document.write( "Question 822676: if four of the angles of the pentagon are x,2x,3x,4x.
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Algebra.Com's Answer #496701 by Edwin McCravy(20056) $\"\"$ $\"About$
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if four of the angles of the pentagon are x,2x,3x,4x.
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document.write( "Since you didn't specify which two opposite sides are parallel,\r\n" );
document.write( "there are four solutions for x.  Only one of them, the first, x=40ｰ\r\n" );
document.write( "defines a convex pentagon. The other three are concave polygons.  A \r\n" );
document.write( "concave polygon has one or more \"sunk-in places\" where one or more \r\n" );
document.write( "of the internal angles are greater than 180ｰ, and are called REFLEX \r\n" );
document.write( "angles.  Examples of pentagons defined by the four solutions are all \r\n" );
document.write( "drawn below.\r\n" );
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document.write( "Since two sides are parallel, one of the sides must be a transversal\r\n" );
document.write( "between those two parallel sides.  This means that two of the interior\r\n" );
document.write( "angles are supplementary.\r\n" );
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document.write( "The sum of the interior angles of a pentagon are\r\n" );
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document.write( "(n-2)ﾗ180ｰ = (5-2)ﾗ180ｰ = 3ﾗ180ｰ = 540ｰ\r\n" );
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document.write( "Let the 5th interior angle be y, then\r\n" );
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document.write( "x+2x+3x+4x+y = 540ｰ\r\n" );
document.write( "       10x+y = 540ｰ\r\n" );
document.write( "           y = 540ｰ-10x\r\n" );
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document.write( "So the 5 angles are\r\n" );
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document.write( "x, 2x, 3x, 4x, 540ｰ-10x\r\n" );
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document.write( "There are 9 cases for the 2 supplementary angles:\r\n" );
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document.write( "Case 1: x and 2x are supplementary:\r\n" );
document.write( "x+2x=180ｰ or 3x=180ｰ or x=60ｰ, so the 5 angles would be\r\n" );
document.write( "60ｰ, 120ｰ, 180ｰ, 240ｰ, -60ｰ not possible\r\n" );
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document.write( "Case 2: x and 3x are supplementary:\r\n" );
document.write( "x+3x=180ｰ or 4x=180ｰ or x=45ｰ, so the 5 angles would be\r\n" );
document.write( "45ｰ, 90ｰ, 135ｰ, 180ｰ, 90ｰ, not possible because an \r\n" );
document.write( "interior angle cannot be 180ｰ, for that would have only \r\n" );
document.write( "4 sides.\r\n" );
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document.write( "Case 3: x and 4x are supplementary:\r\n" );
document.write( "x+4x=180ｰ or 5x=180ｰ or x=36ｰ, so the 5 angles would be \r\n" );
document.write( "36ｰ, 72ｰ, 108ｰ, 144ｰ, 180ｰ, not possible because an interior\r\n" );
document.write( "angle cannot be 180ｰ, for that would have only 4 sides.\r\n" );
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document.write( "Case 4: x and 540ｰ-10x are supplementary:\r\n" );
document.write( "x+540ｰ-10x=180ｰ or -9x=-360ｰ or x=40ｰ, so the 5 angles would be\r\n" );
document.write( "40ｰ, 80ｰ, 120ｰ, 160ｰ, 140ｰ, which IS POSSIBLE!  \r\n" );
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document.write( "Case 5: 2x and 3x are supplementary:\r\n" );
document.write( "x+3x=180ｰ or 4x=180ｰ or x=45ｰ, so the 5 angles would be  \r\n" );
document.write( "45ｰ, 90ｰ, 135ｰ, 180ｰ, 90ｰ, not possible because an interior\r\n" );
document.write( "angle cannot be 180ｰ, for that would have only 4 sides.\r\n" );
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document.write( "Case 6: 2x and 4x are supplementary:\r\n" );
document.write( "2x+4x=180ｰ or 6x=180ｰ or x=30ｰ, so the 5 angles would be \r\n" );
document.write( "30ｰ, 60ｰ, 90ｰ, 120ｰ, 240ｰ, which IS POSSIBLE!\r\n" );
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document.write( "Case 7: 2x and 540ｰ-10x are supplementary:\r\n" );
document.write( "2x+540ｰ-10x=180ｰ or -8x=-360ｰ or x=45ｰ, same as case 5\r\n" );
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document.write( "Case 8: 3x and 4x are supplementary:\r\n" );
document.write( "3x+4x=180ｰ or 7x=180ｰ or x=ｰ, so the 5 angles would be \r\n" );
document.write( "ｰ, ｰ,ｰ, ｰ, ｰ, which IS POSSIBLE!\r\n" );
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document.write( "Case 9: 3x and 540ｰ-10x are supplementary:\r\n" );
document.write( "3x+540ｰ-10x=180ｰ or -7x=-360ｰ or x=ｰ, so the 5 angles \r\n" );
document.write( "would be ｰ, ｰ, ｰ, ,ｰ\r\n" );
document.write( "which IS POSSIBLE!\r\n" );
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document.write( "Case 10: 4x and 540ｰ-10x are supplementary:\r\n" );
document.write( "4x+540ｰ-10x=180ｰ or -6x=-360ｰ or x=60ｰ, same as case 1.  NOT \r\n" );
document.write( "POSSIBLE!\r\n" );
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document.write( "So there are four solutions for x:\r\n" );
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document.write( "1. x=40ｰ, 2x=80ｰ, 3x=120ｰ, 4x=160ｰ, 5th angle=140ｰ\r\n" );
document.write( "2. x=30ｰ, 2x=60ｰ, 3x=90ｰ, 4x=120ｰ, 5th angle=240ｰ\r\n" );
document.write( "3. x=ｰ, 2x=ｰ, 3x=ｰ, 4x=ｰ, 5th angle=ｰ\r\n" );
document.write( "4. x=ｰ, 2x=ｰ, 3x=ｰ, 4x=, 5th angle=ｰ.\r\n" );
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document.write( "Here are typical ways the 5 pentagons could be.  The \r\n" );
document.write( "internal angles which are more than 180ｰ which cause\r\n" );
document.write( "the last 3 pentagons to be concave are indicated with\r\n" );
document.write( "a red arc. \r\n" );
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document.write( "1. 40ｰ, 80ｰ, 120ｰ, 160ｰ, 140ｰ\r\n" );
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document.write( "2. 30ｰ, 60ｰ, 90ｰ, 120ｰ, 240ｰ,\r\n" );
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document.write( "3. ｰ, ｰ,ｰ, ｰ, ｰ\r\n" );
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document.write( "4. ｰ, ｰ, ｰ, ,ｰ\r\n" );
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document.write( "Edwin
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