y=3x+3 and y=3x-10 are parallel.
but
y=3x+3 and y=3x+3 are the same.
Therefore the difference in these examples is the influence of the constant term.
Suggestion: let have the equation
and let have the equation . These are two lines which have the same slope.
Now solve for the points of intersection. Parallel lines will never intersect, so we can deduce that the only point of interception is where and the lines are incidental; otherwise, they are parallel.
The slope m of a line may be converted to an angle θ using:
Suppose we have two slopes and .
If the slopes are parallel or perpendicular, we may state:
Taking the tangent of both sides and applying the double-angle identity for tangent and dividing through by 2, we have:
The root corresponds to parallel slopes.
The root corresponds to perpendicular slopes.