**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.