Preliminaries from Geometry
This chapter contains definitions, axioms, and theorems from geometry which are needed for what follows. They serve to insure that the reader and the authors have a common terminology for the material which is prerequisite to the study of trigonometry, so they are, for the most part, presented as facts without formalism or proof.
1. Rays and Segments
A point P on a (straight) line divides into two half-lines, each of which is a ray with P as its only endpoint. A ray extends infinitely in one direction. Let P and Q be distinct points and be the unique line passing through them. Then the ray PQ designates the ray with P as endpoint which passes through Q. Also, the segment PQ consists of P and Q and all the points between these endpoints on the line By specifying P as the initial point and Q as the final point, the segment PQ becomes the "directed segment"
Directed segments and have the same magnitude if the lengths of the segments are equal, as in Figure 1.
If and are on parallel lines, (or are on the same line), they may have the same direction, as in Figure 2a, or they may have opposite directions, as in Figure 2b.
Figure 2a Figure 2b
If directed segments
have the same magnitude and direction as in Figure 3, this will
be denoted by the notation