document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1203322>Question 1203322</A>: <I><SPAN STYLE=\"word-break: break-all;\"> derive the condition for two lines with slope m1 and m2 to be parallel and perpendicular using the angel between two lines formula. </SPAN>\n" );
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document.write( "theta = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta ALIGN=MIDDLE  ALT=\"theta\"   DISABLED_style= \"max-width:90%; height:auto;\" > = angle between two lines\r<BR>\n" );
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document.write( "One formula to use is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28theta%29+=+abs%28%28m%5B1%5D-m%5B2%5D%29%2F%281+%2B+m%5B1%5Dm%5B2%5D%29%29 ALIGN=MIDDLE  ALT=\"tan%28theta%29+=+abs%28%28m%5B1%5D-m%5B2%5D%29%2F%281+%2B+m%5B1%5Dm%5B2%5D%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Parallel lines will have theta = 0. <BR>\n" );
document.write( "This leads to tan(theta) = tan(0) = 0.<BR>\n" );
document.write( "The right hand side is only zero when the numerator is zero. <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5B1%5D-m%5B2%5D+=+0 ALIGN=MIDDLE  ALT=\"m%5B1%5D-m%5B2%5D+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > which rearranges to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5B1%5D+=+m%5B2%5D ALIGN=MIDDLE  ALT=\"m%5B1%5D+=+m%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Therefore, parallel lines have equal slopes.<BR>\n" );
document.write( "For example, the lines y = 3x+5 and y = 3x+7 are parallel. Each has slope of 3.\r<BR>\n" );
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document.write( "Perpendicular lines will involve theta = 90 degrees.<BR>\n" );
document.write( "Use a unit circle to determine that tan(90) is undefined.<BR>\n" );
document.write( "When it comes to \"undefined\", it means we have 0 in the denominator.<BR>\n" );
document.write( "Division by zero is not allowed.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+%2B+m%5B1%5Dm%5B2%5D+=+0 ALIGN=MIDDLE  ALT=\"1+%2B+m%5B1%5Dm%5B2%5D+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > leads to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5B2%5D+=+-1%2F%28m%5B1%5D%29 ALIGN=MIDDLE  ALT=\"m%5B2%5D+=+-1%2F%28m%5B1%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which means we take the negative reciprocal of m1 to get m2, and vice versa.<BR>\n" );
document.write( "An example pair of perpendicular slopes: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5B1%5D+=+5%2F7 ALIGN=MIDDLE  ALT=\"m%5B1%5D+=+5%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5B2%5D+=+-7%2F5 ALIGN=MIDDLE  ALT=\"m%5B2%5D+=+-7%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Note that any pair of perpendicular slopes multiply to -1. Neither slope can be zero or undefined.<BR>\n" );
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