# Angle between two Lines

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

- Let
*l*_{1}and*l*_{2}be the two non-vertical and non-perpendicular lines with slopes*m*_{1}and*m*_{2}respectively. Let_{1}and_{2}be their inclinations, then*m*_{1}= tan_{1}and*m*_{2}= tan_{2}. There are two angles and - between the lines*l*_{1}and*l*_{2}, given by

tan = ± (*m*_{1}-*m*_{2})(1+*m*_{1}*m*_{2})

## Illustrative Examples

### Example

Find the angle between the lines joining the points

(-1,2), (3,-5) and (-2,3), (5,0).

### Solution

Here, *m*_{1} = slope of the line joining (-1,2) and (3,-5)

= (-5-2)/(3+1) = -7/4 and

*m*_{2} = 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

- 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] - Find the angle between the lines joining the points (0,0), (2,3) and (2,-2), (3,5).
- If A(-2,1), B(2,3) and C(-2,-4) are three points, find the angle between the lines AB and BC.
- Find the angles between the lines x +1 = 0 and 3 x +y -3 = 0.
- Find the angle between the lines which make intercepts on the axes a,-b and b,-a respectively.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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

**1.**(i) 90° (ii) 45° (iii) 60° (iv) 71° 34'

**2.**25° 34'

**3.**33° 42'

**4.**30°

**5.**The acute angle is given by tan =

**6.**45°, 45°, 90°

**7.**3 x +y -17 = 0, x -3 y +11 = 0

**8.**7 x -3 y -10 = 0, 3 x +7 y +4 = 0

**9.**(2 +3) x -y -1 -2 = 0, (2 -3) x -y -1 +2 = 0

**10.**23 x -7y -9 = 0, 7x +23 y -53 = 0.

**11.**Sides are 2 x +y -4 = 0, x -2 y +3 = 0,

2 x +y -14 = 0, x -2 y +13 = 0 and

diagonals are 3 x -y -1 = 0, x +3 y -17 = 0