Drawing a parallel line using the corresponding-angle test

0:00There is a difference between a drawing that looks right and a drawing that is guaranteed. Anyone can rule two lines that look parallel. It takes about four seconds. But looking is exactly what we have established cannot settle it. So here is what we want instead. A procedure. Follow it, and the lines you end up with are parallel because of how you got them. Not because you checked afterwards, and not because they came out looking neat.

0:32There are two such procedures, and they use completely different equipment. One needs a geometry set. The other needs a sheet of paper and nothing else at all. Start with the one that uses instruments. Lay a straight edge down and rule a line along it. Now hold that straight edge exactly where it is. It is not going to move again. Take a set square, which is the right angled triangle out of the box.

1:03Seat one of its edges flat against the straight edge. Now draw along the edge that stands away from it. That is your first line. Keeping the straight edge exactly where it is, slide the set square along it. Slide, not lift. It stays pressed against the straight edge the whole way. And draw again, along the same edge. Two lines now, and we are claiming they keep pace. Now relabel what you have got, because the naming is the whole argument.

1:37There are three lines on the paper. The two you just drew, and the one you started from. The two new ones are the pair we are asking about. And the line you ruled along the straight edge cuts across both of them. Which makes it the transversal. It was the given line a moment ago. Now it is the cutting line. That relabelling is not a trick. It is the reason this construction is worth anything.

2:05Because a transversal cutting two lines is a situation we can already settle. We know exactly which pair of angles decides it. So look at the two crossings that cutting line makes. At the first one, the line you drew came off the short edge of the set square. That edge stands square to the edge you had pressed against the straight edge. So the angle there is ninety degrees, and it is ninety because of the shape of the tool.

2:37At the second crossing, the same edge of the same tool. Ninety again. Now take the pair in matching positions. Above its own line, and right of the cutting line. Ninety and ninety. Equal corresponding angles. And equal corresponding angles make two lines parallel. That is the guarantee, and it is finished. Now turn the set square round, and use its long edge to draw with instead. Same straight edge, same slide, same two drawings.

3:12But this time the lines come off at sixty degrees to the cutting line, not ninety. Draw, slide, draw. And look at what you have. Sixty and sixty. Still equal, and still in matching positions. So the two lines are still parallel, and no right angle came into it anywhere. Which tells you something rather important about what the set square was doing. It was not making right angles. It was repeating an angle, and repeating is all that was needed.

3:46Which puts all the weight on one step, and it is the step that looks least important. Sliding. The straight edge must not move between the two drawings. Suppose it turns. Not much. One degree, which you would never notice. Then the two lines you drew are one degree apart instead of nothing. Draw them five centimetres apart, and they meet. Two point nine metres away. Off the page, off the desk, and across the room.

4:17Make the slip a tenth of a degree, and they meet twenty eight metres away. The drawing looks exactly the same either way. The guarantee is simply gone. Now suppose you want to check the pair you have drawn. There are two honest ways to do it, and both are worth doing. Trace one of the corresponding angles onto a sheet, slide it along, and lay it on the other.

4:44Or set a protractor on each crossing in turn and read the two angles off. Either way you are comparing one corresponding pair, which is the pair that decides it. But be clear about what you have just done. You have confirmed. You have not proved. A protractor read to half a degree cannot separate a pair that meets five pages away. Still, the check is worth understanding, because of how little it needs.

5:15One pair. Not all four. That is the two way test again, and it is doing both of its jobs here. If the pair is equal, the lines are parallel. That direction validates your drawing. If the lines are parallel, the pair has to be equal. That direction is why the check works at all. Without the second direction, a reading that came out unequal would tell you nothing. With it, an unequal reading tells you outright that something slipped.

5:48One comparison, and a verdict either way. That is what a two way test buys you. Now the harder version of the task, and it is the one that actually comes up. Here is a line. And here is a point, sitting above it and off to one side. Draw the line through that point which keeps pace with the first one. There is exactly one such line, wherever you happen to put the point.

6:17Every other direction through that point meets the first line somewhere. Only one of them misses it for ever, and that one is the answer. With instruments it is the same slide as before, stopped in the right place. Seat the set square, slide until its drawing edge touches the point, and draw. Now put the instruments away entirely. A sheet of paper, and two folds. Here is the line, creased across the sheet. And here is the point.

6:50First fold. Bring the line over onto itself, and press the crease through the point. Open it out. That crease meets the line square, and you measured nothing. A fold that lays a line back onto itself has no choice. Only a square crease can do it. Second fold. Bring that new crease over onto itself, pressing through the point again. Open it out. Two creases now, and the line you started with.

7:19And the second crease is the parallel you were after. So why did that work? The answer is one we already have. Look at the first crease. It cuts the original line, and it cuts the second crease. Which makes the first crease a transversal, exactly like the straight edge before it. It meets the original line at a right angle, because that is what the first fold guaranteed. And it meets the second crease at a right angle, because that is what the second fold guaranteed.

7:55Ninety and ninety, in matching positions, so those two lines keep pace. That is the same sentence I said about the set square, word for word. The equipment changed. The argument did not change at all. Which is worth stating on its own, because neither method says it out loud. Here is the general version, and I want to be clear that this wording is mine. If two lines are both at right angles to the same line, then those two are parallel.

8:29That is what the set square did, and it is what the two folds did. One condition, though, and it is the one everybody drops. All three lines have to lie on the same flat surface. Stand a pencil upright on a table. Every line drawn on that table is square to it, and they point every which way. On one sheet of paper, though, two right angles against a common line is all you will ever need.