Geometry: Lines
Angles between lines
We have seen the slope of a line. We can use the slope to determine the angle of the line with the #x#axis.
Angle of inclination
The angle of inclination of a line #k# is the acute or right angle that #k# makes with the #x#axis.
A line with positive slope has a positive angle of inclination and one with a negative slope has a negative angle of inclination. Therefore we have for an angle of inclination #\blue{\alpha}# in radians that \[\frac{\pi}{2} \lt \blue{\alpha} \leq \frac{\pi}{2}\]We have the formula
\[\tan(\blue{\alpha})=\text{slope}\]
Using the angle of inclination of a line we can calculate the angle between two intersecting lines.
Angle between two lines
Stepbystep 
Example 

We calculate the angle between lines #\blue l# and #\green k#. 
#\blue l: y=\sqrt{3}x+1# #\green k: \sqrt{3}x+3y=2# 

Step 1  Calculate the slopes of line #\blue l# and line #\green k#. 
#rc_{\blue l}=\sqrt{3}# #rc_{\green{k}}=\frac{\sqrt{3}}{3}# 
Step 2  Calculate the angle of inclination #\alpha# of the two lines. 
#\alpha_{\blue{l}}=\frac{\pi}{3}# #\alpha_{\green{k}}=\frac{\pi}{6}# 
Step 3 
Calculate the angle between lines #\blue l# and #\green k#. If #\alpha_{\blue l} \gt \alpha_{\green{k}}#, then:

#\frac{\pi}{3}\frac{\pi}{6}=\frac{\pi}{2}# 
The slope of the line #y={{x}\over{\sqrt{3}}}5# is equal to #{{1}\over{\sqrt{3}}}#.
Now for the angle #\alpha# it holds that #\tan(\alpha)={{1}\over{\sqrt{3}}}#.
This means that #\alpha={{\pi}\over{6}}#.
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