Parallel lines are lines in the same plane that never intersect, no matter how far they are extended -- they are always the same distance apart. Perpendicular lines intersect at exactly 90-degree (right) angles. Intersecting lines that are not perpendicular cross at angles other than 90 degrees. These relationships are essential for classifying shapes (rectangles have four right angles, meaning opposite sides are parallel and adjacent sides are perpendicular) and for understanding the coordinate plane (the axes are perpendicular).
Use physical examples: railroad tracks (parallel), the corner of a book (perpendicular), an X shape (intersecting but not perpendicular). Have students identify these relationships in the classroom environment. Draw with rulers and use a corner of paper or a protractor to verify right angles. Practice on grids where parallel and perpendicular lines are easy to see.
You already know that a line extends infinitely in both directions, and you understand right angles — the 90-degree angles that look like the corner of a square. Parallel and perpendicular lines are the two most important *relationships* between lines, and they are defined by what happens (or doesn't happen) when lines extend.
Parallel lines never meet. No matter how far you extend them in either direction, they remain exactly the same distance apart — like two rails on a railroad track. This "same distance apart everywhere" condition is the strict definition. Two lines that get even slightly closer together as they extend will eventually cross, so they are not truly parallel. On a drawing, you can use a ruler to check: measure the perpendicular distance between the lines at two widely separated points; if the distances are equal, the lines are parallel.
Perpendicular lines do meet, and they meet at a precise angle: exactly 90 degrees. You can think of perpendicularity as a right angle carved out by the two lines at their intersection point. The corner of a piece of paper laid against the intersection is the practical test: if the paper corner fits exactly, the lines are perpendicular. Every right angle marks a perpendicular relationship — the sides of a rectangle are perpendicular to each other, which is why rectangles have four right angles.
The key distinction that trips students up is intersecting versus perpendicular. All perpendicular lines intersect — they cross at 90 degrees. But most intersecting lines are *not* perpendicular — they cross at some other angle. "Intersecting" only tells you the lines cross; "perpendicular" makes the additional claim about the exact angle. When you classify quadrilaterals next, you will use these relationships constantly: a rectangle has opposite sides that are parallel *and* adjacent sides that are perpendicular. Squares, rectangles, and right triangles all contain perpendicular pairs; parallelograms (that aren't rectangles) have parallel sides but no perpendicular ones.