document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1203593>Question 1203593</A>: <I><SPAN STYLE=\"word-break: break-all;\"> George took out a piece of paper and placed it on his desk.  He then drew a polygon.  All the angles in the polygon were obtuse.  What type of polygon could could he have drawn?\r<BR>\n" );
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document.write( "A. quadrilateral<BR>\n" );
document.write( "B. rectangle<BR>\n" );
document.write( "C. triangle<BR>\n" );
document.write( "D. hexagon (the correct answer)\r<BR>\n" );
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document.write( "I need to know how to determine this answer (hexagon). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #839262 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=839262\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "An angle is obtuse when the angle is between 90 degrees and 180 degrees excluding each endpoint.<BR>\n" );
document.write( "90 < angle < 180\r<BR>\n" );
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document.write( "Quadruple all sides of that inequality to get 360 < 4*angle < 720<BR>\n" );
document.write( "If a quadrilateral could have all 4 obtuse angles, then the angles would sum to a value anywhere between 360 and 720, excluding those endpoints.\r<BR>\n" );
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document.write( "But recall that the sum of the interior angles of any quadrilateral is always 360 degrees. <BR>\n" );
document.write( "It comes from this formula:<BR>\n" );
document.write( "S = 180(n-2)<BR>\n" );
document.write( "where,<BR>\n" );
document.write( "S = sum of interior angles<BR>\n" );
document.write( "n = number of sides<BR>\n" );
document.write( "When n = 4 it leads to S = 360<BR>\n" );
document.write( "We can see that 360 < 4*angle < 720 wouldn't be possible due to S = 360.<BR>\n" );
document.write( "This contradiction would allow us to rule out choice A.<BR>\n" );
document.write( "Quadrilaterals cannot have all angles that are obtuse.<BR>\n" );
document.write( "The most obtuse angles a quadrilateral can have is 3.\r<BR>\n" );
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document.write( "Choice B is ruled out because all four angles of any rectangle are always 90 degrees.\r<BR>\n" );
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document.write( "Use the formula<BR>\n" );
document.write( "S = 180(n-2)<BR>\n" );
document.write( "to plug in n = 3 and you should get S = 180<BR>\n" );
document.write( "The sum of all three interior angles of any triangle is always 180 degrees.<BR>\n" );
document.write( "Start from 90 < angle < 180 and triple each side to get 270 < 3*angle < 540<BR>\n" );
document.write( "Unfortunately 180 is not in this interval, so we cannot have a triangle with all angles obtuse.<BR>\n" );
document.write( "The most obtuse angles a triangle can have is 1.<BR>\n" );
document.write( "If one angle was obtuse, then the other two angles must be acute. The other two acute angles must be in the interval 0 < angle < 90.<BR>\n" );
document.write( "So we can rule out choice C as well.\r<BR>\n" );
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document.write( "If n = 6, then S = 180*(n-2) = 180*(6-2) = 720 is the sum of all six angles of a hexagon.<BR>\n" );
document.write( "Also if all 6 angles of a hexagon are obtuse, then 90 < angle < 180 leads to 540 < 6*angle < 1080<BR>\n" );
document.write( "The 720 is indeed between 540 and 1080, which means it is possible to have a hexagon with all obtuse angles.<BR>\n" );
document.write( "In fact, any regular hexagon will have each interior angle of 720/6 = 120 degrees\r<BR>\n" );
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document.write( "Bonus Questions:<ol><li>What is the smallest positive integer value of n such that 90 < 180(n-2)/n < 180 is true?</li><li>Can a quadrilateral have exactly 3 right angles, where the 4th angle is not 90 degrees?</li></ol><BR>\n" );
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