Copying an angle, and why triangle congruence proves it works

0:00Here is a band, running along the edge of something. The same shape, over and over, but not the same way up each time. One leans this way, the next one leans the other, and the outline rises and falls. Now look at what is actually repeating. It is one shape, not two. The second one is the first one turned over. Nothing about it is new. So if you can draw the unit once, and copy it exactly, you can draw the whole band.

0:32Which puts the weight on that word: exactly. What does an exact copy actually cost you? Two things, and they are not the same kind of thing. The lengths have to match, and the angles have to match. Get the lengths right and the angles wrong and you have a different shape. Get the angles right and the lengths wrong and you have the same shape at the wrong size, which is also not a copy.

1:00A copy that is right about one and wrong about the other is not a near miss. It is a different figure. So you need both. And one of the two is easy. Lengths are what a compass is for. Open it to a distance, lift it, put it down somewhere else, and the distance came with you. That is the whole trick, and it is exact: nothing was read and nothing was rounded.

1:28Now try the same thing with an angle. Open the compass to the angle. You cannot. There is nothing on the instrument that holds one. A compass remembers one number, and that number is a length. Two figures can have the same angle and share no length at all, so knowing a length tells you nothing about an angle. So we do not transfer the angle. We convert it into something a compass can carry.

1:58Here is the move. Put the compass point on the corner, open it to anything, and cut both arms with one arc. Two marks now, one on each arm. Call them B and C. And here is what you have quietly built: a triangle. Corner to B, corner to C, and B straight across to C. The two arms of that triangle are equal, because one radius made them both.

2:25Which means the only thing left that is not already decided is that third side. Look at what happens to that third side as the angle opens. Narrow angle, the two marks are close together and the distance across is small. Open it up and they move apart, and it grows. Every single time, in the same direction. It never doubles back, so two different angles can never hand you the same distance.

2:56And that is the bridge. At one fixed radius, that one straight distance and the angle determine each other completely. Put numbers on it if you like. Cut the arms at twenty-five and this particular angle gives a distance across of exactly thirty. Thirty is not the angle. But at a radius of twenty-five, thirty and that angle mean the same thing, and you can get from either one back to the other.

3:26The distance is now standing in for the angle. And a distance is exactly what a compass can carry. Over to the new place. A bare ray, and the corner it will start from. Do not change the compass. That opening is load-bearing now. Put the point on the new corner and strike the same arc. It cuts the ray at one point. That is one of the two marks you need.

3:54The other one has to go somewhere on that arc, and where it goes is the whole question. Every point on that arc is the right distance from the corner. Only one of them is at the right distance from the first mark. Now go back and pick up the length that matters. Open the compass to the distance between the two original marks. Not to an arm, to the gap across.

4:22Carry it over. Put the point on the mark on the ray, and cut the arc. There is the second mark. One journey, one length. The arms did not have to travel, because the radius already put them in place at both ends. Now rule the line from the new corner out through that second mark. That is the copy, and nothing was measured anywhere in it. Why is it exact, though, rather than close?

4:52Line up the two triangles and count what they share. First arm to first arm: equal, one radius made them. Second arm to second arm: equal, same radius again. Third side to third side: equal, because you carried it. Three sides matching, and three matching sides force two triangles to be identical in every other way as well. So the angle at the new corner is the angle you started with. Not to the nearest degree. The same angle.

5:28One warning, and it is the thing that goes wrong. The size of that first arc is entirely yours to choose. Small, large, it makes no difference. What matters is that the same opening is used at both ends. Take an angle whose arms you cut at twenty-five, giving a distance across of thirty. Now strike the arc at the new corner at fifty instead, and carry that thirty across anyway.

5:58You do not get a slightly wrong angle. You get a definitely wrong one, and a much narrower one, because the same span reaches across far less of a bigger circle. And if the arc is smaller than half the distance you are carrying, the two never meet and you get no angle at all. Now the part that makes this worth having. Think back to what made two lines parallel.

6:25Run a third line across both of them, and look at the two angles in matching positions. When those two match, the lines are parallel. That was the characterisation. Read it the other way round and it is an instruction. If you can copy an angle, you can build that matching pair on purpose. Which means you can draw a parallel line through any point you like, with a compass and a straight edge and nothing else.

6:56Here it is. A line, and a point somewhere off it. Rule any line at all from that point down to the given line. It does not matter what angle it makes. That crossing gives you an angle. Cut both its arms with one arc. Now go up to your point and strike the same arc, cutting the slanted line above. Carry the one distance across, mark it off, and rule the line through your point and that new mark.

7:25The angle at the top now matches the angle at the bottom, in matching positions. So the new line is parallel to the old one, and you built it by copying an angle. Notice what never happened in that. Nobody measured the distance between the two lines. People often say parallel means staying the same distance apart, and the lines do. But that gap was never an input. Nothing in the construction asked for it or used it.

7:58It came out at whatever it came out at, decided entirely by where you put your point. Move the point further up and the gap gets bigger, and not one step of the construction changes. The construction runs on angles. The equal distance is a consequence. So, two things to go and build. The band from the start. One unit, copied along, turned over every other time. And an eight-pointed star, which looks like a harder problem and is not.

8:33Look at its edges, and you will find them in parallel pairs, opposite sides facing each other across the middle. Every one of those pairs is one copied angle away from the pair before it. Which is the whole of what you have been given here, and it is smaller than it looks. A compass never knew what an angle was. You gave it a length to carry instead, and it brought the angle back for you.