Angle between Two Straight Lines

We will learn how to find the angle between two straight lines.

The angle θ between the lines having slope m$_{1}$ and m$_{2}$ is given by tan θ = ± $\frac{m_{2} - m_{1}}{1 + m_{1} m_{2}}$

Let the equations of the straight lines AB and CD are y = m$_{1}$ x + c$_{1}$ and y = m$_{2}$  x + c$_{2}$  respectively intersect at a point P and make angles θ1 and θ2 respectively with the positive direction of x-axis.

Let ∠APC = θ is angle between the given lines AB and CD.

Clearly, the slope of the line AB and CD are m$_{1}$  and m$_{2}$  respectively.

Then, m$_{1}$  = tan θ$_{1}$  and m$_{2}$  = tan θ$_{2}$

Now, from the above figure we get, θ$_{2}$  = θ + θ$_{1}$  

⇒ θ = θ$_{2}$  - θ$_{1}$

Now taking tangent on both sides, we get,

tan θ = tan (θ$_{2}$  - θ$_{1}$)

⇒ tan θ = $\frac{tan θ_{2} - tan θ_{1}}{1 + tan θ_{1} tan θ_{2}}$, [Using the formula, tan (A + B) = $\frac{tan A - tan B}{1 + tan A tan B}$

⇒ tan θ = $\frac{m_{2} - m_{1}}{1 + m_{1} m_{2}}$, [Since, m$_{1}$  = tan θ$_{1}$  and m$_{2}$  = tan θ$_{2}$]

Therefore, θ = tan$^{-1}$$\frac{m_{2} - m_{1}}{1 + m_{1} m_{2}}$

Again, the angle between the lines AB and CD be ∠APD = π - θ since ∠APC = θ

Therefore, tan ∠APD = tan (π - θ) = - tan θ = - $\frac{m_{2} - m_{1}}{1 + m_{1} m_{2}}$

Therefore, the angle θ between the lines AB and CD is given by,

tan θ = ± $\frac{m_{2} - m_{1}}{1 + m_{1} m_{2}}$

⇒ θ =  tan$^{-1}$(±$\frac{m_{2} - m_{1}}{1 + m_{1} m_{2}}$) 

Notes:    

(i) The angle between the lines AB and CD is acute or obtuse according as the value of $\frac{m_{2} - m_{1}}{1 + m_{1} m_{2}}$ is positive or negative.

(ii) The angle between two intersecting straight lines means the measure of the acute angle between the lines.

(iii) The formula tan θ = ± $\frac{m_{2} - m_{1}}{1 + m_{1} m_{2}}$ cannot be used to find the angle between the lines AB and CD, if AB or CD is parallel to y-axis. Since the slope of the line parallel to y-axis is indeterminate.

Solved examples to find the angle between two given straight lines:

1. If A (-2, 1), B (2, 3) and C (-2, -4) are three points, fine the angle between the straight lines AB and BC.

Solution:

Let the slope of the line AB and BC are m$_{1}$ and m$_{2}$ respectively.

Then,

m$_{1}$ = $\frac{3 - 1}{2 - (-2)}$ = $\frac{2}{4}$= ½ and

m$_{2}$ = $\frac{-4 - 3}{-2 - 2}$= $\frac{7}{4}$

Let θ be the angle between AB and BC. Then,

tan θ = |$\frac{m_{2} - m_{1}}{1 + m_{1} m_{2}}$| = |$\frac{\frac{7}{4} - \frac{1}{2}}{1 + \frac{7}{4}\cdot \frac{1}{2}}$| = |$\frac{\frac{10}{8}}{\frac{15}{8}}$|= ±$\frac{2}{3}$.

⇒ θ = tan$^{-1}$($\frac{2}{3}$), which is the required angle.

2. Find the acute angle between the lines 7x - 4y = 0 and 3x - 11y + 5 = 0.

Solution:  

First we need to find the slope of both the lines.

7x - 4y = 0     

⇒ y = $\frac{7}{4}$x

Therefore, the slope of the line 7x - 4y = 0 is $\frac{7}{4}$

Again, 3x - 11y + 5 = 0     

⇒ y = $\frac{3}{11}$x + $\frac{5}{11}$

Therefore, the slope of the line 3x - 11y + 5 = 0 is = $\frac{3}{11}$

Now, let the angle between the given lines 7x - 4y = 0 and 3x - 11y + 5 = 0 is θ

Now,

tan θ = | $\frac{m_{2} - m_{1}}{1 + m_{1} m_{2}}$| = ±$\frac{\frac{7}{4} - \frac{3}{11}}{1 + \frac{7}{4}\cdot \frac{3}{11}}$ = ± 1

Since θ is acute, hence we take, tan θ = 1 = tan 45°

Therefore, θ = 45°

Therefore, the required acute angle between the given lines is 45°.

● The Straight Line

• Straight Line
• Slope of a Straight Line
• Slope of a Line through Two Given Points
• Collinearity of Three Points
• Equation of a Line Parallel to x-axis
• Equation of a Line Parallel to y-axis
• Slope-intercept Form
• Point-slope Form
• Straight line in Two-point Form
• Straight Line in Intercept Form
• Straight Line in Normal Form
• General Form into Slope-intercept Form
• General Form into Intercept Form
• General Form into Normal Form
  
• Point of Intersection of Two Lines
  
• Concurrency of Three Lines
• Angle between Two Straight Lines
• Condition of Parallelism of Lines
• Equation of a Line Parallel to a Line
• Condition of Perpendicularity of Two Lines
• Equation of a Line Perpendicular to a Line
• Identical Straight Lines
• Position of a Point Relative to a Line
• Distance of a Point from a Straight Line
• Equations of the Bisectors of the Angles between Two Straight Lines
• Bisector of the Angle which Contains the Origin
• Straight Line Formulae
• Problems on Straight Lines
• Word Problems on Straight Lines
• Problems on Slope and Intercept

11 and 12 Grade Math 

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