When you have two straight lines in the plane, there are two alternatives:
1. These two straight lines intersect (as shown in Figure 1a), and
2. These two straight lines do not intersect even at extended length
(as shown in Figure 1b).
Figure 1a. Intersecting
straight lines
Figure 1b. Parallel
straight lines
Definition
Two straight lines in a plane are called parallel if they do not intersect
even at extended length.
The next statement is a postulate, that is a statement, which is accepted without proof:
For a given straight line and a point in the plane outside of this straight line there is
one and only one straight line parallel to the given straight line and passing through the
given point.
Figure 2. The unique parallel
straight line passing through
the given point out the line
Theorem
If a straight line a is parallel to a straight line b and a straight line b is parallel to a straight line c,
then the straight lines a and c are parallel.
Proof
Let us suppose that straight lines a and c are not parallel. Then they have an intersection point. Let us denote as C
this intersection point of the straight lines a and c.
Then we have two straight lines, a and c, passing through the point C and parallel to the straight line b, which is
impossible due to the postulate above.
