If you start with two infinitely long lines, which intersect at a point that is a finite distance in front of you, and straighten them so that they are parallel, then the point of intersection will shoot off to infinity in finite time.
This may seem counterintuitive, but stuff like this happens when you have an infinitely long line and you move it around as a rigid body. For example, if you just think about one line, when you rotate it, a point on the line that is distance "x" away from you sweeps out an arc at a certain speed. Another point at distance "2x" will sweep out an arc twice as long in the same time, so it moves twice as fast. And since the line, by definition of the thought experiment, is infinite, there are points on the line arbitrarily far away from you, which sweep out arbitrarily huge distances at arbitrarily fast speeds when the line is rotated.
This has nothing to do with physical reality because physical reality doesn't contain infinitely long, infinitely rigid physical bodies that respond instantly, all along their infinite length, when you rotate them at the origin. If you point a laser in one direction, and then change the direction, it takes time for the redirected beam to spread out into space, the beam was never infinitely long (because the laser would have had to be switched on forever in order to have created an infinite beam), etc. Anything you actually do will only involve finite distances and changes that take time to travel; you don't need to bring up the curvature of space in order to explain why your thought experiment can't happen in the real world.
The eleven postulates are sufficient to prove 3.11.
Lemma 1 A line and a point not on it, two different lines in a plane, or two parallel lines define a plane.
Two points on a line and a point not on it define a plane by #7. If two lines are different there's a point on the second that's not on the first (by #6), so by the first part they define a plane. By definition two parallel lines are different lines in a plane so define it by the second part.
Lemma 2 If $a,b,t$ are different coplanar lines and $a$ is parallel to $b$ and $t$ is not parallel to $a$ then $t$ is a transversal of $a$ and $b$.
By definition $t$ intersects $a$ so call the point of intersection $A$ defining an angle $\angle at\ne 0$ (by #3). Let $S$ be a point on $b$ then $SA$ defines a line $s$ (by #6) which is a transversal of $a$ and $b$ (by definition). Then $s$ cuts off angles $\angle sb=\angle sa$ (by #10) and $\angle st\ne \angle sa$ (by #4 because they are coincident), so $t$ is not parallel to $b$ by $\angle st\ne \angle sb$ and #10, and is a transversal (by definition).
Proposition If $a,b,c$ are different lines with $a$ parallel to $b$ and $b$ parallel to $c$ then $a$ is parallel to $c$.
If the lines are coplanar then let $t$ be a line intersecting $b$, then applying Lemma 2 twice it is a common traversal of $a,b,c$. By #10 $\angle ta=\angle tb=\angle tc$ and by #11 $a$ is parallel to $c$.
If the lines are not coplanar, then let $C$ be a point on $c$. By Lemma 1 $a$ and $b$ are in a plane $\pi_1$, $b$ and $c$ are in a different plane $\pi_2$, and $a$ and $C$ are in a plane $\pi_3$. By #9 $\pi_2$ and $\pi_3$ intersect in a line $l$ that contains $C$.
$l$ cannot intersect $b$ in any point $B$, otherwise $a$ and $B$ are in both $\pi_1$ and $\pi_3$, so $\pi_1\equiv\pi_3$ by Lemma 1, $b\equiv \pi_1\cap\pi_2\equiv\pi_3\cap\pi_2\equiv l$ which would require $C$ to be on $b$, contradicting that $b$ and $c$ are parallel. So $l$ does not intersect $b$ but it intersects $c$ at $C$. Since $b,c,l$ are coplanar in $\pi_2$, by Lemma 2 they cannot all be different, so $l\equiv c$.
$l$ cannot intersect $a$ in any point $A$, otherwise $b$ and $A$ are in both $\pi_1$ and $\pi_2$, so $\pi_1\equiv\pi_2$ by Lemma 1, contradicting that $a,b,c$ are not coplanar. Since $l\equiv c$ and $a$ are both in $\pi_3$ and do not intersect it follows that $a$ is parallel to $c$.
Best Answer
Parallel lines don't meet. Ever. That's the definition of "parallel" so it does not need any explanation.
One of the axioms (assumptions) of Euclidean geometry is that there are parallel lines. Since that is an assumption, it can't be proved.
There are non-Euclidean geometries in which there are no lines through a given point parallel to a given line, and geometries in which there are many.
Similarly, it's one of the axioms of the integers that they go on forever. The fact that some computer implementations of an
integerdata type can overflow is a property of the implementation, not the integers. And there are ways to manage integers as large as you wish in computer programs without having to deal with infinitely many integers.