Straight and right angles as the landmarks of a full turn

0:00Back to the book, one last time. Your book draws two extreme cases, and they're the two that matter. In the first, the cover has been turned all the way over, right round onto the back, so you can hold the book in one hand. The cover started on top of the base. It ended up underneath. That's a full turn's worth of opening. In the second, the cover is laid open flat on the table.

0:28Stop there, because that one is about to become important. Flat on the table is exactly halfway. The cover has gone half as far round as it could. So: a half turn. And a half turn has a particular look to it, which is what the whole video comes out of. Look at what the book looks like when it's flat. The cover points one way. The base points the other way. And the two of them are in one straight line.

0:59The hinge is still the vertex. The cover and the base are still the two arms. It's still an angle. It just happens to be the angle where the arms have been flattened out until they point in exactly opposite directions. And that has a name. It's called a straight angle. Now, some people object to that. They say: that's just a line, that isn't an angle at all. But it is. There's a vertex. There are two rays leaving it, in opposite directions. That's the definition, and it's satisfied.

1:36What's different is only that the turn happens to be a half turn, so the arms end up collinear. The straight angle is your first landmark. Remember it. Your book draws it, and the drawing is worth looking at properly. Three points on one line: A, then O in the middle, then B. Arrowheads at both ends. And then, at O, a curve. That curve is doing something quite sly. Without it, you'd read that picture as a line and nothing else.

2:09With it, the same ink is a line and an angle at the same time. The curve tells you which reading is wanted. So the angle is written angle A O B, exactly as before. O in the middle, because O is the vertex. One picture, two readings. That's the hinge of this whole section. Now let's do something to it. Take the straight angle A O B, and draw one more ray from O, going upwards. Call the point on it C.

2:42What have you made? Two angles, where there was one. Angle A O C on this side, and angle C O B on that side. And whatever else changes, those two always add up to the straight angle. That's just what cutting something in two means. Now move the ray. Tilt it over towards A. Still two angles. One got smaller, one got bigger, and they still add to the straight angle.

3:11Tilt it the other way. Same story. So: any ray from O splits a straight angle into two angles. Any ray at all. That part's easy. Here's the harder question, and it's the one the section is actually about. Can you place the ray so that the two pieces are equal? And I want you to notice that this is a real question, not an obvious one. Because almost every ray gives you two unequal pieces. Nearly all of them are wrong.

3:44You want the one position where the piece on the left is exactly the same angle as the piece on the right. Now, you could try it by eye. Nudge the ray until it looks even. But looks even isn't good enough here, and your book is quite firm about that. We're about to define something on top of this, and if the halving is only approximately right, everything built on it is only approximately right.

4:13So we need to actually prove the two halves are equal. Not estimate. Prove. And we don't have a protractor. We haven't invented degrees yet. So here's what your book does instead. It folds the paper. Take a rectangular sheet. Along one edge, mark three points: A at one end, B at the other, and O somewhere in between. That edge, from A through O to B, is a straight angle at O. The edge of the paper is doing the work of the line.

4:47Now fold the sheet, hinging it at O, so that the arm O B comes down exactly on top of the arm O A. Line them up carefully. Edge on edge. Then press the fold flat. Now open it out again. There's a crease. It runs from O, up across the sheet, and it wasn't there before. And that crease is the ray you were looking for. Now — why does that settle it? Why is folding a proof rather than a guess?

5:19Think about what the fold actually did. It took the angle on one side of the crease and laid it down on top of the angle on the other side. Vertex on vertex, because you hinged at O. Arm on arm, because you lined up O B with O A. That is superimposition. That's the exact test from last video. And they matched. Nothing stuck out on either side. So the two angles aren't approximately equal, or equal as far as you can tell. They are equal, and the fold is the reason.

5:56Which is a nice reversal, isn't it. Folding sounds rougher than measuring. Here it's the other way round — the fold is exact, and the eye was the estimate. So now we can define the thing. Each of those two equal pieces is called a right angle. And read that definition carefully, because it's not the one most people carry around. A right angle is precisely half a straight angle. That's it. That's the whole definition.

6:30It isn't an angle that looks square. It isn't the corner of a page. Those are ways of recognising one, and they're useful, but they aren't what it means. The corner of a page is a right angle because the page was cut that way — not the other way round. And now turn the definition around, because that's where it gets useful. If a right angle is half a straight angle, then a straight angle is exactly two right angles.

7:01Two of them, side by side, arms flattened out into one line. Every time. Let's stack up what we've got, because they're all the same thing measured in different bites. A full turn brings the arm all the way back to where it started. A half turn is half of that, and it gives you the straight angle. And a right angle is half of the half. So a right angle is a quarter of a full turn.

7:29Quarter, half, whole. Those are your landmarks, and there still isn't a number anywhere. You already recognise the quarter turn, by the way. It's the shape of a capital L. It's the corner of a door frame, the corner of a window, the corner of this screen. It's everywhere in built things, because things that stack and stand tend to be built on quarter turns. Now, one more word, and it comes from what happens if you extend both arms.

8:02Take a right angle, and push each arm out past the vertex, into a full line. You've now got two lines crossing at that point, and four angles round the crossing. And here's the thing — all four of them are right angles. When two lines cross like that, we say they are perpendicular. Now, a warning, because this catches people out. Perpendicular does not mean vertical. It isn't about upright, and it isn't about the page.

8:34It's a relation between two lines: it says how they meet, not which way they point. Tilt the whole picture over. Spin it round. Both lines are now slanting. They're still perpendicular, because the four angles at the crossing are still right angles. Nothing about the crossing changed. Your book then sets a puzzle on a grid of dots, and the puzzle is a good one because the hint is the whole method.

9:05You've got a square grid. A dot is marked A, and another is marked B. First task: make a straight angle at A. That means finding a line through A that hits grid dots on both sides of it. And there's a move for that. Take the segment from B to A, and just keep going — extend it past A until you land on another dot. Call it C.

9:31Now A has dots on both sides of it, in one straight line. Angle B A C is a straight angle. Second task, and this is the clever bit: make a right angle at A. You already know how. A right angle is half a straight angle. You've got a straight angle sitting right there. So find the line through A that halves it. And on a square grid you can do that exactly, without folding anything, because the grid gives you the perpendicular direction for free.

10:04If your segment went across three and up one, the perpendicular goes across one and down three. Swap the steps, flip one sign. Try it. That's a rule you'll meet again in a few years with a lot more machinery around it. And now, with two landmarks fixed, you can sort every angle there is, still without a single number. Smaller than a right angle: your book calls those acute.

10:33Exactly a right angle: that's its own group, right in the middle. Bigger than a right angle, but smaller than a straight angle: obtuse. And your book asks you a question about those two words that I like a lot. Why those words? Acute, in ordinary English, means sharp. And a small angle is a sharp point. Obtuse means blunt. And a wide angle is a blunt corner. So they're not arbitrary labels to memorise. They're descriptions, and they're descriptions of exactly the right thing.

11:09Now here's yours, and it's from the end of the section. Draw a triangle. Count its acute angles — there are three, one at each corner. Now join the midpoints of its three sides. That puts a smaller upside-down triangle inside it. Count the acute angles now: there are twelve. Do it once more, inside that middle triangle, and there are twenty-one. Three, twelve, twenty-one. So how many in the next one? And more importantly — why does it go up by nine each time?

11:44Here's a hint: every new stage puts three new points onto lines that were already there, and each of those points has two new rays leaving it. Work out how many new acute angles each new point brings, and the nine will explain itself. Comments below. Next time: we finally put numbers on those landmarks, and find out why anybody would cut a full turn into three hundred and sixty pieces.