Making parallels by folding

Take a square sheet of paper (newspaper is fine). The opposite edges are parallel. The edges that meet at a corner are perpendicular.

Fold the sheet in half across, and open it. Counting the two edges, there are now 3 parallel lines. Fold it again (in half, then in half again) and count: 5. Again: 9. Again: 17. Do you see the pattern? Each time, the spaces double.

Now make one fold the other way (top to bottom). That crease is perpendicular to all the others. Then fold along a diagonal: can you make a fold parallel to the diagonal?

With instruments: draw a line l. Place a set square on it and draw a line along its upright edge. Slide the set square along l and draw another. Both lines make 90° with l.

Are they parallel? Yes: they make the same angle (90°) with l, in the same position. That is what makes them parallel; the next lessons explain why.

Harder: draw a line parallel to l through a given point A, by folding. First fold a crease t through A, perpendicular to l. Then fold a crease m through A, perpendicular to t. Now l and m are parallel, because both are perpendicular to t.