# Angle of Intersection of Two Lines

Let and be two coplanar and non-parallel lines with inclination and respectively, as shown in the given diagram. The angle of intersection of lines and is the angle through which line is rotated counter-clockwise about the point of intersection so that it coincides with .

The angle is the angle of the intersection of lines and measured from to . The angle is also the angle of intersection of lines and measured from to . If the lines are not perpendicular, then one angle between them is an acute angle.

__Theorem__**:** The angle of the intersection of two non-vertical lines and from to is given by , where and are the slopes of lines and respectively.

**Proof:** Let and be two coplanar and non-parallel lines with inclination and respectively, as shown in the given diagram. It is clear from the diagram that

Since and are the inclination of lines and respectively, their slopes are , . Putting these values of , in equation (i), we have

If the lines and are perpendicular, then , and using above formula, we have

This is the condition for two lines to be perpendicular.