Parallel lines. Both have the same slope, or both have undefined slopes. A line with an undefined slope is a valid defined line.

 

We say a similar thing about perpendicular lines. If one line has undefined slope, and the other line has 0 slope, the lines are perpendicular.

 

The textbook ignores undefined slopes. This may do more harm than good. Students must understand the difference between division by zero, which is undefined, and infinity, which is very well defined (the length of a line). The undefined slope is not an infinite slope.

 

Perpendicular lines have slopes that are the negative reciprocal of each other. We can understand the negative sign by if one line has a positive slope, so that it is increasing, the other must be decreasing, and so have a negative slope.

 

To prove this, rotate a line by 90°. The equation y = mx is x = y/m. When we rotate the axis, we have to flip the x axis to get it back, and so our equation becomes x =- y/m.