document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1043390>Question 1043390</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The interior angles of a convex polygon form arithmetic progression with common difference 4° .Determine the sides of the polygon if its largest interior angle is 172°. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #658549 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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/cgi-bin/show-face.mpl?user=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=658549\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The sum of the measures of the exterior angles of a polygon is always <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=360%5Eo ALIGN=MIDDLE  ALT=\"360%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That is simpler and easier to see and remember than anything about interior angles.<BR>\n" );
document.write( "Each exterior angle is supplementary to the adjacent interior angle,<BR>\n" );
document.write( "so the measures (in degrees) of the exterior angles also form an arithmetic progression.<BR>\n" );
document.write( "Starting from the smallest exterior angle, the terms of that progression are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D=180-172=8 ALIGN=MIDDLE  ALT=\"a%5B1%5D=180-172=8\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B2%5D=8%2B4=12 ALIGN=MIDDLE  ALT=\"a%5B2%5D=8%2B4=12\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B3%5D=8%2B2%2A4=16 ALIGN=MIDDLE  ALT=\"a%5B3%5D=8%2B2%2A4=16\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and so on, up to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D=8%2B4%28n-1%29 ALIGN=MIDDLE  ALT=\"a%5Bn%5D=8%2B4%28n-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The sum of the first <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > terms of an arithmetic progression is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bn%5D=n%28a%5B1%5D%2Ba%5Bn%5D%29%2F2 ALIGN=MIDDLE  ALT=\"S%5Bn%5D=n%28a%5B1%5D%2Ba%5Bn%5D%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "In this case, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bn%5D=n%288%2B8%2B4%28n-1%29%29%2F2 ALIGN=MIDDLE  ALT=\"S%5Bn%5D=n%288%2B8%2B4%28n-1%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and it equals <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=360 ALIGN=MIDDLE  ALT=\"360\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "because the sum of the measures of the exterior angles of any polygon is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=360%5Eo ALIGN=MIDDLE  ALT=\"360%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%288%2B8%2B4%28n-1%29%29%2F2=360 ALIGN=MIDDLE  ALT=\"n%288%2B8%2B4%28n-1%29%29%2F2=360\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%2816%2B4n-4%29%2F2=360 ALIGN=MIDDLE  ALT=\"n%2816%2B4n-4%29%2F2=360\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%2812%2B4n%29=360%2A2 ALIGN=MIDDLE  ALT=\"n%2812%2B4n%29=360%2A2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12n%2B4n%5E2=720 ALIGN=MIDDLE  ALT=\"12n%2B4n%5E2=720\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12n%2B4n%5E2-720=0 ALIGN=MIDDLE  ALT=\"12n%2B4n%5E2-720=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3n%2Bn%5E2-180=0 ALIGN=MIDDLE  ALT=\"3n%2Bn%5E2-180=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%5E2%2B3n-180=0 ALIGN=MIDDLE  ALT=\"n%5E2%2B3n-180=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28n-12%29%28n%2B15%29=0 ALIGN=MIDDLE  ALT=\"%28n-12%29%28n%2B15%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28n=12%29 ALIGN=MIDDLE  ALT=\"highlight%28n=12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The number of sides of the polygon is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%2812%29 ALIGN=MIDDLE  ALT=\"highlight%2812%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "NOTEs:<BR>\n" );
document.write( "The problem does not ask for the measures of any other interior angles, but they range from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=128%5Eo ALIGN=MIDDLE  ALT=\"128%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=172%5Eo ALIGN=MIDDLE  ALT=\"172%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "We cannot find what the lengths of the sides are,<BR>\n" );
document.write( "because there are many different polygons with those same angle measures.<BR>\n" );
document.write( "The possibilities are endless.<BR>\n" );
document.write( "Here is what one of them looks like:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-4.5%2C5.5%2C-1.3%2C8.7%2C%0D%0Agreen%28line%280%2C0%2C0.99%2C0.139%29%29%2Cline%282.056%2C0.527%2C0.99%2C0.139%29%2C%0D%0Aline%282.056%2C0.527%2C3.097%2C1.283%29%2Cline%283.913%2C2.493%2C3.097%2C1.283%29%2C%0D%0Aline%283.913%2C2.493%2C4.2%2C4.123%29%2Cline%283.62%2C5.908%2C4.2%2C4.123%29%2C%0D%0Aline%283.62%2C5.908%2C1.989%2C7.276%29%2Cline%28-0.419%2C7.444%2C1.989%2C7.276%29%2C%0D%0Aline%28-0.419%2C7.444%2C-2.635%2C5.835%29%2Cline%28-3.174%2C2.776%2C-2.635%2C5.835%29%2C%0D%0Aline%28-3.174%2C2.776%2C-1.005%2C0%29%2Cred%28line%280%2C0%2C-1.005%2C0%29%29%2C%0D%0Agreen%28arrow%280%2C0%2C5.446%2C0.765%29%29%2Cgreen%28arrow%280%2C0%2C5.5%2C0%29%29%2C%0D%0Agreen%28arc%280%2C0%2C8%2C8%2C-8%2C0%29%29%2Clocate%284.1%2C0.7%2Cgreen%288%5Eo%29%29%2C%0D%0Ared%28arc%280%2C0%2C1%2C1%2C-180%2C-8%29%29%2Clocate%28-0.2%2C1.3%2Cred%28172%5Eo%29%29%2C%0D%0Aarc%28-1.005%2C0%2C0.9%2C0.9%2C-128%2C0%29%2Clocate%28-1.3%2C1.2%2C128%5Eo%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );