Proof: To show that the relation " is parallel to" is an equivalence relation we must show that it is reflexive, symmetric, and transitive. [br]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. [br]If a line [i]l[/i] is parallel to another line [i]m[/i], we know that [i]m [/i]is also parallel to [i]l[/i]. [size=100]That is, the relation "is parallel to" is symmetric. [br][/size]If a line [i]l [/i]is parallel to a line [i]m[/i] and [i]m[/i] is parallel to line [i]n, [/i]then we know that [i]l [/i]is parallel to [i]n[/i]. Thus, "is parallel to" is transitive.