document.write( "Question 234283: What is the maximum possible number of acute angles in a pentagon? \n" ); document.write( "
Algebra.Com's Answer #172777 by Edwin McCravy(20056)\"\" \"About 
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What is the maximum possible number of acute interior (vertex) angles in a pentagon?\r
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document.write( "The sum of the interior angles of any polygon is\r\n" );
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document.write( " (N-2)180° \r\n" );
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document.write( "where N is the number of sides (and interior angles).  \r\n" );
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document.write( "In the case of a pentagon, N=5, so the sum of the \r\n" );
document.write( "5 interior angles is\r\n" );
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document.write( "(5-2)180° = (3)180° = 540°.\r\n" );
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document.write( "Let the angles be A, B, C, D, and E.  Then\r\n" );
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document.write( "A + B + C + D + E = 540\r\n" );
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document.write( "Let's see if all 5 can be acute:\r\n" );
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document.write( "A < 90°\r\n" );
document.write( "B < 90°\r\n" );
document.write( "C < 90°\r\n" );
document.write( "D < 90°\r\n" );
document.write( "E < 90°\r\n" );
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document.write( "Adding all those inequalities:\r\n" );
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document.write( "        A < 90°\r\n" );
document.write( "        B < 90°\r\n" );
document.write( "        C < 90°\r\n" );
document.write( "        D < 90°\r\n" );
document.write( "        E < 90°\r\n" );
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document.write( "A+B+C+D+E < 450°\r\n" );
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document.write( "No, since that sum must equal 540°,\r\n" );
document.write( "not less that 450°. But we could make\r\n" );
document.write( "4 of them, A, B, C, and D acute, \r\n" );
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document.write( "        A < 90°\r\n" );
document.write( "        B < 90°\r\n" );
document.write( "        C < 90°\r\n" );
document.write( "        D < 90°\r\n" );
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document.write( "  A+B+C+D < 360°\r\n" );
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document.write( "and so E would then have to be more than 180°,\r\n" );
document.write( "which is called a \"reflex angle\".\r\n" );
document.write( "for instance\r\n" );
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document.write( "Edwin
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