A transversal creates two matching sets of four angles

0:00Here are two lines, and the question is whether they keep pace with each other for ever. You cannot settle that by looking, and you cannot settle it by measuring the gap in two places. A drawing is about the width of your hand, and the claim is about the whole of for ever. So we stop asking about the two lines on their own, and do something that looks like a detour.

0:25We draw a third line, straight across both of them. Which seems like the wrong move. Two lines were hard enough, and now there are three. But the third line brings something the first two never had between them, which is angles. Angles can be compared exactly, and that is the whole reason any of this works. Let me build the figure slowly, because everything after this is read straight off it.

0:54Two straight lines, one above the other, both running up towards the right. Look at them carefully. They are not quite pointing the same way. The upper one climbs a little less steeply than the lower one, and that is deliberate. Nothing here assumes those two keep pace. If we assumed it, there would be nothing left to establish. Now the third line, cutting down through both of them. A line that cuts across two others like this is called a transversal.

1:26It only has to cut both. It can lean any way it likes, steep or shallow, and it is still a transversal. The transversal meets the upper line at a point, and the lower line at another point. At each of those points four arms of chalk run outwards, and four angles open up between them. Four at the top, four at the bottom. Eight angles in all. They need names, so we number them, and the numbering is worth learning as a walk.

2:00At the upper crossing, one is above and to the left, two is above and to the right. Then three below and right, and four below and left. Round the point. At the lower crossing, exactly the same walk again: five, six, seven, eight. Same four positions, same order, one crossing sitting below the other. Which brings the first question actually worth asking about them. Could all eight of those angles be different sizes?

2:32There are eight, and they are spread around two separate points on the board. Nothing obviously stops it, and eight different sizes would mean eight separate things to carry away. So try it. Pick a size for the first one and start filling the others in. The moment you write a number at the top left, another angle in this figure is settled for you. Not suggested. Not made likely. Settled.

3:00And once you see which angle that is, the answer to the question turns up on its own. Look at the upper crossing on its own, with the rest of the figure put away. Angle one sits above and to the left. Directly across the point from it sits angle three. Those two face each other across the crossing, and they are the same size. Always. That is not new. Any two lines crossing anywhere make two pairs like it.

3:31So one and three are one such pair, and two and four are the other. Now bring the lower crossing back. It is the same arrangement again, further down. Five and seven face each other, and so do six and eight. Four pairs, two at each crossing, and every one of them is forced to be equal. Now write that down as a count, because the count is the whole answer.

3:59Eight angles on the board. Four pairs, and both members of a pair carry the same size as each other. So each pair contributes one size between the two of them, never two. Four pairs, one size apiece. At most four different sizes in the entire figure. Which means all eight different is not merely unlikely. It cannot be done. Half of what you are looking at is a copy of the other half.

4:28Eight angles were never eight facts. There is a second way to that same four, and I think it is the better one. This route is mine rather than the usual one, so take it as an extra rather than the standard. Cover the lower crossing and look only at the top of the figure. Two lines crossing at a point give four angles in at most two sizes. One size, and its partner that fills up the straight line beside it. That is all one crossing can ever do.

5:02Now uncover the bottom. Also one crossing. Also at most two sizes. Two crossings, at most two sizes each, and that is at most four sizes altogether. The same ceiling as before, except that now you can see exactly where the four came from. Second question, and it is sharper than it first looks. Take five of the eight. Six, five, four, three and two. Could those five all be different sizes?

5:35Run your eye down the list and hunt for two that are already tied to each other. There they are. Four and two face each other across the upper crossing. So those two are equal, which leaves at most four different sizes among the five. And that list was not special in any way. Any five you care to name will do the same thing. Four pairs to draw from and five angles to pick. One pair has to hand over both of its members.

6:07Which turns the question round, and the turned round version is the one worth having. Not what are the eight. How few different sizes can they possibly come in? Four is the most, and you have now seen why twice over. Two is easy to arrange, and one is available as well, when every angle in the figure is square. Three is the interesting one. It can happen, but only in a single way.

6:36A square crossing has all four of its angles the same size, so it contributes one size instead of two. One crossing square and the other not, and the figure shows exactly three sizes. So the whole range is one, two, three or four. Never five, and never eight. Now split the eight up, in the way the figure has been asking you to all along. The transversal makes four angles with the upper line. That is one set.

7:08And four more with the lower line. That is the other. One, two, three, four in the first set. Five, six, seven, eight in the second. The transversal belongs to both sets, because it is in both crossings. Each of the other two lines belongs to one set only. So the third line did not create a new situation for you at all. It took a situation you already understood, and handed it to you a second time.

7:41Which lets us match them up, one angle from each set. Angle one sits above the upper line, and to the left of the transversal. Angle five sits above the lower line, and to the left of the transversal. Same position, different set. Two angles matched like that are called corresponding angles. Slide the lower crossing straight up along the transversal, and five arrives in the corner where one is sitting.

8:11The full matching is two with six, three with seven, four with eight. Four pairs in all. Be careful here, because the test has two halves and one of them is usually dropped. Left of the transversal is only half of it. Above its own line is the other half, and a pair has to pass both. So where does all of that leave the question we started with? One and five are corresponding angles. Are they equal?

8:43Nothing so far says they are, and here is the demonstration that nothing does. Hold the upper line still, hold the transversal still, and turn only the lower line. Angle one has not moved at all. Angle five is changing while we watch. Corresponding means sitting in matching positions. So far it does not mean equal. There is one position of the lower line where the two sets fall into step together.

9:13Whether that is a fact or merely a good drawing is exactly what has to be settled next.