document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=756594>Question 756594</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The 3 angles of a triangular flag are x, 4 times the complement of x, one-third the supplement of x. Find the angle measures </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #460307 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20855)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=460307\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> let angles be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha ALIGN=MIDDLE  ALT=\"alpha\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=beta ALIGN=MIDDLE  ALT=\"beta\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=gamma ALIGN=MIDDLE  ALT=\"gamma\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "if we have a triangle, we know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha%2Bbeta%2Bgamma=180 ALIGN=MIDDLE  ALT=\"alpha%2Bbeta%2Bgamma=180\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > angles of a triangular flag are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha=x ALIGN=MIDDLE  ALT=\"alpha=x\"   DISABLED_style= \"max-width:90%; height:auto;\" >, \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=beta ALIGN=MIDDLE  ALT=\"beta\"   DISABLED_style= \"max-width:90%; height:auto;\" > equal <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > times the complement of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and we know that complementary angles are angles whose measures sum to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=90 ALIGN=MIDDLE  ALT=\"90\"   DISABLED_style= \"max-width:90%; height:auto;\" >°, then the complement of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=90-x ALIGN=MIDDLE  ALT=\"90-x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "so, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=beta=4%2890-x%29 ALIGN=MIDDLE  ALT=\"beta=4%2890-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "and if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=gamma ALIGN=MIDDLE  ALT=\"gamma\"   DISABLED_style= \"max-width:90%; height:auto;\" > equal to one-third the supplement of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=180-x ALIGN=MIDDLE  ALT=\"180-x\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=gamma=%281%2F3%29%28180-x%29 ALIGN=MIDDLE  ALT=\"gamma=%281%2F3%29%28180-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "now substitute all in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha%2Bbeta%2Bgamma=180 ALIGN=MIDDLE  ALT=\"alpha%2Bbeta%2Bgamma=180\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B4%2890-x%29%2B%281%2F3%29%28180-x%29=180 ALIGN=MIDDLE  ALT=\"x%2B4%2890-x%29%2B%281%2F3%29%28180-x%29=180\"   DISABLED_style= \"max-width:90%; height:auto;\" >....solve for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B360-4x%2B60-x%2F3=180 ALIGN=MIDDLE  ALT=\"x%2B360-4x%2B60-x%2F3=180\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=420-3x-x%2F3=180 ALIGN=MIDDLE  ALT=\"420-3x-x%2F3=180\"   DISABLED_style= \"max-width:90%; height:auto;\" >...both sides multiply by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1260-9x-x=540 ALIGN=MIDDLE  ALT=\"1260-9x-x=540\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1260-10x=540 ALIGN=MIDDLE  ALT=\"1260-10x=540\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1260-540=10x ALIGN=MIDDLE  ALT=\"1260-540=10x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=720=10x ALIGN=MIDDLE  ALT=\"720=10x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=720%2F10=x ALIGN=MIDDLE  ALT=\"720%2F10=x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=72=x ALIGN=MIDDLE  ALT=\"72=x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "now find angles\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha=x ALIGN=MIDDLE  ALT=\"alpha=x\"   DISABLED_style= \"max-width:90%; height:auto;\" > => <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28alpha=72%29 ALIGN=MIDDLE  ALT=\"highlight%28alpha=72%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=beta=4%2890-x%29 ALIGN=MIDDLE  ALT=\"beta=4%2890-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > => <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=beta=4%2890-72%29 ALIGN=MIDDLE  ALT=\"beta=4%2890-72%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > => <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28beta=72%29 ALIGN=MIDDLE  ALT=\"highlight%28beta=72%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=gamma=%281%2F3%29%28180-x%29 ALIGN=MIDDLE  ALT=\"gamma=%281%2F3%29%28180-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > => <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=gamma=%281%2F3%29%28180-72%29 ALIGN=MIDDLE  ALT=\"gamma=%281%2F3%29%28180-72%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > => <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28gamma=36%29 ALIGN=MIDDLE  ALT=\"highlight%28gamma=36%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "check the sum:\r<BR>\n" );
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document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha%2Bbeta%2Bgamma=180 ALIGN=MIDDLE  ALT=\"alpha%2Bbeta%2Bgamma=180\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=72%2B72%2B36=180 ALIGN=MIDDLE  ALT=\"72%2B72%2B36=180\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=180=180 ALIGN=MIDDLE  ALT=\"180=180\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );