Are railway lines parallel or perpendicular?

Answer 1

Railway lines are parallel. 2 lines are said to be parallel when they are contained in the same plane and do not intersect. This is the definition. That parallel lines exist is an assumption (postulate) of Euclidean geometry:

Parallel lines are like the rails of a train track, and you might think of defining them this way, as lines that are the same distance apart everywhere. The problem with this kind of definition is it assumes both tracks are straight. Though this seems an obvious possibility, when you go into the vast universe it is not that obvious. Parallel lines puzzled the best mathematicians for centuries until it was realized that we must assume they exist (you can't prove they exist from simpler postulates). The problem with parallel lines lies in the notion that the lines have infinite extent.

Euclid used a somewhat different parallel postulate in trying to avoid the notion of the infinite. He observed that when two parallel lines are intersected by a third line, called a transversal, then if you measure two angles formed by these three lines, on the same side of the transversal and between the parallels, they will add to (that is, they will be supplementary). Such angles are called same-side interior angles.Another important concept is perpendicular. By definition, two lines are perpendicular if they intersect at right angles. That is, two perpendicular lines form 4 right angles. Segments and rays can also be perpendicular. This means they intersect in at least one point, and the two lines containing them are perpendicular.

We use perpendicular segments to measure the distance from a point to a line, a point to a plane, or the distance between two parallel lines or planes. The ties of a railroad track are perpendicular to the rails and of the same length. This common length is the distance between the rails. (If parallel lines exist, then railroad tracks in space can go on forever.)

There are three theorems about perpendicular lines that you should know. We will not attempt to prove them here, but if you think about them they should be rather obvious.We can use this fact to define the distance from a point to a line: That distance is the length of a segment perpendicular to the line with the given point as one of its endpoints and the other endpoint on the line. In fact, a similar notion holds in 3 dimensions. If we have a plane and a point not on that plane, then there is only one line through the point perpendicular to the plane, and the length of the segment determined by that point and the intersection of the perpendicular line with the plane is defined as the distance from the point to.