Constructing from two sides and the angle between them

0:00Up to now, every triangle we built started from three lengths. Three sides, two arcs, a crossing. And sometimes no crossing at all. Now something changes. One of the three numbers stops being a length. Instead of a third side, you are handed an angle. Two sidelengths, and the angle sitting between them. Which means a new tool. A protractor, for setting an angle you have been told. And it turns out this changes rather more than the equipment.

0:33Three lengths could refuse to close. Two lengths and an angle never can — and by the end you will see exactly why. First, a word that is doing real work here. Included. Here is a triangle. A, B and C. Take two of its sides. Say A B and A C. Both of them run out of the same corner, and that corner is A. So the angle at A is the included angle for those two sides. The one wedged between them.

1:06Now change the pair. Take B A and B C instead. Those two run out of B, so the included angle moves to B. It is not a fixed corner of the triangle. It is whichever corner the two sides you were given happen to share. Three examples, and each is written the same way. A length, an angle, a length. Three centimetres, sixty degrees, four centimetres. Six centimetres, thirty degrees, five centimetres.

1:39Two centimetres, forty degrees, three centimetres. Now look at where the angle is marked in each one. Bottom left. Then bottom right. Then right up at the top. It moves about, and it moves for a reason. It sits wherever the two marked sides meet. So the first job with any of these is not to draw anything. It is to find the corner that both given sides come out of.

2:09Let us build one properly. A B is five centimetres. The angle at A is forty-five degrees. A C is four centimetres. Read that carefully. Two of those are lengths, and they share the corner A. The third is the angle at that shared corner. So it is a length, an included angle, and another length. Exactly the shape of the three examples. And notice what you have not been told. Nothing about B C at all.

2:40Nothing about the angle at B, or the angle at C. Three measurements out of a possible six — and we are about to watch the other three appear without anyone asking for them. Step one. Draw the base. A segment five centimetres long, with the ruler. Mark the ends A and B. Either of the two given sides could have been the base, by the way. Five is not the base because it is longer. It is the base because we drew it first.

3:13Step two. Put the protractor at A, and set forty-five degrees from A B. Draw the second arm out of A along that direction. Make it long. Longer than you think you need. You do not know yet where it ends, and you do not have to. Two arms out of A now, with a definite angle holding them apart. Step three. Measure four centimetres along the new arm from A, and mark the point.

3:43That is C. Notice there was no searching. You measure, you mark, and you are done. The arm is a straight line out of A, so there is exactly one point on it four centimetres along. Step four. Join B to C. And that is the triangle. Rub out the rest of the arm, or leave it. The figure is finished either way. Four steps, and not one of them was a check.

4:11Now look at what is on the paper. Six measurements. Three sides and three angles. Three of them we chose. A B five, the angle at A forty-five, A C four. And three of them we did not. B C came out at about three point five seven centimetres. The angle at B came out at about fifty-two and a half degrees. The angle at C, about eighty-two and a half.

4:39Nobody asked for any of those. They arrived because the other three were fixed. Half the figure was specified. The other half followed, and had no say in the matter. Here is the part worth stopping on. When three lengths were given, one of them could beat the other two, and the whole thing collapsed. So ask what could go wrong here. And then go looking for it. The base is a straight segment. Nothing can stop you drawing five centimetres.

5:11The angle is set at A. Nothing can stop you opening a protractor to forty-five degrees. The second arm is a ray. It runs on and on, so a point four centimetres along it certainly exists. And once C exists, joining B to C is drawing a segment between two points that are already there. There is nothing left to check, because there was never a third condition. The third side is not something you were asked for. It is whatever the two arms leave behind.

5:48So here is a question worth putting to yourself. Is there any pair of lengths, with any included angle, that fails? Take a moment before I answer it. The answer is no, and the reason is the one we just walked through. As long as the angle is a genuine one — bigger than nothing, and smaller than a straight line — the two arms really do separate. Both marked points exist, and two points always have a segment between them.

6:19I put every whole angle from one degree to a hundred and seventy-nine against every pair of whole lengths up to twenty. Seventy-one thousand six hundred cases, and not one failure. And forty thousand random ones on top of that. Still none. But push at the edges, because that is where you find out what the condition really is. Watch the angle open, slowly, from almost nothing. Near zero, the two arms lie almost on top of each other, and the triangle is a sliver.

6:56As the angle opens, the third side grows. It never pauses, and it never turns back. Near a straight line, the arms point almost opposite ways, and it is a sliver again. Now take those two ends exactly. At nothing at all, both arms are the same ray, and the three corners are in a line. At a straight angle, the arms point opposite ways, and the corners are in a line again.

7:26Neither of those is a very flat triangle. There is no triangle — no area, nothing enclosed. And that is not the construction failing. It is you not having given it an angle. One more, and this one looks as though it might cause trouble. Three centimetres, a hundred and twenty degrees, eight centimetres. A hundred and twenty is wider than a square corner. The angle opens right back on itself.

7:57So does the method mind? Not in the least. Base of eight. Protractor at one end, set to a hundred and twenty. Arm out along it, three centimetres, mark, and join. There is the triangle. Long, thin, and perfectly closed. The side that came out is about nine point eight five centimetres — longer than either side we were handed. Which makes sense, once you have watched the angle open. The wider it goes, the further apart the two far ends end up.

8:33So: two ways of building a triangle, and they behave nothing like each other. Three lengths. Three conditions, all competing for one point, and sometimes no such point exists. Two lengths and the angle between them. Two conditions, laid out one after the other, with nothing left to compete. One last caution, and it matters. The angle has to be the one between the two given sides. Hand someone two sides and an angle somewhere else, and that is a different problem — one that can have two answers, or none at all.

9:11Included is not a decoration on the word angle. It is the entire reason this works. Three measurements went in. Three came out. And at no point did the figure have the chance to say no.