Theorems 1.1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15

**Theorem
1.1**: If (*x*_{1}, *y*_{1}) and (*x*_{2},
*y*_{2}) are any two points on the line whose equation is *y* =
*mx* + *b*, then the slope between them is *m*.^{1}

**Theorem 1.2**:
A vertical line has infinite
slope^{2}. A line is
not vertical if and only if its equation can be written in the form

**Theorem
1.3**: There is a uniquely determined line going through a
given point with a given slope. If the given point is (*x*_{1},
*y*_{1}), and the line is
vertical^{3}, then the
equation is

If the line is not vertical, then the equation can be written as

which can be simplified to

where

**Theorem
1.4**: Two points uniquely determine a line. If the points are
(*x*_{1},* y*_{1}) and (*x*_{2}, *y*_{2}) then the equation is

if the two points are vertical, and if not then it is of the form

where

and

**Theorem
1.5**: If (*x*_{1}, *y*_{1}) and (*x* _{2},
*y*_{2}) are points on the line whose equation is * y* = *mx* + *b * then the distance between them
is^{T1}

**Theorem
1.6**: If two distinct lines are parallel, then they do not
meet. If they are not parallel, then they meet at a uniquely
determined point. If one of the lines is vertical, its equation will
be of the form

If the other one is not vertical, it will have an equation of the form

and the point of intersection will be

If neither of the lines are vertical, they will have equations of the form

The coordinates of the point of intersection
^{4} are

and

**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.^{5}

**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.

**Theorem
1.9**: Given a line whose equation is

and a point whose coordinates are

the equation of the unique line through the point, perpendicular to the line, is

**Theorem
1.10**: Given a line whose equation is

and a point whose coordinates are

the coordinates of the foot of the point in the line are

**Theorem
1.11**: The perpendicular distance from the point

to the line

is

**Theorem
1.12**: Parallel lines stay the same distance apart. If the
equations of the lines are

then the perpendicular distance from any point on one line to the
other line is^{T2}

If the lines are vertical, then their equations are

and the distance between the lines is

**Theorem
1.13**: A line is parallel to one of two parallel lines if and
only if it is parallel to the other.

**Theorem
1.14**: If a line is perpendicular to one of two parallel lines
then it is perpendicular to the other.

**Theorem
1.15**: If two lines are perpendicular to the same line then
they are parallel.

**Theorem 1.16:** If a line is parallel to one of two perpendicular lines, it is perpendicular to the other.