Parallel Lines

Imagine a train moving along a straight track. For safe movement, the two rails must remain the same distance apart. These rails can be modeled as parallel lines, which are lines in the same plane that never intersect, even when extended.

One property of nonvertical parallel lines is that they have the same slope. We can show this by considering the lines on a coordinate graph. If right triangles are formed along the lines using horizontal and vertical sides, the corresponding acute angles are equal because the lines are parallel.

Each triangle also has a right angle, so the triangles are similar by the AA criterion. This means the corresponding sides are proportional, giving both triangles the same rise-to-run ratio. Since slope equals rise over run, the two lines have the same slope.

Because the lines are distinct, they must have different y-intercepts. Otherwise, they would represent the same line. So, two distinct nonvertical lines are parallel if and only if they have the same slope and different y-intercepts.

Engineers use this idea to design and inspect tracks, keeping the rails parallel so train wheels fit safely.