document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=999526>Question 999526</A>: <I><SPAN STYLE=\"word-break: break-all;\"> find the counterexample to show that each conjecture is false  for if<BR>\n" );
document.write( " <A and <B are complementary angles then they share a common side </SPAN>\n" );
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document.write( "by definition:<BR>\n" );
document.write( "Complementary Angles are 2 angles the sum of whose measures is 90°\r<BR>\n" );
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document.write( "the counterexample to show that each conjecture is false  for if<BR>\n" );
document.write( " <A and <B are complementary angles then they share a common side:<BR>\n" );
document.write( "the counterexample: adjacent angles (two angles are adjacent when they have a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=common ALIGN=MIDDLE  ALT=\"common\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+side ALIGN=MIDDLE  ALT=\"+side\"   DISABLED_style= \"max-width:90%; height:auto;\" > and a common vertex (corner point) and don't overlap)\r<BR>\n" );
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document.write( "some more examples:\r<BR>\n" );
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document.write( "a counterexample is an exception to a proposed general rule or law\r<BR>\n" );
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document.write( " for example, consider the proposition \"all students are lazy\"<BR>\n" );
document.write( "because this statement makes the claim that a certain property (laziness) holds for all students, even a single example of a diligent student will prove it false<BR>\n" );
document.write( " thus, any hard-working student is a counterexample to \"all students are lazy\"<BR>\n" );
document.write( " more precisely, a counterexample is a specific instance of the falsity of a universal quantification (a \"for all\" statement)\r<BR>\n" );
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document.write( "or\r<BR>\n" );
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document.write( "the sum of two integers is always positive: a counterexample =><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%2B3=-1 ALIGN=MIDDLE  ALT=\"-4%2B3=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "the product of two mixed numbers is never a whole number:<BR>\n" );
document.write( "a counterexample => (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+1%2F4 ALIGN=MIDDLE  ALT=\"+1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >)(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+2%2F5 ALIGN=MIDDLE  ALT=\"+2%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" >)=<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( "all four-sided figures are rectangles: a counterexample =>parallelogram with no <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=90 ALIGN=MIDDLE  ALT=\"90\"   DISABLED_style= \"max-width:90%; height:auto;\" > degree angles\r<BR>\n" );
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document.write( " </SPAN>\n" );
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