Question 485890: The measures of the interior angles of a convex 9-gon form an arithmetic sequence and, measured in degrees, all are distinct integers. What is the measure of the largest possible angle if all the angles are obtuse?
Answer by richard1234(7193)   (Show Source):
You can put this solution on YOUR website! The sum of the angles in a convex 9-gon is 7(180) = 1260. Hence the middle term is 1260/9 = 140 (the median of the angle measures). We can fix the smallest angle to be the smallest integer greater than 90, such that all the other angle measures are integers.
If a1 is the smallest angle, a5 = 140 and a9 is the largest angle, then we have an arithmetic sequence
a1 _ _ _ a5 _ _ _ a9
In other words, a5 = 140 = a1 + 4d. The optimal value of d is therefore 12, a1 = 140 - 4(12) = 92. The measure of a9 is 92 + 8(12) = 92 + 96 = 188 (degrees).
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