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Using Slopes to Find the Angle between Lines

Date: 11/02/2005 at 18:58:29
From: Tom
Subject: Acute Angle Measure

What formula do you use to find the acute angle between two 
intersecting lines?  I know there is a formula, but I just can't find it.

Date: 11/02/2005 at 20:12:38
From: Doctor Peterson
Subject: Re: Acute Angle Measure

Hi, Tom.

I'll assume that you know the equations of the two lines, from which 
you can get their slopes.  Suppose they are m1 and m2.  Since the
slope of a line is the tangent of its angle of elevation, the angle
between the lines is the difference in the two angles.  Thus, if theta
is the angle between, and A and B are the angles of elevation of the
two lines,

                           tan(A) - tan(B)     m1 - m2
 tan(theta) = tan(A - B) = ---------------- = ---------
                           1 + tan(A)tan(B)   1 + m1*m2

Here I used the identity

  tan(x-y) = [tan(x) - tan(y)]/[1 + tan(x)tan(y)]

found in our FAQ 

So the angle you are looking for is the inverse tangent of 
(m1 - m2)/(1 + m1*m2).

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum 
Associated Topics:
High School Trigonometry

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