Theorem 1.7: If we are given a line and a point not on the line, then there is a unique line, going through a given point, parallel to the given line.

Proof: If we are given a line, then that line has a slope, even if it is infinite by Theorems 1.1 and 1.2. Any line parallel to the given line has the same slope as the given line by definition. If we are given a point not on the line and we are given the line, then we are given a point and a slope, so by Theorem 1.3, there is a unique line through the point having the same slope as the given line. Since it has the same slope as the given line, it will be parallel to it by definition.

next theorem (1.8)