Acute, obtuse and reflex: one classification covering every angle

0:00We've done all the hard work for this one already, so this video is mostly about tidying up. Two measures are settled. A straight angle is a hundred and eighty degrees. A right angle is ninety. And we didn't look those up. We got them by halving a full turn, twice. Now, here's the way I want you to picture what comes next. Imagine all the angles there could possibly be, laid out in a row, from nothing at all up to a complete turn.

0:32And on that row, put two marks. One at ninety. One at a hundred and eighty. Those two marks cut the row into pieces. And the names we're about to meet are just the names of the pieces. So there's nothing to memorise here. There's a row, and there are two marks on it. Start below ninety. An angle bigger than nothing, but smaller than a right angle, is called acute.

1:01And notice how your book writes it: more than zero degrees and less than ninety degrees. Both of those are strict. More than. Less than. Neither end is included. That matters more than it sounds, and here's why. Ninety degrees is not acute. It's a right angle — it has its own name, and it isn't in this group. So don't think of acute as everything that isn't obtuse. Think of it as a band, with an open end at each side.

1:34Your book gives you three of them, drawn at completely different tilts, so you can't recognise them by orientation. Forty degrees, at a vertex P. Fifty degrees, at a vertex T. And seventy-five degrees, at a vertex F. That last one is only fifteen degrees short of a right angle, and it still counts. All three are acute, because all three are under ninety. That's the only test. And remember where the word came from — acute means sharp. Look at the forty degree one. It's a sharp point.

2:12Now look at the seventy-five. Much blunter, but still on the sharp side of the landmark. The word describes the group, not each individual case. Now the next piece of the row. Between the two marks. An angle bigger than a right angle but smaller than a straight angle is called obtuse. More than ninety degrees, less than a hundred and eighty. Strict at both ends again. Your book draws two: a hundred and ten degrees at a vertex I, and a hundred and thirty at a vertex W.

2:50And obtuse means blunt, which is what those look like. Wide, open corners. So now we've got acute below ninety, the right angle exactly at ninety, obtuse between ninety and a hundred and eighty, and the straight angle exactly at a hundred and eighty. Four names. And at this point most people would stop, because that feels like a complete set. But your book doesn't stop. It asks a question instead, and it's the best moment in the section.

3:25Have we covered every possible measure? Have a proper look at that row before answering, because the answer is genuinely no. Look at what we've named. Everything from zero up to a hundred and eighty. And now look at where the row actually ends. Three hundred and sixty. A full turn. So there's a whole half of the range — everything from a hundred and eighty up to three hundred and sixty — with no name on it at all.

3:55And it isn't empty. Those are real angles. A door swung most of the way open. The hands of a clock at ten past twelve going the long way round. We've been ignoring them since the very first video in this chapter, when I said two rays cut the space around a point into two angles, and we'd meet the big one later. This is later. An angle bigger than a straight angle but smaller than a full turn is called a reflex angle.

4:29More than a hundred and eighty degrees. Less than three hundred and sixty. And now the row is covered. Nothing between zero and three sixty is left without a name. Your book draws two of them, and I want you to notice something about how they're drawn. They have a great big curve sweeping the long way round the vertex. And crucially, they carry no number. There's no measure printed on them at all.

4:57The sweep is the entire piece of information. The curve is doing the job the number would do. Which tells you something important about reflex angles: you can't spot one from the arms alone. You have to be told which way round to go. Let's collect the whole scale in one place. Acute: more than zero, less than ninety. Right angle: exactly ninety. Obtuse: more than ninety, less than a hundred and eighty.

5:30Straight angle: exactly a hundred and eighty. Reflex: more than a hundred and eighty, less than three hundred and sixty. And three hundred and sixty itself has a name too — your book calls it the whole angle. The complete turn. Now, an honest note, because sharp students always spot this. Look at the two ends of that scale. Zero and three hundred and sixty. Acute starts above zero, so zero itself isn't acute. Reflex stops below three sixty, so those two ends aren't claimed by the five groups.

6:09That's not a mistake in your book. It's what strict bounds do, and it's worth noticing rather than glossing over. Now the idea that makes reflex angles actually useful, and your book has an exercise built on it. There's a figure with a vertex T and several rays leaving it, and one of the questions asks for angle P T W. And another asks for angle W T P. Hang on. Those are the same three letters, just written backwards.

6:43And we said, ages ago, that the order of the outer letters doesn't matter — that angle D B E and angle E B D are the same angle. So why are these two separate questions? Because a pair of arms doesn't give you one angle. It gives you two. The one you go round the short way, and the one you go round the long way. And those two always add up to a full turn. Three hundred and sixty degrees. Always.

7:14So if the small one is a hundred and ten, the reflex one is three sixty take away a hundred and ten. Two hundred and fifty. Which means the drawing has to tell you which one is meant, and the only thing that can tell you is the curve. Small curve tucked into the corner: the small angle. Big curve swinging round the outside: the reflex one. So when you draw an angle, and especially when you draw a reflex angle, the curve isn't optional decoration. It's carrying the answer.

7:52Your book then asks you to make one of each on a grid of dots, and it's a good exercise because the grid stops you cheating. An acute angle is easy. Pick a dot, and send two lines out at slightly different slopes. Anything under ninety. A right angle you already know how to build exactly, from the last-but-one video — swap the two steps over and flip a sign.

8:19An obtuse angle: start from the right angle you just made, and swing one arm a bit further round. And a reflex angle is the interesting one, because you draw exactly the same two arms as an acute angle. The arms don't change. What changes is the curve you draw at the vertex — it has to sweep the long way round. That's the whole lesson of this section in one picture: same lines, different angle, and the marking decides.

8:53Here's a nice problem, because it makes you use the landmarks as tools rather than as labels. There's a straight line through a point E, running from B on one side to R on the other. And two rays rise from E: one to S, and one to T. You're given two facts. Angle T E R is eighty degrees. And angle S E R is ninety degrees — that one's a right angle.

9:21First question: what is angle B E T? Well, B, E and R are on one straight line, so angle B E R is a straight angle. A hundred and eighty degrees. And angle T E R is eighty of those. So angle B E T is a hundred and eighty take away eighty. A hundred degrees. Second question: what is angle S E T? Same trick, different landmark. Angle S E R is ninety, and angle T E R is eighty, and T sits inside.

9:58So angle S E T is ninety take away eighty. Ten degrees. And that's the point of the exercise: two subtractions, from two different landmarks, in the same figure. A hundred degrees is obtuse. Ten degrees is acute. Same picture, both groups. Then your book gives you two drawing briefs, and one of them has a hidden feature I want you to catch. The first: draw a capital M, where the two outer angles are forty degrees each and the middle angle is sixty.

10:34Fine. Forty, sixty, forty. Draw it, measure it, check it. The second: draw a capital Y, where the three angles are a hundred and fifty, sixty, and a hundred and fifty. Now, add those three up before you draw anything. A hundred and fifty plus sixty plus a hundred and fifty. Three hundred and sixty. That's a full turn. Which is not a coincidence — in a Y, all three angles meet at the same point, and go all the way round it.

11:08So the three numbers had to add to three sixty, or the Y would be impossible to draw. Now go back and add the M's numbers. Forty plus sixty plus forty is a hundred and forty. Nowhere near three sixty. And that's fine, because the M's three angles are at three different vertices. They're not going round anything. Same-looking brief. Completely different constraint. Worth spotting before you start drawing rather than after.

11:41Two last problems, and then a real one for you. First: a wheel with twenty-four spokes, evenly spaced. Your book uses a particular one; any twenty-four-spoke wheel does the same job. What's the angle between two neighbouring spokes? Twenty-four equal parts of a full turn. Three hundred and sixty divided by twenty-four. Fifteen degrees. Now the harder half: what's the largest acute angle you can find between two of those spokes?

12:15Every angle in that wheel is a whole number of fifteens. Fifteen, thirty, forty-five, sixty, seventy-five, ninety, and up. Acute means strictly under ninety. Ninety itself is a right angle, so it's out. So the answer is seventy-five degrees — five spokes apart. And now yours, and I'd like you to actually do this one, because it has a surprise in it. Find an acute angle where doubling it is still acute, tripling it is still acute, and quadrupling it is still acute — but multiplying it by five makes it obtuse.

12:56Here's how to think about it. Four times your angle has to stay under ninety. So your angle is under twenty-two and a half. And five times it has to get past ninety. So your angle is over eighteen. Which means the answer isn't a number. It's a range: anything strictly between eighteen and twenty-two and a half degrees. In whole degrees, that's nineteen, twenty, twenty-one or twenty-two. Four answers, all correct.

13:28Check one of them all the way through and tell me which you picked. Comments below. That's the end of Chapter Two. You started with a dot that has no size, and you've ended with every possible angle sorted into five named groups, using nothing but halving, counting and two landmarks.