# Parallel and Perpendicular Lines

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..

## 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 m1 = m2.

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 a1x+b1y+c1=0 and a2x+b2y+c2=0 are parallel.

Then the slopes of these lines must be equal. Therefore, we have -a1/b1 = -a2/b2 or a1/a2 = b1/b2

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

Okay. About perpendicular lines..

## 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 m1m2=-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 a1x+b1y+c1=0 and a2x+b2y+c2=0 are perpendicular, then -a1/b1 x -a2/b2 = -1, or a1a2 + b1b2 = 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

1. Two lines, whose slopes are m1 and m2 are parallel if m1 = m2
2. Two lines, whose slopes are m1 and m2 are perpendicular if m1.m2 = -1
3. Two lines, whose equations are a1x + b1y + c= 0 and a2x + b2y + c= 0 parallel if a1/a2 = b1/b2
4. Two lines, whose equations are a1x + b1y + c= 0 and a2x + b2y + c= 0 perpendicular if a1a2 + b1b2 = 0.