Question 1207909
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Prove that if two nonvertical lines have slopes whose product is -1, then the lines are perpendicular.
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<pre>
The slope of a straight line is a value of the tangent function of the angle between this line and x-axis.

So, we can write  {{{m[1]}}} = {{{tan(alpha)}}},  {{{m[2]}}} = {{{tan(beta)}}},  where  {{{alpha}}}  and  {{{beta}}}  are the angles 
that these straight line form with x-axis.


The angle between these straight lines is  {{{beta-alpha}}}.  Let's calculate the tangent of this angle


    {{{tan(beta-alpha)}}} = {{{(tan(beta)-tan(alpha))/(1+tan(beta)*tan(alpha))}}}.


The denominator  {{{1+tan(beta)*tan(alpha)}}}  is  {{{1 + m[1]*m[2]}}} = 1 + (-1) = 0,  so  {{{tan(beta-alpha)}}}  is not defined.


But the tangent function of an angle is not defined if and only if the angle is +/- 90 degrees.


Thus we proved that the angle between our lines is +/- 90 degrees.


It means that the lines are perpendicular.
</pre>

At this point, the proof is complete and the problem is solved.