## To understand why the slope of a horizontal line is zero and the slope of a vertical line is undefined, we need to look at the definition of slope and how it is calculated.

The slope of a line measures the rate at which it is ascending or descending. It represents the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line.

When we have a horizontal line, it means that the line is parallel to the x-axis and does not have any vertical change. In other words, as we move along the line horizontally, the y-coordinate remains the same. If there is no vertical change (or rise), the ratio of the vertical change to any given horizontal change (run) will always be zero. Therefore, the slope of a horizontal line is zero.

On the other hand, a vertical line is perpendicular to the x-axis and does not have any horizontal change. As we move up or down along the line, the x-coordinate remains constant. In this case, there is no horizontal change (or run). When we try to calculate the slope using the formula, which is the ratio of the vertical change to the horizontal change, we end up dividing the vertical change by zero. Division by zero is undefined in mathematics, meaning it does not produce a meaningful result. Therefore, the slope of a vertical line is considered undefined.

In summary, a horizontal line has a slope of zero because there is no vertical change, and a vertical line has an undefined slope because there is no horizontal change.