1. One common misconception in developments of Geometry is that one can justify the fact that the slope of a line is well defined independent of the choice of points used to compute it through the use of silmilar triangles. In this development, similar triangles are practically the last topic, where we can show that the slope of a line is well defined as the first result using simply algebra. As a result, we have that similar triangles is very heavily dependent on the fact that the slope of a line is well defined, not the other way around.
2. There is a strong tradition of using infinity in Geometry. In projective geometry one uses points and lines at infinity. We shall see that there are instances where it is most appropriate to use infinite values when we find ourselves dealing with quantities that have zero denominators. Remember, infinity is not a real number, and we will avoid arithmetic with infinity except to say that infinity is the reciprocal of sero.
3. The fact that vertical lines have infinite slope is an annoying inconvience. It means that we can't use computations involving slopes for infinite lines. Thus, if a result uses computations involving slopes, then we will have to deal with the case of vertical lines separately. Fortunately, the equation of a vertical line is actually simpler than the general equation, so the inconvenience of having to fashion a separate argument for vertical lines is ameliorated by the fact that the argument for the simpler equations of vertical lines are usually simpler than the general case.
4. Notice that if the slopes are the same, these formulas say that parallel lines meet at infinity.
5. This is one of the most commonly used equivalents to Euclid's Parallel Postulate and is known as Playfair's Postulate.