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.
A line of slope $m$ makes a signed angle with the $x$-axis equal to $\theta = \arctan(m)$ (expressed in radians), and this angle has value between $-\pi/2$ and $+\pi/2$.
For two lines of slopes $m_1$, $m_2$, which make signed angles equal to $\theta_1 = \arctan(m_1)$ and $\theta_2 = \arctan(m_2)$, the "angular distance" between those two lines will be either $|\theta_1 - \theta_2|$ or $\pi - |\theta_1 - \theta_2|$, whichever is smaller. The alternative $|\theta_1 - \theta_2|$ is used when that number is $\le \pi/2$, and if that number is $> \pi/2$ then the other alternative $\pi - |\theta_1 - \theta_2]$ will be $\le \pi/2$.
Once you have computed the angular distance between the two lines, set it to be less than whatever threshold you desire.
Best Answer
Lines can not be equal because they are not numbers. I think there are same lines.
Two lines defined parallel if they are the same or (if our were placed in the plain) they have no common points.
We need it for the following important property:
If $a||b$ and $b||c$ then $a||c$.
If $a\equiv b$, but $a\not||b$ we obtain that the following is wrong.
$a||b$ and $b||a$ then $a||a$.
If so it's wrong. I don't like it!