An angle is an amount of turn, not a pair of drawn arms

0:00Last time we made a ray. One starting point, and one endless direction. Now put two of them down, and make them share that starting point. That's it. That's an angle. Two rays, out of one shared point. Nothing else is required. And notice the word shared. If the two rays start at different places, you haven't made an angle. You've just made two rays. So the sharing is the whole condition.

0:30But here's the thing I want you to hold on to for the next ten minutes, because everything in this video comes out of it. What you can see is two arms. What an angle actually is, is a turn — one that already happened before you looked. The arms are just the record of where it started and where it stopped. First, the parts, because they get named and you'll need the names.

0:57The shared point has a name. It's called the vertex. And the two rays leaving it have a name. They're called the arms. Your book draws this as a point B, with one arm running up to a point D, and the other running down to a point E. So: vertex B. Arm B D. Arm B E. And notice those two arms are rays, written the way we learned last time — the starting point first, and the starting point is the vertex.

1:28Which makes sense. Both arms start there. That's what shared means. Now, how do you write an angle down? There's a symbol for the word angle. It's a little corner mark, like this, and you read it out loud as angle. And then you give it three letters. For the figure we just drew, it's angle D B E. Look at where B went. B is the vertex, and it's in the middle.

1:56That's not a style choice. It's the only part of the name that tells you where the corner is. The outer two letters just say which arms — one point from each. So you could equally write angle E B D. Same corner, same two arms, same angle. Only the middle letter is load-bearing. Which means if somebody writes angle B D E, they haven't written the same thing at all. They've put the corner at D.

2:27There's one more mark, and it's small and easy to skip past. At the vertex, between the two arms, your book draws a tiny curve. That curve isn't decoration either. It's pointing at which angle you mean. Because two rays out of a point don't make one angle. Strictly, they cut the whole space around that point into two. There's the one you obviously meant — and there's the rest of the way round, going the other side.

2:58Both of them are angles. We'll meet the big one properly later in the chapter, when it gets its own name. For now, the little curve is how the figure says: this one, not that one. So when you draw an angle, draw the curve. It's part of saying which. Now, you'll often see angles written with just one letter. Angle B. And that's allowed — sometimes. Here's the test, and it's the whole of this section.

3:28You can use the vertex on its own only if there is exactly one angle at that vertex to mean. In our figure, exactly two rays leave B. Two rays make one angle. So angle B is unambiguous, and you may write it. Now your book gives you a figure where that breaks. There's a point P, and three rays leave it — one through A, one through B, and one through C.

3:55Now say angle P to me. You can't, and here's why. Three rays give you three different angles at that same vertex. There's the one between the first two arms. The one between the second and third. And the wide one between the first and the third. All three of them are angle P. So angle P names nothing. That's why the figure has to say angle A P B, and why it cannot be shortened.

4:24The rule underneath: one letter is a shortcut, and a shortcut is only safe when there's nothing to confuse it with. Right. Now the part that actually matters, and your book gets there with a nice picture. It shows the same book being opened, six times over. Case one through case six. In case one the cover is barely lifted. By case six it's swung right open, nearly flat. Six pictures, and every one of them is an angle. The hinge of the book is the vertex, and the cover and the base are the two arms.

5:04Now here's the question your book asks, and it's a better question than it looks. Which of those six is the biggest angle? You have no ruler. You have no protractor. There isn't a single number anywhere on the page. And you can still answer it instantly. Case six. So — how did you do that? Let's take it slowly, because the answer to that question is the definition we're heading for.

5:34Compare case one and case two. Case two is bigger. Why? Not because it's drawn larger — it isn't. Not because its arms are longer — they're the same book. It's bigger because the cover has been swung further round. Same again for two and three, and three and four, all the way along. Each picture has more turning in it than the one before. And more turning is what you were reading as a bigger angle.

6:03So you were already using a definition before anybody gave you one. You measured the angle by how far the hinge had been opened. That's it. That's the whole idea. So here it is, stated properly, the way your book states it. The size of an angle is the amount of turning about the vertex that separates the two arms. Read that again, because of what it doesn't mention. It doesn't mention how long the arms are.

6:35It doesn't mention how much of the page the drawing takes up. It doesn't mention the space in between, or shading, or area. Only the turn. And that's what lets you say two angles drawn at completely different scales are equal — one tiny, one filling the page — and be exactly right. If the same amount of turn separates the arms, it's the same angle. The drawing is just how big you happened to draw it.

7:07Your book draws the turn itself, and this is the figure the whole topic rests on. Start with one ray. Call this the initial position. Now turn it about the vertex. Keep the starting point pinned, and swing the rest of it round. Stop it wherever you like. That's the final position. And now you have an angle, made of exactly two things: where you started, and where you stopped.

7:35The curved arrow between them is the part we usually forget to draw. It's the turn. It's the actual quantity. So an angle isn't really a shape sitting on the page. It's a record of a movement. The two arms are the before and the after. The angle is what happened in between. And once you've got that, angles stop being a thing on paper. Anything with a hinge is an angle, right now, in front of you.

8:06A pair of scissors. The screw in the middle is the vertex. The two blades are the arms. Opening them is the turn. A pair of compasses. The joint at the top is the vertex, and the two legs are the arms. Spectacles, where the arm folds against the frame. A stapler. A wallet. A laptop lid. A step-ladder, which is a hinge that has to hold a person up.

8:34A door, opening. The hinge is the vertex, and one arm is the door frame, which never moves. That last one's worth pausing on, because one of the arms isn't drawn anywhere. It's just the wall. You can still talk about the angle, because the turn is real whether or not somebody drew the line it started from. Hold on to that. Later in the chapter it's how a ramp has a slope and a leaning tower has a lean.

9:06Now the misconception, and your book kills it with a joke rather than a rule. There's a little cartoon. One child is holding a hinged V-shape, arms sticking out. A second child takes a pair of scissors and cuts both arms shorter. Snip, snip. And announces, proudly, that the angle is now smaller. It isn't. Nothing happened to it at all. Look at what the snip changed. It changed how far along each arm the drawing goes.

9:39And look at what it didn't change. The vertex is where it was. Neither arm turned. Not by a hair. The amount of turn between them is identical, so the angle is identical. This is the single most common mistake in the whole chapter — long arms read as a big angle — and there's even a note to your teacher about it printed later on. So here's the guard against it. If nothing rotated, nothing changed. Length is not turn.

10:13Let's finish with a counting question, because it makes you use the definition rather than the picture. Mark three points, not on one line. Join every pair. That gives you three lines. And how many angles can you name? One at each point. Three altogether. Now four points, with no three of them on one line. Join every pair again. Six lines now. And angles? Don't guess. Build it. Go and stand at one of the four points.

10:45From there, three rays head off, one to each of the other three points. An angle needs two arms, so pick any two of those three rays. There are three ways to do that. So three angles at that point. And every one of the four points is in the same situation. Four points, three angles each. Four times three. Twelve angles. Notice what you just did there — you counted a thing you never drew a single curve for.

11:16Here's yours. Five points, no three on one line. How many angles? Same argument, one step further. Comments below. Next time: how to tell which of two angles is bigger — with no protractor, and no numbers at all.