document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1074021>Question 1074021</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The ratio of interior and exterior angles of a polygon is 6:4. Find the sum of the angles of the polygon. <BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #688793 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=688793\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> THE SHORT WAY:<BR>\n" );
document.write( "The ratio given means that it is a polygon<BR>\n" );
document.write( "with congruent angles, and at least 5 sides, because<BR>\n" );
document.write( "if the angles were not all congruent,<BR>\n" );
document.write( "there would be two or more different ratios,<BR>\n" );
document.write( "and for 3- and 4-sided polygons with congruent angles,<BR>\n" );
document.write( "the ratios are 1:2 and 1:1 respectively.<BR>\n" );
document.write( "So, for this polygon, At every vertex,<BR>\n" );
document.write( "the ratio of exterior to interior angle,<BR>\n" );
document.write( "and the ratio of the sums of exterior and interior angles is 6:4.<BR>\n" );
document.write( "The sum of all 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( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >= the sum of all interior angles,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2F360%5Eo=6%2F4 ALIGN=MIDDLE  ALT=\"x%2F360%5Eo=6%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=6%2A360%5Eo%2F4 ALIGN=MIDDLE  ALT=\"x=6%2A360%5Eo%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=540%5Eo%29 ALIGN=MIDDLE  ALT=\"highlight%28x=540%5Eo%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
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document.write( "ANOTHER WAY (with insights):<BR>\n" );
document.write( "The interior and exterior angles at each vertex are supplementary,<BR>\n" );
document.write( "meaning that their measures add up to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=180%5Eo ALIGN=MIDDLE  ALT=\"180%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, a larger interior angle would be paired with a smaller exterior angle.<BR>\n" );
document.write( "If the ratio is 6:4 for the pair of angles at every vertex,<BR>\n" );
document.write( "all interior angles in that polygon have the same measure,<BR>\n" );
document.write( "and the polygon is equiangular, just like a regular polygon<BR>\n" );
document.write( "Except for an equilateral triangles (with just 3 angles)<BR>\n" );
document.write( "and rectangles or squares (with just 4 angles),<BR>\n" );
document.write( "the exterior angles are smaller than the interior ones<BR>\n" );
document.write( "in all regular polygons.<BR>\n" );
document.write( "So, if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E ALIGN=MIDDLE  ALT=\"E\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the measure of each exterior angle in degrees,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%286%2F4%29%2AE ALIGN=MIDDLE  ALT=\"%286%2F4%29%2AE\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the measier of each interior angle,<BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%286%2F4%29E%2BE=180 ALIGN=MIDDLE  ALT=\"%286%2F4%29E%2BE=180\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Solving for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E ALIGN=MIDDLE  ALT=\"E\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%286%2F4%2B1%29E=180 ALIGN=MIDDLE  ALT=\"%286%2F4%2B1%29E=180\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2810%2F4%29E=180 ALIGN=MIDDLE  ALT=\"%2810%2F4%29E=180\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E=180%2A4%2F10 ALIGN=MIDDLE  ALT=\"E=180%2A4%2F10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E=72 ALIGN=MIDDLE  ALT=\"E=72\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The exterior angles are the change in direction<BR>\n" );
document.write( "as each vertex as you go around the polygon,<BR>\n" );
document.write( "So their measures add up to a whole turn, or <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( "If the polygon has <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > angles,<BR>\n" );
document.write( "then the sum of the measures of the exterior angles (in degrees) is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=72n=360 ALIGN=MIDDLE  ALT=\"72n=360\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=360%2F72 ALIGN=MIDDLE  ALT=\"n=360%2F72\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=5 ALIGN=MIDDLE  ALT=\"n=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, the polygon is a pentagon.<BR>\n" );
document.write( "The formula for the sum of interior angles for a polygon with<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > sides (and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > angles) is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28n-2%29%2A180%5Eo ALIGN=MIDDLE  ALT=\"%28n-2%29%2A180%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=5 ALIGN=MIDDLE  ALT=\"n=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > , that sum is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285-2%29%2A180%5Eo=3%2A180%5Eo=highlight%28540%5Eo%29 ALIGN=MIDDLE  ALT=\"%285-2%29%2A180%5Eo=3%2A180%5Eo=highlight%28540%5Eo%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
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