Thread: Parallel Proof

Prove that nonvertical lines are parallel if and only if their slopes are equal.

Lines can be written in the form , where is the gradient and is the intercept.

So we could write the equation of two lines as

and .


You should know that parallel lines never cross. These two lines will cross when their equations are equal. So







.


Obviously there is no solution when the denominator is , in other words, where .


So the only time when there will not be a point of intersection is when the two gradients are equal.

In other words, the lines are parallel when their gradients are equal.