document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=703979>Question 703979</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I need some help. Can someone please assist?\r<BR>\n" );
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document.write( "Two lines in a plane can intersect forming four angles, (some may have the same measure). What is the greatest number of angles we can form using three lines?\r<BR>\n" );
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document.write( "AND:\r<BR>\n" );
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document.write( "The same question using four lines.\r<BR>\n" );
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document.write( "Bless your heart for the help. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #433943 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=433943\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I thought hard too, and this is what I came up with.<BR>\n" );
document.write( "At one crossing of 2 lines, there are four angles, and the wording hints that we should count all four.<BR>\n" );
document.write( " <BR>\n" );
document.write( "With 3 lines, each line has at most <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3-1=2 ALIGN=MIDDLE  ALT=\"3-1=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > crossings<BR>\n" );
document.write( "(less if one or both of the other lines are parallel).<BR>\n" );
document.write( "If we multiply 3 (lines) times <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > crossings per line,<BR>\n" );
document.write( "you get at most 6 crossings, but each crossing is counted twice,<BR>\n" );
document.write( "one time from the point of view of one of the crossing lines,<BR>\n" );
document.write( "and another time from the point of view of the other crossing line.<BR>\n" );
document.write( "So there are really at most <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%2F2=3 ALIGN=MIDDLE  ALT=\"6%2F2=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > crossings<BR>\n" );
document.write( "(less if all 3 lines were to cross together at the same point).<BR>\n" );
document.write( "That would mean at most <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2A4=highlight%2812%29 ALIGN=MIDDLE  ALT=\"3%2A4=highlight%2812%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > angles.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C0%2C10%2C0%2C10%2C%0D%0Aline%283%2C0%2C3%2C10%29%2Cline%280%2C7%2C10%2C7%29%2Cline%280%2C0%2C10%2C10%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "With four lines there are at most <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2A%284-1%29%2F2=6 ALIGN=MIDDLE  ALT=\"4%2A%284-1%29%2F2=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > crossings.<BR>\n" );
document.write( "That would give you at most <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%2A4=highlight%2824%29 ALIGN=MIDDLE  ALT=\"6%2A4=highlight%2824%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > angles.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C0%2C10%2C0%2C10%2C%0D%0Aline%283%2C0%2C3%2C10%29%2Cline%280%2C7%2C10%2C7%29%2C%0D%0Aline%280%2C0%2C10%2C10%29%2Cline%280%2C8%2C10%2C0%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "EXTRA:<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;\" > lines could have at most <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%28n-1%29%2F2 ALIGN=MIDDLE  ALT=\"n%28n-1%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > crossings and<BR>\n" );
document.write( "that would make a maximum of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4n%28n-1%29%2F2=2n%28n-1%29 ALIGN=MIDDLE  ALT=\"4n%28n-1%29%2F2=2n%28n-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > angles\r<BR>\n" );
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document.write( "NOTE:<BR>\n" );
document.write( "Maybe you should try the artofproblemsolving.com forums.<BR>\n" );
document.write( "Some of the people there are future Math Olympics competitors.  </SPAN>\n" );
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