document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=524132>Question 524132</A>: <I><SPAN STYLE=\"word-break: break-all;\"> There is a famous theorem in Euclidean geometry that states that the sum of the interior angles of a triangle is 180 degrees.<BR>\n" );
document.write( " a)Use the theorem about triangles to determine the sum of the angles of a convex quadrilateral. Hint: Draw a convex quadrilateral and draw a diagonal. <BR>\n" );
document.write( " b) Use the result in Part(1) to determine the sum of the angles of a convex pentagon.<BR>\n" );
document.write( " c) Use the result in Part(2) to determine the sum of the angles of a convex hexagon.<BR>\n" );
document.write( " d)Let 'n' be a natural number with 'n'> or = to 3.  Make a conjecture about the sum of the angles of a convex polygon with 'n' sides and use mathematical induction to prove your conjecture.\r<BR>\n" );
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document.write( "** I've figured out a, b, and c.  I've also created a conjecture for d. Here's what I have: \"Let 'n' be a natural number with 'n'> or = to 3.  For any convex polygon with 'n' sides, the sum of the angle of the polygon is 180(n-2)\". I have my basis class which is when n=3 then the sum = 180 degrees, but I don't know how to prove my induction step. I know (k+1) must be substitued in for 'n' at some point but not sure when and what to do.  Please help.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #347599 by </B><A HREF=/tutors/aboutme.mpl?userid=stanbon><B>stanbon(75887)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=stanbon><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=347599\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  Make a conjecture about the sum of the angles of a convex polygon with 'n' sides and use mathematical induction to prove your conjecture. <BR>\n" );
document.write( "----<BR>\n" );
document.write( "Prove that an n sided quadrilateral <BR>\n" );
document.write( "has sum of interior angles = (n-2)180 degrees<BR>\n" );
document.write( "------------------------------------------------------<BR>\n" );
document.write( "Step #1: <BR>\n" );
document.write( "For n = 3, sum of angles = (3-2)180 = 180::::that is true<BR>\n" );
document.write( "----<BR>\n" );
document.write( "Step # 2:<BR>\n" );
document.write( "Assume true for n = k sides, i.e, <BR>\n" );
document.write( "sum of interio angles = (k-2)180<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "Step # 3:<BR>\n" );
document.write( "Prove true for n = k+1 sides<BR>\n" );
document.write( "----<BR>\n" );
document.write( "sum for k+1 = sum for k sides + 180<BR>\n" );
document.write( "sum for k+1 = (k-2)180 + 180<BR>\n" );
document.write( "Factor out the 180 to get:<BR>\n" );
document.write( "sum for k+1 = (k-2+1)(180)<BR>\n" );
document.write( "Therefore, sum for k+1 sides = [(k+1)-2)]180<BR>\n" );
document.write( "Q.E.D.  </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );