How do you prove that two parallel lines never cross? By definition this is implied, but how do you prove it for any pair of parallel lines? In other words, how do prove that 2 parallel lines will never cross. Thanks.
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$\begingroup$Try to solve the set of two equations describing the two parallel lines:
$$ \begin{align} y & = m x + b_1 \\ y & = m x + b_2 \end{align} $$
This results in a contradiction:
$$ b_1 = b_2 $$
$\endgroup$ $\begingroup$y=mx+c is the form of a line in cartesian coordinates, where m is the gradient and c is the y intercept, now if two lines are parallel, the gradient is same for both but c will vary. Now say L1 (line 1): y=mx+c L2: y=mx+d where c=/=d
suppose there is a common point say (a,b)
from L1: a=mb+c from L2: a=mb+d
Can you go on from here and get a contradiction? Hence showing no common point exists.
$\endgroup$ $\begingroup$In order to determine If two parallel lines intersect, you could put both lines in matrix form and solve the matrix determining if there were no solutions.
$\endgroup$ $\begingroup$Take two parallel lines that are cut by a transversal. Since the lines are parallel, the consecutive interior angles of intersection are supplementary. Specifically, if the two parallel lines intersect, we have formed a triangle with angles that add up to more than $180$ degrees, which is a contradiction.
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