The Angle Between Two Lines

Angle Between Two Lines Formula

In a plane when two non-parallel straight lines intersect each other, it forms two opposite vertical angles. One of them is acute i.e. less than 90 degrees and the other one is obtuse that is more than 90 degrees. We will be calculating the angle between two non-perpendicular lines as the angle between two perpendicular lines would be 90 degrees and that of parallel lines will be zero. Finding the value of these angles depend on the slopes formed by the intersecting lines. 


Formula to Find the Angle Between Two Lines:

Consider two nonparallel lines which have slopes m1 and m2 and Ɵ is the angle between the lines, then the formula for finding the angle between the two lines would be: 

tan Ɵ=(m2-m1)/(1+m1m2)


Derivation of the Formula:

Consider the below figure here we can see that we have a plane with x-axis and y-axis.

Two lines L1 and L2 are intersecting each other. Forming one acute and the other obtuse angle. Let us consider the acute angle as Ɵ. (image will be uploaded soon)

Step 1: At first, We have to show that 

            θ  =  θ2 - θ1

Step 2: For proving above, Let us consider ∆ABC

By angle sum property we can state that

θ + θ1 + x  = 180 ……..(1)

 x + θ2  = 180…………. (2) (as angle  x and θ2  forms a linear pair)

Step 3: From equation 1 and 2 we can equate them

θ + θ1 + x  = x + θ2 = 180 

Subtracting x from both sides 

θ + θ1 + x - x = x + θ2- x 

 we get

θ + θ1 =  θ2

Step 4: Subtracting θ1 from both sides

θ + θ1 - θ1  =  θ2 - θ1

We get,

θ  =  θ2 - θ1

Step 5: Applying tangent on both sides

tan θ = tan ( θ2 - θ1)

Using the formula of tangents we get 

tan θ = tan θ2 - tan θ1/1+tan θ1tanθ

From the inclination of a line, we know that tan θ = m

Hence we can substitute tan θ1 = m1 and tan θ2  = m2 we get,

tan Ɵ= |(m2-m1)/(1+m1m2)|


Note: The value of tan Ɵ will always be positive. 


If the Lines are Perpendicular

(image will be uploaded soon)

When the two lines are perpendicular angle between them will be 90° i.e Ɵ=90°

  • 1/tan Ɵ = 0

  • 1+m1m2/m1-m2 = 0

  • 1+m1m 2 =0

  • m1m2 = -1

The product of their slope is -1. It shows that the lines are perpendicular.


If the Lines are Parallel

(image will be uploaded soon)

The two lines are parallel means, the angle between them is zero Ɵ = 0°

  • i.e tan Ɵ =0

  • m1-m2/1+m1m2=0

  • m1-m2=0

  • m1=m2

The slopes are equal. It shows that the lines are parallel                 


Calculating Angle Between two Lines in Coordinate Geometry

How to Find Angle Between Two Lines

Let us consider three points are given on the x-axis and y-axis whose coordinates are given.

Consider a line whose endpoints have coordinates (x1 y1) and (x2 y2). 

The equation of the slope will be

m = y2-y1/x2-x1

m1  and m2 can be calculated by substituting this in the above formula then the values of m1 and m2 can be substituted in the formula given.

tan θ = ± (m1 – m2 ) / (1- m1*m2)

Let us understand it by solving examples.


Solved Examples:

Example 1:

If P (2, -1), Q (5, 3) and R (-2, 6) are three points, find the angle between the straight lines PQ and QR.

Solution:

The slope of PQ is given by

  • m = ( y2 – y1 ) / (x2 – x1)

  • m =( 3 – (- 1) ) / (5 – 2 ))

  • m= 4/3

Therefore, m1=4/3

The slope of QR is given by

  • m= (6 - 3 ) / (−2−5)

  • m= 3/-7

  • Therefore, m2 = 3/-7

Substituting the values of m2 and m1 in the formula for the angle between two lines we get,

  • tan θ = ± (m2 – m1 ) / (1- m1 m2)

  • tan θ = ± (3/-7) – (4/3) ) / (1- (3/-7)(4/3))

  • tan θ = ± (37/33)

Therefore,  θ = tan -1 (37 / 33)

Example 2:

 Find the angle between the following two lines.

Line 1: 4x -3y = 8

Line 2: 2x + 5y = 4

Solution

Put  4x -3y = 8 into slope-intercept form so you can clearly identify the slope. 

  • 4x -3y = 8

  • 3y = 4x - 8

  • y = 4x / 3 - 8/3

  • y = (4/3)x - 8/3

Put 2x + 5y = 4 into slope-intercept form so you can clearly identify the slope.

  • 2x + 5y = 4

  • 5y = -2x + 4

  • y = -2x/5 + 4/5

  • y = (-2/5)x + 4/5

The slopes are 4/3 and -2/5 or 1.33 and -0.4. It does not matter which one is m1 or m2. You will get the same answer.

Let m1  = 1.33 and m2 = -0.4

  •  tan θ = ± (m1 – m2 ) / (1+ m1*m2)

  •  tan θ = ± (1.33 - (- 0.4)) / (1- (1.33)*(-0.4))

  •  tan θ = ± (1.73) / (1- 0.532)

  •  tan θ = ± (1.73 ) / (0.468)

  •  tan θ= 3.696

θ = tan-1(3.69)

Example 3:

Find the acute angle between y = 3x+1 and y = -4x+3

Solution:

 m1= 3 and m2 = -4

  •  tan θ = ±  (m1 – m2 ) / (1+ m1*m2)

  •  tan θ = ± (3-(-4) ) / (1+ 3*-4)

  •  tan θ = ±  (7 ) / (1+(-12))

  •  tan θ = ± (7 ) / (-11)

  •  tan θ = ± (7/11)

  •  tan θ = 0.636

θ = tan-1(0.636)

FAQ (Frequently Asked Questions)

1. How do you Find the Angle Between two Straight lLnes?

Answer

When the two lines are given l1 and l2, they are intersecting each other and angle between them is Ɵ then we can find it by the formula,

tan θ = ±  (m1 – m2 ) / (1+ m1m2)

 Here m1 and m2 are the slopes of the given lines.

And slopes m1 and m2 can be found using the formula, 

 m  = y2 - y1 / x2 - x1 

where x1,  y1,  x2 and y2 are the coordinates of the points of lines.

2. To Find the Slope of a Line?

Answer:

To find the slope, you have to divide the difference of y-coordinates of 2 end-points on a line by the difference of x-coordinates of the same endpoints. 

Suppose A(x1 y1) and B(x2 ,y2) then their slope will be

 m = y2-y1 / x2-x1 

Here, x1 and x2 are x-coordinates and y1 and y2 are y coordinates on X-axis and Y-axis respectively.