document.write( "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? \n" ); document.write( "

Algebra.Com's Answer #748866 by ikleyn(52784) $\"\"$ $\"About$

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document.write( "The sum of all 9 interior angles is  (9-2)*180° = 1260°.\r\n" );
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document.write( "9 is a nice number.\r\n" );
document.write( "Let's take the 5-th angle, which is exactly midway between the 1-st angle and the 9-th angle.\r\n" );
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document.write( "Since the angle measures form an arithmetic progression, the 5-th angle measure is exactly   = 140°.\r\n" );
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document.write( "Also, we can write   =  = 140°.\r\n" );
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document.write( "They want us to define the maximum possible positive integer \"d\", providing  positive integer, too.\r\n" );
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document.write( "So, what a multiple of 4 is closest to 140, still lesser than 140?  - It is 136°.\r\n" );
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document.write( "Then d =  = 34°.   \r\n" );
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document.write( "    Is it the solution ?      - No.\r\n" );
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document.write( "    Why ?     - Because it does not provide CONVEXITY of the nonagon.\r\n" );
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document.write( "    I will not explain why it is so - you can easily check it on your own.\r\n" );
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document.write( "    What to do ?    - You need to take a look on the problem from the other end.\r\n" );
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document.write( "What is the interval between 140° and 180° ?   - It is 40°.\r\n" );
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document.write( "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°.\r\n" );
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document.write( "It gives you  d =  =  = 9°.\r\n" );
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document.write( "It leads you to the ANSWER on the problem's question: the smallest possible angle in this nonagon is  140° - 4*9° = 104°.\r\n" );
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