**Theorem 2.6**: If two
lines are
parallel, then all of
the points on one
line lie on the
same side of the other
line.

**Proof**: If the two
lines are
vertical, then they
have equations

by Theorem 1.3. If one substitutes the coordinates of any point on one line into the equation for the second line, one will get a constant answer, so all of the points on one line satisfy the same inequality for the second line, and by definition, they will all be on the same side of the line.

If the lines are not vertical, then since they are parallel, by definition, their slopes are the same, and by Theorem 1.2, we can assume that their equations look like

or

so again, if you substitute the coordinates of any point on one line into the equation for the second line, you will always get the same value on the left side, so all of the points will satisfy the same inequality for the second line.