Comparing two angles without measuring either of them

0:00Your book opens this section with animals. Four of them, each with its mouth open by a different amount. An eel, a fish, a chick, and a crocodile. And each open mouth is an angle. The hinge of the jaw is the vertex, and the two jaws are the arms. So: put them in order, smallest to largest. You just did it, didn't you. In about a second, without thinking about it.

0:28And you were right. The eel is barely open. The crocodile is widest. Which tells you something useful: your eye is already quite good at this. For most pairs of angles, looking is genuinely enough. You don't need any equipment. So this video isn't about replacing your eye. It's about what to do when your eye runs out. Here's where it runs out. Your book draws two angles, at a point O. One made by rays to X and Y. The other made by rays to A and B.

1:05Which is bigger? Have a proper look before I go on. I'll be honest with you: I can't tell either. And neither can your book — it says so. They're close enough that looking doesn't settle it. And staring harder won't help. Now, this is the moment where a lot of people reach for a protractor. But hold on. We don't have degrees yet. Degrees are coming, but not for another two videos.

1:33And here's the interesting thing — you don't need them. You can settle which of two angles is bigger completely, with no number anywhere in the room. Three ways. Let's do all three. Method one, and it's the one your book leads with. Superimposition. That's a long word for a simple instruction: pick one angle up and put it down on top of the other. Your book uses angle A B C and angle P Q R. Two angles, drawn apart on the page.

2:08Now lift the second one. Imagine it's on tracing paper. And here's the step people get wrong, so watch it carefully. The first thing you line up is not the drawings. It's the vertices. Q goes exactly on top of B. Corner on corner. That comes first, always. Then, keeping the vertices together, turn the tracing paper until one arm lies exactly along one arm. Q P sits on top of B A. Now the two angles are sharing a corner and sharing an arm.

2:44And now, and only now, there's something to read. Look at what's left. Two arms, both leaving that shared corner, and one of them is inside the other. That's the answer. The one on the inside is the smaller angle. In your book's picture, the arm Q R falls inside angle A B C. So angle P Q R is the smaller one, and angle A B C is greater.

3:13And notice how little you needed for that. You didn't measure. You didn't count anything. You laid one turn on top of another and looked at which one went further. That word further is doing all the work. Further round from the shared arm means more turn, and more turn means a bigger angle. Which is the definition from last time, being used rather than recited. Now the important case, because it's where the definition gets sharp.

3:46What if neither arm falls inside the other? What if they land exactly on each other? Then the two angles are equal. That's what equal means here. Both arms coincide. There's no daylight between them anywhere. And your book gives you a worked pair for this: angle A O B and angle X O Y. You put O on O, you put the arm through A onto the arm through X, and the arm through B lands exactly on the arm through Y.

4:17Equal. Say it arm by arm — that's the check. But look at how those two are drawn, because this is deliberate. O A and O B are drawn short. O X and O Y run much further out. They don't look like the same size at a glance. They are the same angle. Because when you laid them together, the turns coincided — and the turn is what an angle is.

4:44Here's the same conclusion again, without lifting anything, because I want you to see it twice. Forget the tracing paper. Just think about the turn. In the first angle, to get from one arm to the other, you swing round by some amount. In the second angle, you swing round by some amount too. If those two amounts of swing are the same, the angles are equal. Full stop. That's really all superimposition is doing. Laying them on top of each other is just a way of checking whether the two turns are the same turn.

5:22The tracing paper is the tool. The turn is the reason. And that's why the same method works on angles drawn at completely different scales. Now, your book sets a trap here, and it sets it on purpose. There's a figure with a vertex O, and several rays leaving it. One through X. One through A. One through Y. And along one of those arms, two points: B, and then C, further out. Same ray. C is just drawn further along it.

5:56Now the question: which is bigger, angle X O B, or angle X O C? Give it a second. The honest answer is that it's a trick question, and the trick is worth failing once. They're the same angle. Both of them start at the arm through X. Both of them turn round to the same ray. The only difference is which point on that ray we picked to name it with.

6:24B and C are on one ray from O. So angle X O B and angle X O C are two spellings of one angle. Nothing turned. Only the labelling moved. There is a note printed to your teacher, later in this chapter, saying students very commonly think the longer arms make the bigger angle. This figure is the antidote. Same turn, different pen distance, same angle. Right. Now a genuinely harder problem, and it's the one that leads somewhere.

7:00Your book shows two cranes — the birds — each with its beak open, and each drawn inside its own little round frame. Which beak is open wider? And this time you cannot use method one, because you cannot lift either of them. They're printed on the page, in separate pictures. There's no tracing paper in your hand, and no cutting the book up. So the problem has changed shape. It isn't which is bigger any more. It's: how do you compare two angles that both refuse to move?

7:35And the answer is lovely. If the angles can't come to each other, send something to them. Take a circle of something see-through. Tracing paper, plastic, anything. Lay it on the first crane's angle, and put the centre of the circle exactly on the vertex — the corner of the beak. Centre on vertex. Same rule as before, corner first. Now look at where each arm crosses the rim of the circle.

8:06Two places. Mark them both. Your book calls them A and B. And now stop and notice what just happened, because it's the whole idea. The angle was a pair of arms at a corner. Now it's a pair of marks on a rim. You've copied the turn onto something portable, and you did it without measuring a single thing. There are no numbers on that circle. Not one. It's blank.

8:35Now pick the circle up and carry it over to the second crane. Centre on the second vertex. One marked point, A, laid along one of its arms. And now just look at where that crane's other arm falls. In your book's picture, it lands inside — between the mark at A and the mark at B. Inside means less turn. So the second crane's beak is open by less. The first crane wins.

9:04And you compared two angles that never moved, and never touched, using a blank disc. Now here's the sentence I most want you to keep from this video. That circle did not measure anything. It has no marks on it, no scale, no numbers. What it did was turn a comparison of two turns into a comparison of two arcs on one rim. Which means the only thing standing between you and a protractor is marks on the edge of that circle.

9:36That's the next video. And now it should feel like a small step, rather than a new idea. Method three, and this one you should actually build, because it takes about a minute. Two straws, and a paper clip through both of them at one end. That's it. That's a working pair of rotating arms. The paper clip is the vertex. The two straws are the arms. And now you can set any angle you like and hold it there.

10:07Your book calls this a pair of rotating arms, and it's more useful than it looks. You can carry an angle around the room with it. Set it against a door, walk to a window, compare. Which is superimposition again, except the angle now has a handle. And here's the test your book finishes on, which I think is the best thing in the section. Cut an angle-shaped slit in a piece of card. A V-shaped notch, at a fixed opening.

10:39Now take your rotating arms and try to push them through it. Three cases. Open the arms wider than the slit — they jam. They won't go. Close them narrower than the slit — they go through, but loosely, with a gap on one side. Set them to exactly the slit's angle — and they pass through, flush, touching all the way along. So the slit is an equality tester. It says yes to exactly one angle and no to every other one.

11:11And now the part that matters. Your straws might be long or short. Doesn't matter. As long as the arms are short enough to fit through the card, whether they pass depends on the angle and nothing else. Which is the same lesson as the snipped arms, arriving from a completely different direction. Length is not turn. Turn is the angle. Here's yours. Take a rectangular sheet of paper and fold it along its diagonal, corner to the corner across from it. Then open it out again.

11:47At one of those corners, the crease now makes an angle with the long side, and another angle with the short side. Are those two angles equal? And can you settle it with no protractor, using something from this video? Try it on a sheet that isn't square, and then try it on one that is. Tell me what changes. Comments below. Next time: the two turns you already know by heart, without ever having been taught them — the straight angle, and the right angle.