Angle between Two Lines

The angle between two lines refers to the acute angle formed at their intersection. If the lines are not parallel or identical, they will form two pairs of vertical angles. We typically refer to the smaller (acute) angle.

The following diagram gives the formula to find the angle between two lines. Scroll down the page for more examples and solutions.

 

Angle Between Two Lines in 2D (Using Slopes)
If the lines are given in the form y = m₁x + c₁ and y = m₂x + c₂, where m₁ and m₂ are their slopes.

The angle θ between two lines with slopes m₁ and m₂ is given by:

\(tan\theta=\left|\frac{m_{1}-m_{2}}{1+m_{1}m_{2}}\right|\)

Then,

\(\theta=arctan\left|\frac{m_{1}-m_{2}}{1+m_{1}m_{2}}\right|\)

Note: This formula applies when 1 + m₁m₂ ≠ 0. If 1 + m₁m₂ = 0 (i.e. m₁m₂ = -1), the lines are perpendicular, and θ = 90° or \(\frac{\pi}{2}\)

How to find the angle between two straight lines?
Examples:

1. Find the acute angle between y = 2x + 1 and y = -3x - 2 to the nearest degree.
   
2. Find the acute angle between 3x - 2y + 7 = 0 and 2y + 4x - 3 = 0 to the nearest degree.
   
3. Find the acute angle between y = x + 3 and y = -3x + 5 to the nearest degree.
   

Finding Angle Between 2 Lines
How to find the angle between two lines using the formula?

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