Think of a plane as a flat surface extending infinitely in all directions, one and only one line goes through two distinct points and in that plane, denoted by , which also extends infinitely.
The line segment or segment between and is the set consisting , and all points on line lying between and , denoted by , and its length denoted by . These two points also determine two rays, either starting from or . In other words, a ray is determined by its starting point, the vertex, and any other point on it.
We define lines and to be parallel if either
a) , which means the line is parallel to itself, or,
b) and does not intersect
Assumed properties we accept as facts without any further justification are called axioms. Using the definition of parallel lines, we have these axioms: