Theorem 1.8: If we are given a point and a line, there is a unique line, through the point, perpendicular 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. The slope of any line perpendicular to the given line is the negative reciprocal of the slope of the given line. If we are given a point 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 whose slope is the negative reciprocal of the slope of the given line, and will be perpendicular to it by definition. For the purposes of this proof, we will consider zero and infinity to be negative reciprocals of each other.

next theorem (1.9)