Using slopes to prove that lines are parallel is the easiest way to o the same. Else we need to find the equation of the lines and solve them to determine any points of intersection. If no points of intersection can be found, the lines are parallel.
For line through (6,2);(8,4) and (7,3) and (9,5)
The slope is (4 - 2)/(8 - 6) = 2/2 = 1
and (5 - 3)/(9 - 7) = 2/2 = 1
The slope is equal, so the lines are parallel.
The advantage of using slopes to prove parallelism is the following: since we have the points the lines are passing through, we can determine the slopes using identity: m = (y2-y1)/(x2-x1) and then, all we have to do is to determine their relative positions by comparing their slopes. If they are equal, then the lines are parallel.
We can consider, of course, the method of solving the system formed by the equations of the lines. If the system has no solution, then the lines are not intercepting.
Slope of line through the points (6,2) and (8,4) is:
m1 = 2/2
m1 = 1
Now, we'll consider the next two points (7,3) and (9,5).
m2 = (5-3)/(9-7)
m2 = 1
Comparing, we'll get m1 = m2 = 1
Therefore, the given lines are parallel.