If and are two points on a line, the slope is given by

.

A horizontal line has slope zero, because the rise is zero. A vertical line has infinite slope, because the run is zero. A line at a 45° angle has slope 1, because the rise and run are equal.

You can rotate the entire figure through 360°. You will find that the lines remain parallel and their slopes, although changing with rotation, remain equal.

Consider two parallel lines. Why do they have the same slope?

Let be a triangle with on one line and a right angle at , and be a triangle with on the other line and a right angle at . Since the two lines are parallel, they make the same angle with the axis, and so and are similar. Therefore their sides are proportional: and for some number . Then , so the slopes are equal.

Some notation: in math, "" (the Greek letter capital delta), often stands for "difference". So is the difference in the coordinates, while is the difference in the coordinates. So another way to write the slope is as .

Finally, the slope is , by the definition of tangent as opposite over adjacent.