Corresponding angles are equal exactly when the lines are parallel

0:00Last time we numbered the eight angles a third line makes with two others. And we matched them up, one from each crossing, by the position each one sits in. Angle one and angle five. Same side of the cutting line, same side of its own line. Corresponding angles, and so far that name is about position only. Nothing we have said makes a corresponding pair the same size. Watch. Turn the lower line, and the upper angle does not move while the lower one changes.

0:36So we have a matching, and no equality to go with it. This is where that gets settled, and settling it takes two statements rather than one. Start again with almost nothing. One straight line, drawn across the board. And one more cutting through it at a slant. A transversal. They meet at a point. Call it X, because we will need to refer back to it. Four angles open up at X, and we already know they come in two sizes.

1:08Now the plan, said in advance, so you can see where this is going. We are going to draw a second line, somewhere below, using nothing but a copy of an angle. No ruler laid alongside the first one. No checking the gap at both ends. One angle, copied. And then we look at what kind of line that has got us. First, the angle we are going to copy. Take the one above the line and right of the transversal.

1:39Measure it. Sixty degrees. Its neighbour along the straight line has to make up the rest, so that one is a hundred and twenty. And across the point from each of those sits its equal partner. Four angles, two sizes. Sixty, and a hundred and twenty. Hold on to that count, because the count is the target. When the second line arrives it will bring four more angles with it.

2:08If the finished figure still shows only two sizes, something has gone right. Now mark a second point further down the transversal. Call it Y. It can be anywhere along there, and that freedom is worth noticing. Lay a sheet of tracing paper over the first crossing, and trace the sixty degree angle. Now slide the sheet down the transversal until the traced corner sits on Y. Slide, never turn. That is the whole trick, and it is why tracing works at all.

2:43A sheet that is only slid carries every direction on it unchanged. If you would rather use numbers, a protractor does the same job. Lay off sixty degrees at Y. Either way, draw the second line along that copied arm. There it is. A second line, made from one copied angle and nothing else at all. Look at what we have got. It certainly looks parallel to the first one.

3:13And the count came out right. Sixty and a hundred and twenty, and nothing else anywhere. Eight angles in the figure now, and still only two sizes between the lot of them. But wait, because we have to be careful here, and the care is the whole point. Looks parallel is exactly the evidence we spent nine minutes last time refusing to accept. A pair a twentieth of a degree out also looks parallel, and it meets, forty six pages away.

3:47So let me state what the construction is pointing at, and then be honest about its standing. If a pair of corresponding angles is equal, then the two lines are parallel. That is the first claim, and it does real work. It turns a check you carry out at one point into a statement about two endless lines. Copy one angle. Get a parallel line. That is a construction, not a hope.

4:15And notice how little it asked for. One pair, not four. Once the two lines keep pace, the other three pairs have no choice but to follow. The drawing supports this claim. It does not prove it, and I will come back to that. Now turn the question round, because the other direction is a different question. Start this time with two lines you already know keep pace with each other.

4:43How do you know? Because somebody marked them. A small arrowhead on each. That mark is not a measurement. It records a decision somebody already made. Draw a transversal across both, and pick out a corresponding pair. Trace the upper one, slide the sheet down, and lay it over the lower one. It fits. Not nearly fits. Fits. We started from parallel this time, and the equality came out at the end.

5:15So here is the second claim, and read it beside the first one. If two lines are parallel, then every pair of corresponding angles is equal. Those two are not the same statement, however alike they sound. One takes equal angles and hands you parallel lines. The other takes parallel lines and hands you equal angles. The arrows point opposite ways, and neither one gives you the other for free.

5:45You want both, and the reason will be clear in a couple of minutes. For now, notice that they are stated separately because they are separately useful. Here is an instruction, and I would like you to try it before I say anything else. Two lines that are not parallel. Look carefully. The gap widens as you go right. Draw a transversal across both, so that one pair of corresponding angles comes out equal.

6:16Go on, move it about. Steeper, shallower, further along. Let me put the two readings up and sweep the line for you. Watch the two numbers as it swings. They move together, and they never touch. And here is why, which is a good deal stronger than the drawing being fiddly. The two readings differ by exactly the angle between the two lines. Twelve degrees, the whole way across. The transversal never enters into that difference at all.

6:50It was decided before you drew it, by the two lines and nothing else. So there is no clever position to hunt for. A sweep cannot close a gap it does not touch. Which gives the third statement, and it is the one people actually reach for. If two lines are not parallel, their corresponding angles are never equal. That is not a fourth thing to learn. It is the first claim, said backwards.

7:19Equal makes parallel, and not parallel makes not equal, are one statement in two shapes. But it is the shape that catches a wrong answer, so it has earned its own box. Now put both arrows on one picture, and look at what you have bought. Equal corresponding angles are enough to make two lines parallel. And they are required. No parallel pair anywhere can manage without them. Enough, and required. A test that works in both directions.

7:53That pair of properties has a name, and it is worth knowing what the name buys. Necessary and sufficient. One condition that settles the question either way round. With the second claim alone, you could confirm something you had already been told. With both, you can decide a case that nobody has told you anything about. So let us decide one. Two lines, a transversal, and two measurements. At the upper crossing, this angle is a hundred and twenty degrees.

8:28At the lower crossing, this one is seventy. Careful. Those two are not a corresponding pair, so we cannot compare them yet. The one we want is the neighbour of the first, along its straight line. A hundred and eighty, take away a hundred and twenty. Sixty. Sixty against seventy. Not equal, so the two lines are not parallel, and that is settled. Ten degrees apart, and no redrawing of the transversal will ever hide it.