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a slope of zero. horizontal is undefined
Horizontal lines always have a slope of zero. (i.e completely flat, level surfaces have a slope of zero). However a line does not have to have a slope of zero in order to be a line.
For vertical lines, when you try to figure out the slope, you get zero in the denominator - in other words, a division by zero.
zero is horizontal, undefined is vertical
The slope of any vertical Line is undefined because anything divided by zero is undefined.