I recently talked about finding out the angle between two lines. A useful application of this formula is to determine whether two lines are parallel or perpendicular. Here’s how..

## 1. Parallel Lines

Suppose two lines are parallel. Then the angle between them must be 0. That is, θ=0, which makes tanθ = 0.

Therefore, our formula tanθ = \(|\frac{m_1 – m_2}{1+m_1m_2}|\), leads us to **m _{1} = m_{2}**.

In other words, the slopes of the two parallel lines must be equal. This seems obvious, as two parallel lines must make the same angle with a transversal (i.e. the X-axis)

Conversely, if the slopes of two lines are equal, then they must be parallel.

There’s one small result which you might want to remember:

Suppose the lines a_{1}x+b_{1}y+c_{1}=0 and a_{2}x+b_{2}y+c_{2}=0 are parallel.

Then the slopes of these lines must be equal. Therefore, we have -a_{1}/b_{1} = -a_{2}/b_{2} or **a _{1}/a_{2} = b_{1}/b_{2}**

That is, the coefficients of x and y are proportional (the equations must be in the form as shown).

Okay. About perpendicular lines..

## 2. Perpendicular Lines

In this case θ = 90°, or cotθ = 0. Again, using our formula we have, \(|\frac{1+m_1m_2}{m_1 – m_2}|\) = 0

This leads us to **m _{1}m_{2}=-1**. That is, the product of the slopes of two perpendicular lines must be equal to -1

Conversely, if the product of the slopes of two lines equals -1, then the lines must be perpendicular.

And, analogous to the previous result, if the lines a_{1}x+b_{1}y+c_{1}=0 and a_{2}x+b_{2}y+c_{2}=0 are perpendicular, then -a_{1}/b_{1} x -a_{2}/b_{2 }= -1, or **a _{1}a_{2} + b_{1}b_{2} = 0**.

That’s it for now. I’ll talk a bit more about equations of parallel and perpendicular lines while covering examples (so don’t miss them).

## Lesson Summary

- Two lines, whose slopes are m
_{1}and m_{2}are parallel if m_{1}= m_{2} - Two lines, whose slopes are m
_{1}and m_{2}are perpendicular if m_{1}.m_{2}= -1 - Two lines, whose equations are a
_{1}x + b_{1}y + c_{1 }= 0 and a_{2}x + b_{2}y + c_{2 }= 0 parallel if a_{1}/a_{2 }= b_{1}/b_{2} - Two lines, whose equations are a
_{1}x + b_{1}y + c_{1 }= 0 and a_{2}x + b_{2}y + c_{2 }= 0 perpendicular if a_{1}a_{2}+ b_{1}b_{2}= 0.