Proof Exercise 7.3.18
In the Euclidean plane, parallelism is an equivalence relation for lines.
Proof: To show that the relation " is parallel to" is an equivalence relation we must show that it is reflexive, symmetric, and transitive.
A line in an affine plane, like the Euclidean plane, is said to be parallel to itself. Thus the relation "is parallel to" is reflexive.
If a line l is parallel to another line m, we know that m is also parallel to l. That is, the relation "is parallel to" is symmetric.
If a line l is parallel to a line m and m is parallel to line n, then we know that l is parallel to n. Thus, "is parallel to" is transitive.