If the gradients are not the same, then there will be a point, which intersects. That contradicts the statement that theclines have no points in common.
Explanation:
Parallel lines are lines that have the same gradient.
Because parallel lines have a fixed distance between each other,it means they will never meet, no matter how long they are extended.
Since parallel lines have the same slope by definition, they will never intersect. Parallel lines continue, literally, forever without touching.
Prove:
Suppose that there are two lines, line l and line m, which have no points in common. Both lines are in the same plane.
If the gradients are not exactly the same, then there will be a point, which intersects. That contradicts the first statement (which says l and m have no points in common), therefore the gradients must be the same in order for l and m to have no points in common.