In Hilbert's Grundlagen der Geometrie (`Foundations of Geometry'; the original edition in 1899) there is the following argumentation.

Let $\alpha$ be an arbitrary plane, $a$ a line in $\alpha$ and $A$ a point in $\alpha$ which lies outside $a$ . If we draw in $\alpha$ a line $c$ which passes through $A$ and intersects $a$ and then through $A$ a line $b$ such that the line $c$ intersects the lines $a$ , $b$ with equal alternate interior angles (``unter gleichen Gegenwinkeln''), then it follows easily from the theorem on the outer angles, that the lines $a$ , $b$ have no common point, i.e., in a plane $\alpha$ one can always draw otside a line $a$ another line which does not intersect the line $a$ .

The Parallel Axiom reads now:

IV (Euclidean Axiom). Let $a$ be an arbitrary line and $A$ be a point outside $a$ : then in the plane determined by $a$ and $A$ there exists at most one line which passes through $A$ and does not intersect $a$ .

Explanation. According the the preceding text and on grounds of the Parallel Axiom we realize, that there is one and only one line which passes through $A$ and do not intersect $a$ ; that is called the parallel of $a$ through $A$ .

The Parallel Axiom means the same as the following requirement:

When two lines $a$ , $b$ in a plane do not meet a third line $c$ of the same plane, then also they do not meet each other.

The theorem on the outer angles is the following: An outer angle of a triangle is greater than both non-adjacent angles of the triangle. Using this one may indirectly justify the assertion in the first cited paragraph.

Introducing the Parallel Axiom simplifies the foundations and facilitates the construction of geometry significantly.

If we associate the Parallel Axiom to the Congruence Axioms, then we obtain easily the following well-known fact:

Theorem 31. If two parallels intersect a third line, then the corresponding angles and the alternate interior angles are congruent, and conversely: the congruence of the corresponding or alternate interior angles implies that the lines are parallel.

Bibliography

1

D. HILBERT: Grundlagen der Geometrie. Neunte Auflage, revidiert und ergänzt von Paul Bernays. B. G. Teubner Verlagsgesellschaft, Stuttgart (1962).