Angle between two Lines

The angle between two non-vertical and non-perpendicular lines

• Let l₁ and l₂ be the two non-vertical and non-perpendicular lines with slopes m₁ and m₂ respectively. Let ₁ and ₂ be their inclinations, then m₁ = tan ₁ and m₂ = tan ₂. There are two angles and - between the lines l₁ and l₂, given by
  tan = ± (m₁-m₂)(1+m₁m₂)
                                

Illustrative Examples

Example

Find the angle between the lines joining the points
(-1,2), (3,-5) and (-2,3), (5,0).

Solution

Here, m₁ = slope of the line joining (-1,2) and (3,-5)
   = (-5-2)/(3+1) = -7/4 and
 m₂ = slope of the line joining (-2,3) and (5,0)
   = (0-3)/(5+2) = -3/7
Let be the acute angle between the given lines, then
tan = .
          =
Hence the acute angle between the lines is given by
   tan = 37/49

Exercise

1. Find the angle between the following pairs of lines:
   (i) 3 x -7 y +5 = 0 and 7 x +3 y -11 = 0
   (ii) 3 x +y -7 = 0 and x +2 y +9 = 0
   (iii) y = (2 -3) x +9 and y = (2 + 3) x +1
   (iv) 2 x -y +3 = 0 and x +y -2 = 0
   [Hint. (iv) It will be found that acute angle is given by tan = 3
   which gives as 71° 34', by using tables of natural tangents]
2. Find the angle between the lines joining the points (0,0), (2,3) and (2,-2), (3,5).
3. If A(-2,1), B(2,3) and C(-2,-4) are three points, find the angle between the lines AB and BC.
4. Find the angles between the lines x +1 = 0 and 3 x +y -3 = 0.
5. Find the angle between the lines which make intercepts on the axes a,-b and b,-a respectively.
6. Find the measures of the angles of the triangle whose sides lie along the lines x +y -5 = 0,  x -y +1 = 0 and y -1 = 0.
7. Find the equations of the two straight lines passing through the point (4,5) which make an acute angle of 45° with the line 2 x-y +7 = 0.
8. Find the equations of the two straight lines passing through the point (1,-1) and inclined at an angle of 45° to the line 2 x -5 y +7 = 0.
9. A vertex of an equilateral triangle is (2,3) and the equation of the opposite side is x +y +2 = 0. Find the equations of the other two sides.
10. One diagonal of a square lies along the line 8 x -15 y = 0 and one vertex of the square is at (1,2). Find the equations of the sides of the square passing through this vertex.
11. If (1,2) and (3,8) are a pair of opposite vertices of a square, find the equations of the sides and the diagonals of the square.

Answers