Drawing parallel lines using perpendiculars

The idea

Lines perpendicular to the same line make equal 90° corresponding angles with it, so they are parallel; a parallel line is drawn by sliding a set square along a line, or by folding a perpendicular to a perpendicular.

Draw two parallel lines

You have a ruler line l and a set square, a triangle with a 90° corner. How could you draw two lines that never meet?

Which way would you try?

• Slide the set square along l and draw along the same edge twice
• Draw two lines by eye and hope
• Draw along two different edges of the set square

Slide the set square

Put the set square on l. Slide it, draw, slide again. Then try the same with paper folding.

The same 90° every time

Each line you draw makes 90° with l. Seen from l, the lines are crossed by a transversal and the corresponding angles are all 90°, so they are equal. Equal corresponding angles mean the lines are parallel.

Lines perpendicular to the same line make equal 90° corresponding angles with it, so they are parallel. We draw a parallel line by sliding a set square along a line, or by folding a perpendicular to a perpendicular.

• Line | Angle with l
• p | 90°
• q | 90°
• r | 90°

Notes

Lines perpendicular to the same line make equal 90° corresponding angles with it, so they are parallel. We draw a parallel line by sliding a set square along a line, or by folding a perpendicular to a perpendicular.

Check yourself

The lines p and q are both perpendicular to l. How can we be sure that p and q are parallel?

l is a crease. We fold t perpendicular to l through A, then fold m perpendicular to t through A. Why is m parallel to l?

A set square slides along l and draws the lines p, q and r, all perpendicular to l. How many pairs of parallel lines do you get?

Answer: 3

All three lines make 90° with l, so any two of them are parallel: p and q, p and r, q and r. That is 3 pairs.

A set square slides along l. At the second place you tilt it a little, then draw. Are the two lines parallel?

• The corresponding angles they make with l are equal, both 90°. — correct. Yes! Equal corresponding angles mean parallel lines.
• They are drawn close together.. Being close does not matter. Parallel lines can be far apart, and close lines can still meet.
• l is longer than p and q.. The length of l has nothing to do with it. The angles do.
• Because m passes through A.. Passing through A only fixes where m goes. What makes it parallel is the 90° angles with t.
• Because m was folded after t.. The order of the folds does not make lines parallel. Equal corresponding angles do.
• l and m are both perpendicular to t, so their corresponding angles with t are both 90°. — correct. Yes! Equal corresponding angles with t, so l and m are parallel.
• Yes, both were drawn with the same set square.. The set square must keep its edge perpendicular to l each time. If it tilts, the angle changes.
• Yes, because both start on l.. Starting on l does not make lines parallel. Equal corresponding angles do.
• No. The angles with l are no longer equal. — correct. Right. One line makes 90° with l and the other does not, so they are not parallel and will meet.