SOLUTION: The interior angles of a convex nonagon have degree measures that are integers in arithmetic sequence. What is the smallest angle possible in this nonagon?

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Question 1132033: The interior angles of a convex nonagon have degree measures that are integers in arithmetic sequence. What is the smallest angle possible in this nonagon?
Found 3 solutions by Alan3354, ikleyn, greenestamps:
Answer by Alan3354(69443) About Me  (Show Source):
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The interior angles of a convex nonagon have degree measures that are integers in arithmetic sequence. What is the smallest angle possible in this nonagon?
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There are 9 angles.
n is the smallest.
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It's the sum of n, n+1, n+2 ... n+8
The sum is 1260 degrees.

Answer by ikleyn(52781) About Me  (Show Source):
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.
The sum of all 9 interior angles is  (9-2)*180° = 1260°.


9 is a nice number.
Let's take the 5-th angle, which is exactly midway between the 1-st angle and the 9-th angle.


Since the angle measures form an arithmetic progression, the 5-th angle measure is exactly  1260%5Eo%2F9 = 140°.


Also, we can write  a%5B5%5D = a%5B1%5D%2B4%2Ad = 140°.


They want us to define the maximum possible positive integer "d", providing a%5B1%5D positive integer, too.


So, what a multiple of 4 is closest to 140, still lesser than 140?  - It is 136°.

Then d = 136%5Eo%2F4 = 34°.   



    Is it the solution ?      - No.

    Why ?     - Because it does not provide CONVEXITY of the nonagon.

    I will not explain why it is so - you can easily check it on your own.



    What to do ?    - You need to take a look on the problem from the other end.



What is the interval between 140° and 180° ?   - It is 40°.


So, in the interval of 40° we should place 4 intervals / (gaps) between the 5-th and 9-th angles - leaving the 9-th angle still lesser than 180°.


It gives you  d = %28176%5Eo-140%5Eo%29%2F4 = 36%5Eo%2F4 = 9°.


It leads you to the ANSWER on the problem's question: the smallest possible angle in this nonagon is  140° - 4*9° = 104°.



Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


The sum of the interior angles of a nonagon is 180(9-2) = 1260. The average measure of an angle is 1260/9 = 140.

Since the measures of the nine angles are in arithmetic sequence, the smallest is 140-4n and the largest is 140+4n, where n is the common difference between the angle measures.

For the polygon to be convex, the largest angle has to be less than 180 degrees:

140%2B4n+%3C+180
4n+%3C+40
n+%3C+10

Since the angle measures are integers, n=9, and the measure of the smallest angle is 140-4n = 140-4(9) = 104 degrees.