaining interior angle is
1620%5Eo-1578%5Eo=highlight%2842%5Eo%29 .