http://www.mathsisfun.com/algebra/line-parallel-perpendicular.html
"Parallel Lines
How do you know if two lines are parallel?
Their slopes are the same!"
The lines must also have different y-intercepts, else they are the
same line.
So, if you know that their slopes are the same, then you don't
(automatically) know if they are parallel lines or not.
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Edit:
On parallel vertical lines:
"a vertical line is parallel to another vertical line."
No, for example, x = -2 is a vertical line, and
3x + 6 = 0 is another vertical line. These lines
are not parallel to each other because they are the
same line.
Distinct vertical lines are parallel to each other.