**Theorem
1.12**: Parallel
lines stay the same
distance apart. If the
equations of the lines are

then the perpendicular distance from any point on one line to the other line is

If the lines are vertical, then their equations are

and the distance between the lines is

**Proof**: If (*x*_{1}, *y*_{1}) is a point on the
line whose equation is *y* =
*mx* + *b*_{1}, then by Theorem
1.3,

and the result follows from Theorem 1.11.

If the lines are
vertical then a
point on * x* = *a*_{1}
would have the form (*a*_{1}, *y*_{1}). The line which is
parallel to the first
vertical
line would also be
vertical, and the
line from (*a*_{1},
*y*) perpendicular
to the other vertical
line would be
horizontal, and its
equation would be

The point of intersection between this line and the vertical line

would be

The distance between
(*a*_{1}, *y*_{1}) and (*a*_{2}, *y*_{1})
would be

by a simple application of the distance formula.