 Slope gives us a way to quantify this geometric idea of steepness and what that means is that we can start to apply all of the tools of algebra to many of the problems in geometry. This does require that we find the algebraic equivalent of geometric ideas. So let's begin with a couple of simple ones like horizontal and vertical. So suppose I have a line. If I want to talk about the slope of the line, I'll need two points. So one of these points will call, I don't know, X1, Y1. And maybe I'll take another point, X2, Y2. Now the use of the variables means that while these points have definite coordinates, I'm not going to commit myself to what those coordinates are, at least not yet. But if the line is horizontal, I do have to make a commitment. So let's make this line horizontal. Nope, too far. So now we have a horizontal line and now we have to commit ourselves. If the line is horizontal, the vertical distances from the x-axis have to be the same. So Y2 must be Y1. So the slope of the horizontal line will be, if you remember slope is rise over run, the rise is zero and the run is, well, who cares, it's going to be zero. Or you could drag in the slope formula and calculate it. Since Y2 is equal to Y1, then the numerator is going to be zero and who cares what the denominator is. And this proves the following result, the slope of a horizontal line is zero. What if we have a vertical line? So again, I have one point, X1, Y1, and another point, X2, Y2. Again, I'm not committing myself, except I have to. If the line is vertical, then X1 has to be equal to X2. And that's because I have to go over some distance and I can go up different amounts, but I can't go over any further or any less and still get a vertical line. Because the line is vertical, I know that X1 is equal to X2. So this denominator, X2 minus X1 is, and we got a problem because we're trying to divide by zero, this is undefined. So this proves another important result, the slope of a vertical line is undefined. What about parallel lines? Suppose we have two parallel lines. We can find the slope of one line by choosing two points and determining the rise and the run between the points. We can find the slope of the other line in the same way, but if we choose points on the other line so that one set of points is directly above the other set of points, we see when the runs are the same. The rises also have to be the same. Otherwise, the two lines won't be parallel. And this leads to the following theorem. Parallel lines have the same slope. What if two lines are perpendicular? So let's consider that the slope of one line is going to be m equals rise over run. Now, if the lines are supposed to be perpendicular, then the rise of the second line must be negative the run of the other. And similarly, the run of this line must be the same as the rise of the other. So that means if the lines are perpendicular, the slope of the other line must be m prime minus run over rise. And this leads to the following result. The slopes of perpendicular lines are negative reciprocals. For example, let's take a line that passes through the points 2, 1, and 5, negative 3. And let's find the slope of a line parallel and the slope of a line perpendicular. Well, in order to find that, we need to find the slope of the line. Negative 4 thirds. Since parallel lines have the same slope, then the slope of a parallel line will be the same negative 4 thirds. What about a perpendicular line? The slopes of perpendicular lines will be negative reciprocals. So the slope of the perpendicular line will be negative 1 over the slope. So that's negative 1 over negative 4 thirds. So we'll invert and multiply. And we get the slope of the perpendicular line 3 fourths.