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

x = a1   and   x = a2

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

y = mx + b1   and   y = mx + b2

or

y - mx = b1   and   y - mx = b2

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.

next theorem (2.7)