Constructing from two angles and the side between them

0:00There is more than one way to be handed a triangle. One way is two sides, with the angle sitting between them. Two lengths and an angle. Here is the other way round. Two angles, and one side. Forty degrees at one end, five centimetres of base, fifty degrees at the other end. It reads like the same kind of instruction, and it very nearly is. But there is one difference between them, and it is a large one.

0:30That first construction can never fail. This one can. And what decides whether it fails is not the thing you would expect. First, which side is the included one. Take a triangle, and pick two of its angles. Say the ones at A and at B. The side running between those two corners is A B, so A B is the included side. Now pick a different pair. The angles at B and at C.

1:00Those two stand at the ends of B C, so the included side moves to B C. Same triangle, different pair, different side. So the side is not fixed either. It is whichever side has both of the chosen angles standing on it. Which makes the first job an easy one to get wrong, and worth doing before you draw anything. Three of them, and each is written the same way.

1:28An angle, a length, an angle. Forty degrees, five centimetres, fifty degrees. Thirty degrees, four centimetres, forty-five degrees. Twenty degrees, six centimetres, fifty degrees. That last one is drawn awkwardly on purpose. Its given side is neither at the bottom nor horizontal. Which is exactly the point. You cannot look for the flat side underneath and call that the base. Find the side that both marked angles are standing on. That is the base, whichever way up it happens to sit.

2:08So let us build one properly. A B is five centimetres. The angle at A is forty-five degrees, and the angle at B is eighty. Step one. Draw the base, five centimetres, and mark the ends A and B. Step two. Protractor at A, and an arm at forty-five degrees, going up from the base. Step three. Protractor at B, and an arm at eighty degrees — measured from B A this time, not from A B.

2:39Both arms lean inwards, towards each other. Extend them until they cross, and that crossing point is C. One length and two angles went in, and there is the triangle. Now look carefully at where C came from. You did not measure to it. You did not mark it. You did not choose it. It is simply the place where the two arms happened to meet. And it arrives carrying three more measurements, none of which anybody asked for.

3:11The angle at C came out at fifty-five degrees. A C came out at about six centimetres, and B C at about four point three. The third corner sits about four and a quarter centimetres above the base. Three in and three out, then. But this time the crossing was not guaranteed. Here is where the two constructions part company. With two sides and the angle between them, you lay both lengths off from one corner, one after the other.

3:44Both far ends exist, so the segment joining them exists. Nothing in that can refuse. This construction is different, because the third corner is not something you place. It is something you wait for. Two arms go out from the two ends of the base, and either they meet or they do not. And sometimes they simply do not. So the real question is which pairs of angles work, and which pairs do not.

4:16Start with the failures you can see at a glance. Draw a base, and set both arms leaning outwards, away from each other. Extend them as far as you like. They separate faster the further out you go. There is no crossing point out there, and there is never going to be one. Try again with one arm square to the base — a right angle — and the other leaning out.

4:42The square one goes straight up. The other leans away from it. No meeting again. Now make both arms square to the base. Two right angles. Two lines square to the same base run alongside each other for ever. They are parallel, and parallel lines never meet. Those are obvious. The interesting question is where the boundary between them sits. So pin one angle down and move the other one.

5:13Forty degrees at A, and it stays at forty for the rest of this. Start the arm at B at sixty degrees. The two arms cross low down, under three centimetres above the base. Open it to ninety, and the crossing climbs. A hundred and thirty, and the crossing is fourteen centimetres up — already off the paper. A hundred and thirty-nine point nine nine, and it is over a hundred metres above the base.

5:42It is not settling anywhere. Every time you get ten times closer, it goes ten times further away. So push it all the way to a hundred and forty. And the crossing point does not go somewhere enormously far off. It stops existing. Look at the two arms at that moment. One leaves A at forty degrees. The other leaves B at a hundred and forty. They are pointing in exactly the same direction.

6:13They are parallel. They run up the page side by side, and the gap between them never closes. That is not the last case that works. It is the first one that fails. Which is worth getting the right way round. At a hundred and forty there is no triangle, and above it there is no triangle either. Now where did a hundred and forty come from? You can read it straight off that picture, without measuring anything.

6:45The two arms are parallel, and the base cuts across both of them. A line cutting across two others like that is called a transversal. The angle at A and the angle at B both sit inside the pair of parallel lines, and on the same side of the base. And two angles in that position always total a straight angle. A hundred and eighty degrees. So if the first is forty, the one at the boundary is a hundred and forty.

7:16But do not let a hundred and forty become a magic number. It is the boundary for forty, and forty only. Which leaves one thing unexplained. What happened to the length? You were handed a five centimetre base. Surely a longer one gives the two arms more room to find each other. It does not. Watch. Here is a pair of angles that works, on a short base. The arms cross.

7:46Stretch the base and they still cross — higher up, further apart, but they cross. Now a pair that fails, on that same short base. Stretch it and see. Stretching slides the two arms apart. What it does not do is tilt either of them. I ran every whole pair of angles on bases from half a centimetre to ten metres. Thirty-two thousand pairs, and not one of them changed its answer.

8:16So the rule uses the two angles and nothing else, and it is a single comparison. Add them together. Under a hundred and eighty degrees, the arms meet and the triangle exists. A hundred and eighty or more, and they never meet at all. Try it. Thirty-five and a hundred and fifty make a hundred and eighty-five. No triangle. Seventy and thirty make a hundred, and that works. Fifty and a hundred and fifty make two hundred, and that does not.

8:49Ninety and eighty-five make a hundred and seventy-five — and that one works, right angle and all. A right angle in the pair is not the problem. It takes two of them, or one angle big enough to make up the difference. And one last thing. Sixty and seventy on a five centimetre base, then on a seven: the base changes, the third angle stays at fifty. Two angles pin the shape down completely. What they leave open is the size.