Math Circle Theorems Part 1 July 2026

[00:00:03]So with circle theorems I just going to make the assumption everybody here knows a little bit about angles that's sea you know an angle like for this angle is roughly wrong 30° so angles are measured in degrees right so it could be any angle it could be 30° 60° 81° and of course you can't forget this from primary school the right angle the one that's 90° degrees the the little box now right so you're going to be seeing that a lot today like I said I'm not going to um to spend too much time on this this is sea work everyone should know angles and know angles are measured in degrees well for this particular topic all right so the first thing I want to do is settle in everyone was asked to choose a team to be on for the summer right um I know not everyone read the messages and whatnot. But now we just going to finalize it. There are four teams for the summer and we're going to be playing for points. So, you're going to be representing your teams. Make a choice. We're going to run a vote just now. Just look at the pictures. It's either you're on the red team, the mercenaries, the blue team, the samurai warriors, the green team, the forest majors, or the yellow team, the Vikings. Just have a look at it for if you haven't made up your mind as yet.

[00:01:30]Take a few seconds, make up your mind, and whichever team you decide on now, you're stuck with that team for the rest of the salmon middle. All right, so I'm going get the recording on there. The radius is from the center of the circle to the circumference. The diameter is from one side of the circumference through the center to the other side.

[00:01:58]Remember this dot here is the center.

[00:02:00]The diameter must go through the center.

[00:02:02]And it's real important to remember that the diameter is twice the length of the radius. So if the radius is seven, the diameter would be 14 cm.

[00:02:17]All right. Now, a lot of people don't know what a chord is. Uh, believe it or not, a chord, a lot of people mix up a chord with a diameter. Now, a chord starts on one side of the circumference and goes to the other side of the circumference, but it doesn't go through the center. You see, a diameter goes from one side of the circumference to the other side, right? But it has to go through the center in order to be the diameter. The the chord is is a wannabe [clears throat] diameter. He doesn't actually go through the center. Right?

[00:02:58]So there's some basic parts here. Now one thing I want to concentrate on is this word arc. This is going to confuse a few people. Right? Next one. Let me just take a few seconds to describe that. The full distance around a circle is called the circumference. Basics.

[00:03:20]An arc is part of the circumference.

[00:03:26]For example, let me get um let me just highlight something here for you guys.

[00:03:32]Let's say I start here and go over here.

[00:03:36]Is that red line the full circumference?

[00:03:40]No. This red line is what? An arc. An arc is part of a circumference.

[00:03:52]Right? The whole thing is the circumference. An arc is just a portion.

[00:03:58]Now, one thing I want to specify is that in most textbooks when they specify what an arc is, they usually draw the ark to be what? Small.

[00:04:12]Because look at this. This red line here is small. So, you call it an arc. But suppose I'm just saying suppose, right?

[00:04:22]I made this arc longer like I went all the way here. It's still not the full circumference. That red line is not the full circumference. Is it still an arc?

[00:04:37]Yes. Once it's [snorts] piece of the circumference, and we're not talking about the whole thing, it is an ark. And all arcs will have a beginning. And all arcs will have an end. Like for example, the ark starts here and finishes here.

[00:04:56]Now you could say it the other way. You could say the ark starts here and finishes over here. It doesn't really matter. If one side is the starting point, the other side is the finishing point. Right? Like I said, there's nothing ready to take down here. I do I just want you to listen for the time being. I will make it real real real specific when I want you to write and draw something. For the time being, just listen. All right? I'm a person who says things over and over and over. I'm pretty annoying with it. All right? So, let's start with our first rule in circle theorems. Everyone write rule one. There are seven rules you need for this particular topic. Right now, we're only going to do rule one. And when you finish writing rule one, I want everyone to take their compass or that circle draw that you were supposed to have in class. And I want everyone to draw a decent size circle for me about I would say one/ird the size of of your copy book page. It could be a little larger.

[00:06:07]It could be half. Yeah, it doesn't have to be exact. Now look at the circle. I want you to have two colors in it. You notice the circle goes all the way around, but you see an arc, a red arc. Now, if you don't have a red ink, not a problem. Once the ark is a different color, and you notice the ark has a starting point. I'm calling it A. So, put a dot there. And the ark has a finishing point, B. Make sure and label it just like this. point A and point B. I I I haven't explained what rule one is as yet. You just draw everything on the screen there as it is, as neatly as you can. Make sure and write rule one. And when you're finished, raise your hands so I will know who is finished and I'll be able to gauge when to start my explanations.

[00:07:06]Now don't worry if you don't catch it on the first rounds. I'm a person who says things more than once, right? There's no need to worry or panic. So I want you all to keep your eyes on point A. That's the beginning of the ark. Right now look what's going to happen from point A. I'm going to draw a line. You all see that yellow line. You notice what happens?

[00:07:31]That line starts at A and it hits the circumference. Make sure the line touches the circumference at any point on the circumference. Anywhere on top the circle. I know everybody's going to choose a different slightly different point, but draw a line from point A and let it hit somewhere here. The let it touch the circumference, but it must touch the circumference. This rule doesn't work unless it touches the circumference. Now look what's going to happen. The line is going to start at point A. It's going to hit the circumerence and it's going to bounce back to point B. Let me show you that.

[00:08:17]In other words, this these two yellow lines, it starts at the beginning of the ark. It hits the circumference and bounces back to the end of the ark.

[00:08:31]Remember the beginning of the ark is point A. The end of the [clears throat] ark is point B.

[00:08:38]Right? Now, if you use your imagination, use your imagination here. If I draw this a stick man, doesn't it look like this stick man? Like the two yellow things are his legs. So it comes like the stickman is standing on the ark where one let me draw his face here in case you're having problems seeing that.

[00:09:02]Right. So it looks like the stick man is standing on the ark where one leg is at the beginning of the ark and the other leg is at the end of the red ark. Okay.

[00:09:17]So it look I want you all to keep that that um word in the back of your head.

[00:09:22]Standing. He's standing on the ark.

[00:09:26]Okay, I think everyone has to agree that there must be an angle in between here. Now, I don't know the exact size of the angle. It could be 29°, it could be 30°, but you could be sure that there will be an angle here. Like if you I'm not asking you to use your pro your protractor but let's say I'm just guessing. Let's say the size of this angle is 28°.

[00:09:58]I'm just saying if here's the amazing thing about circles.

[00:10:05]If I were to draw another line starting at point A. I'm going to draw another line starting at point A. But look what's going to happen.

[00:10:15]It starts at point A, but it that pinkish or reddish line. It starts at point A, but it hits the circumerence at a different [snorts] point and then it bounces back to B.

[00:10:33]I I color coded it. You all notice a slight similarity between the yellow legs and the red legs.

[00:10:45]Both yellow and red legs start at point A at the beginning of the ark and end at point B the end of the ark. The only difference between these two pair of legs is that both of them hit the circumerence at different points. That's the only only difference. Example, for those who are having trouble visualizing it, not a problem. I could always draw back the stick man here. This is the yellow stick man. And I could probably draw this one.

[00:11:25]You look angry. All right. So, it's two pairs of legs. Now, here's the amazing thing about this. In maths, this angle will automatically be 28 degrees. It's like magic. But it only works if both of them start at the beginning of the ark, the same ark, and both of them end at the same exact ark.

[00:11:52]It doesn't matter if they are if if it's hitting different points on the circumference. It's magic. I like to call it magic, right? So, would it work for a third pair of legs? If we had a third pair of legs starting at point A, look at it. I have some blue legs here.

[00:12:14]Starting at point A, it hits the circumference here lower, much lower.

[00:12:20]But the point is it's starting at A.

[00:12:22]It's hitting the circumference.

[00:12:24]If it bounces back to B, will over here still be 28°?

[00:12:32]And the answer is yes. That will also be 28°.

[00:12:39]Once again, think of it like as magic.

[00:12:45]Right now, how do you put this rule in words? The examiner will ask you to put this in words.

[00:12:54]you you might know what it is, but when it's time to write it out in a paragraph, some of you might not be able to do that. So, this is the rule here.

[00:13:04]Angles standing on the same arc. Notice the blue legs, the yellow legs, and the red legs are all standing on the same arc.

[00:13:16]That's this red ark in my example. And touching the circumference have the same size. We see that all the angles are automatically equal. So if this one was 33, I'm just saying if the rest of them would automatically be 33 as well. I want you all to draw this diagram as neatly as possible. If you want to draw the stick man, feel free. But take down the diagram as neatly as you can. take down the rule one in words and then take down this rule where the eyes must be looking in the same direction. I haven't explained that as yet. Just take down everything on the screen and I'll explain the eyes thing just now for you guys. Okay.

[00:14:04]When you're finished, raise your hand so I know how to gauge the class.

[00:14:10]I don't know if you all are into like um drawing or or what you call it um sketching drawings. You know sometimes when somebody draw a eye it is look like this or when you go in a textbook a science textbook you just draw an eye like that. I'm not too sure if you all see the eye there but if I put in a nose and a mouth and some teeth here right?

[00:14:35]You all see a eye right? So this person is looking this way.

[00:14:40]So y'all see um that that's how they draw eyes in textbook, right? They draw it like this, right? So oops, sorry. The rule only works if the eyes are looking in the same direction.

[00:14:54]But what eyes? Well, what what is sir talking about? You don't see any eyes.

[00:14:59]But look at these angles here. Right? If I draw this here and I draw this here and I draw this here and it looking like the eyes, this looks like eyes. This eye is looking this way. This eye is looking this way. This eye is looking this way.

[00:15:19]I want you all to put in that same thing. Convert all of them to eyes. Take the the angle and just put a little loop, half a loop there. The rule only works if the eyes are looking in the same general direction. So look at it.

[00:15:35]The blue eye, the yellow eye and the red eye are all kind of looking downwards well towards the ark. So this rule applies. It's okay to say this is rule one and only in that scenario all the angles will be exactly the same. All right.

[00:15:58]Right. Now to to ready really already grasp these rules and very few people grasp all of these rules 100%. Right.

[00:16:09]That is normal. I custom with it. I'm going to explain the rule once like a few times here and then you'll see me do a whole set of examples on it and we'll keep practicing the rules. All right?

[00:16:20]You will get it before you leave here.

[00:16:21]So don't panic right if you didn't catch it the first rounds. Now it's very important to note that the angles must touch the circumference. Now you might be saying, "Oh God, sir say that so much time. That's obvious. We know. We know it must for this rule to work. It must touch the circumference."

[00:16:40]There's a good reason I'm saying that and you're going to see why everybody's going to forget about that in a few minutes. Okay? So please forgive me if I say it over and over. Let's move to the next slide. Another version of rule one.

[00:16:57]You all notice the slight difference between this rule. This is still rule.

[00:17:01]We still on rule oneh. But you notice in the last example the ark was underneath.

[00:17:09]Suppose the ark is on top.

[00:17:13]Could we still have a rule one?

[00:17:15]Obviously it doesn't matter where rule where the ark is. The point is for it to be rule one, it must start at the beginning of the ark, hit the circumference, and bounce back to the end of the ark. So, everyone draw this diagram for me with the ark, the red ark on top. Okay, when you're finished, raise your hands so I know how to explain the slightly different version of rule one. In fact, it doesn't matter where the ark is. The arc could be at the top underneath to the left side to the right side. Okay, and down right look at it.

[00:18:05]A line starts at point A, the beginning of the ark. It hits the circumference and it's going to bounce back to point B. There must be an angle here. Now, I don't know what the angle is. Let's just That looks like a small angle. It looks approximately 11°. I just approximate it, right? So, it starts at the beginning of the ark, which is the red.

[00:18:29]We're concentrating on the red ark. It hits the circumference. Key thing, it must hit the circumference and bounce back to the end of the ark. How about we have another line? Look at that orange line. It started at point A, the beginning of the ark. It hits the circumference again. And guess what?

[00:18:50]It's going to bounce back to the end of the red arc. And it does, right? Well, no surprise. Obviously, there must be an angle here. I guarantee you if the blue one is 11°, I bet all my money, all $5 the the orange one is also 11° right now. I know this is definitely rule one.

[00:19:21]If I apply the eyes, if I change this into an eye by putting the eyeball here, the iris, however you call it, and both eyes looking the same general direction, both of them looking towards the ark. So, it's okay to say, hey, [snorts] this is rule one. And both of them are looking in the same direction.

[00:19:45]If the eyes are not looking in the same direction, cuz the examiner will try to fool you. He will give you something looking like rule one. It'll be like an illusion. It will look like rule one.

[00:19:57]But if you look really really properly, you go realize the eyes not actually looking in the same direction. Like one eye will be looking upwards and the other one will be looking in the opposite direction downwards. And don't let the examiner catch you with that. If the eyes are not looking in the same direction, it's not rule one. And you cannot assume both angles are the same. So everyone draw this diagram. There's no need to retake this down. You already have this note here. I just need you to draw this diagram exactly like how I have it with the eyes and the little dotted arrows and everything. Okay, finish. You know, you know what to do.

[00:20:42]Yeah, we'll feel flat.

[00:20:44]So look at the circle on the left hand side of the screen.

[00:20:49]It has an arc, a red arc. It starts at A. It finishes at B. So what's going to happen? It's going to start off just like rule one. I'm going to draw a line.

[00:20:59]Look at that yellow line. It starts at point A. It hits the circumference again. Just like rule one. And it's going to bounce back to point B. See?

[00:21:13]starts at point A, hits the circumference, and bounces back to point B, which is the end of the ark.

[00:21:22]There must be an angle here. Let's just assume the angle is 21°. Just put in 21.

[00:21:28]Let's I mean, it looks like 21 on my diagram. So, like I said, at first glance, people are going to roll their eyes and say this is the same thing as rule one. But it can't be. If one is named rule one and one is rule two, there must be some difference between rule one and rule two. I get into it.

[00:21:47]Don't worry, be patient. Be patient. Right.

[00:21:54]Right.

[00:21:56]Right. So, what's going to happen now?

[00:21:59]We're going to start back at point A.

[00:22:01]Everyone focus on point A again.

[00:22:06]You're going to draw a line, but this time, look at the green line.

[00:22:10]The line started at point A, but it didn't hit the circumference this time.

[00:22:17]Remember in rule one, the other pair of legs would have hit the circumference.

[00:22:24]This one hits the center instead. And it bounces back to the end of the ark, which is B. So you see the subtle difference. One pair of legs is touching the circumference. That's up here. And the green pair of legs is touching the center now. It must touch the center.

[00:22:45]Here's the cool thing. The one at the center. Doesn't this look a little bigger? And this in this angle is definitely looking bigger than the yellow angle. The cool thing about it, the green one, the one that touching the center, this always be twice as large.

[00:23:03]Always, always, always. It's like magic.

[00:23:06]So this green angle will be 42° cuz if you take 21 and multiply it by two you get 42. If over here was 10 over here would be 20. If over here was 50 down here would be 50. The point is the one at the center always be twice as large.

[00:23:25]Or another way of saying it the one at the circumference the one who's touching the circumerence is half the value. Right? Any way you want to see it, say it, different textbooks will say the same thing two different ways. Okay? But the eyes must be looking in the same direction. If I change change this green one into an I and this yellow one into an eye, you notice both eyes are looking towards what? And both eyes looking downwards.

[00:23:57]Both of them looking towards the same arc. It's only rule two if the eyes are looking in the same direction. I'll put it in words just now for you guys. You don't have to worry. I will put it in words for you all. I just want you all to understand the concept, right?

[00:24:14]So, will it work if it's upside down?

[00:24:18]Obviously, cuz if you take your your book and you turn rule two upside down, isn't it still rule two? So, this rule should work if it's right side up, I guess, or upside down. So, I'm going to draw the upside down version here, right?

[00:24:35]Starts at B. Well, it doesn't matter which one you call the beginning or the end, right? In this case here, the beginning is B and the end of the arc is A. It's the same arc. It hits the circumference and bounces back to the end. There must be an angle here. Let's assume this angle is 50°.

[00:24:53]If we have another angle starting at point A, but it hits the circumference instead, it sorry, it hits the center instead and bounces back. Look at this angle here.

[00:25:09]This will be 100°. The one at the center will always be twice the size of the one touching the circumference. And if I convert this to an I and I convert this to an I am and both eyes looking towards the same arc or in the same general direction. So this is rule two and this is how you put rule two in words.

[00:25:36]Write down rule two here in words. And yes take down back this note here. The eyes looking in the same direction also applies to rule two just like [clears throat] how it applied to rule one. Right? So the subtle once again the subtle difference between rule one and rule two is one pair of legs touches the circumference and the other pair of legs touches the center. It must must must must must touch the center. It can't be 1 millm away from the center. It has to hit exactly the center, right?

[00:26:14]When you're finished, take it out everything. You all know the drill.

[00:26:17]Raise your in the next time. Now, there's a little trick to rule two that the examiner likes to play on you. I'm going to say this again, right? For it to be rule two, the angles must be looking the same direction. Keep that in the back of your head cuz annoying sir is saying it over and over. He must have a reason, right?

[00:26:44]Well, this is nugget. Quick question. Is this rule two? Is this diagram here rule two?

[00:26:54]Yes or no? Let me know on the chat.

[00:26:57]Remember, only I could see your answers.

[00:26:59]Well, not on YouTube. Everybody on YouTube if you send an answer, everyone could see it.

[00:27:03]>> [laughter] >> But what I'm asking on Zoom are is this rule two?

[00:27:10]Well, the answer is obvious, right? For those who read who read what's on the screen, it is rule two. But how? You're not seeing two pairs of legs. You're not seeing four legs, but two legs are overlapping. Look at this. If here, if I call here A and I call here B, this red leg is starting at A. It's hitting the circumference. Look at it. It hits the circumference and bounces back to B, the end of the ark. So, let me just assume this angle is 10°. I'm just saying if.

[00:27:52]Now, look at the blue leg. You have a blue leg starting at A.

[00:27:57]It hitting what? The center. That dot is supposed to represent the center. In the exam, he will put the letter O. And then it kind of bounces back and overlaps.

[00:28:09]I I should probably put it as a dash line so you could see what the blue and red or pink leg at the same time. You notice it have two legs here. our blue leg kind of resting on a pink leg, right? So the blue angle starts at A, it hits the center and bounces back to B. It's just in this case here because two legs are kind of overlapping. You don't clearly see four legs, but it's still rule two. So that means this blue angle is 20°. The one at the center is twice the one at the circumference. And if you're drawing your little eyes, look at this.

[00:28:58]This eye is looking downwards and this eye is looking downwards. Both of them looking at the arc AB. They're looking the same general direction. Doesn't have to be the exact exact exact same direction. Just the general direction must be the same. So this here is a trick question but then have another trick after this. So everyone draw this diagram and take down the note. Right?

[00:29:23]Right. So I hope you all seen it there.

[00:29:24]I think that's the best I could explain this. I don't think I'll be able to explain it better than this. Right. If I explain it a different way I think it'll be the same explanation. It's just two one is touching the center and one is touching the circumference. And here you have the overlap of legs.

[00:29:42]Right? It doesn't always have to be two distinct pair of legs. You don't have to see four. Right. Draw it. When you're finished, raise your hand.

[00:29:55]Right.

[00:29:58]Right.

[00:30:00]Um, but this is just the upside down. I'm not going to make you draw this one.

[00:30:05]Right. This is just the up. This is just two different versions. This is not a trick, by the way. Um you notice one is touching what? One of them is going to the center and one of them is going to what?

[00:30:19]The one is going to the circumference and one is going to the center. Right?

[00:30:23]The the thing about this one, the one that's going to the circumference is just a little bit lower. Most people are accustomed seeing the one going to the circumference being very high or kind of directly over the center one. But but that's not the actual rule, right? For it to be rule two, one pair of legs has to go to the center. This is the center. Look the dot there. And the other one has to go to the circumference. and both eyes if you draw a eye here and a eye here and both of them looking in the same general direction towards this arc here. So this is rule two. In other words, if this one was 10°, the one at the center would be 20°. And this one here is the upside down version of this one. Like if you were to take this and spin it so it's upside down, it's the same thing. Now, I'm not going to make you draw this, right? I just wanted to point it out there. We'll be doing we're going to have plenty examples today like to work this out um to practice these things.

[00:31:33]Sorry. You're going to be seeing these things over and over and then I'm going to remind you of this diagram. In any case, so do not take this down. It's not necessary. I want to get to the trick question with rule two. That's my main thing.

[00:31:51]All right. So, I have a question for you. Let's say for a million dollars, which obviously I'm not going to pay you, but let's pretend that I would I would part with a million dollars.

[00:32:04]If this green angle is 100°, you'll need a calculator, right?

[00:32:15]What's the size of this yellow angle?

[00:32:20]First of all, is this rule two?

[00:32:26]Anyway, send your answers.

[00:32:29]Think about it. I I don't care if you get it wrong. The only thing I care about is that you try your best and produce an answer.

[00:32:38]I prefer to see a wrong answer than no answer at all. I just need to know you're trying on that side. So, everyone draw that diagram and then you could send your answer on the chat. What's the size of the yellow angle for a million faith?

[00:32:55]Let me explain the the build up of getting this question wrong. Let me explain the thought process. So, you open your exam paper and you see this craziness.

[00:33:08]All you're going to see for the time being is this. One pair of legs touching the center.

[00:33:18]Another pair of legs touching what? The circumference. And that's what the majority of you all um seeing there. At the end of the day if you had to glance that as a unexperienced person the first thing that going to click in your head I know sir tell me something about one touching the center one angle touching the center one angle touching the circumference and he say some madness about one of them being twice as large as the other. Right? So the one touching the center that's this guy. He's the center guy. Isn't he the one who is always twice as large? You can turn back in your notes. The one touching the center is always be twice as large. So naturally you will say 200 because 200 is twice as large as 100. I mean technically that is what I kind of tell you. And some of youall would have just guessed 50 because you know one of them have to be half the other value. So you just say all right I just going to say 50. So, some of you all would say 200 and some of you all would say 50.

[00:34:30]Well, you have a little problem there, right? Let me explain why.

[00:34:35]Remember, for it to be rule two, the eyes must be facing the same direction. So, if I take this angle and convert it to an eye and this eye, isn't the yellow one staring downwards?

[00:34:54]and is in the green one staring upwards.

[00:34:58]Are they looking the same direction? Yes or no?

[00:35:02]Is that a situation where two eyes are steering the same direction?

[00:35:11]To me, that's the complete opposite.

[00:35:14]That's a case where two eyes look in the complete opposite direction. That is not rule two. That is not rule two. Not yet.

[00:35:23]But if you use a little bit, if you think outside the box, you will realize, wait, now this is rule two. You all remember something from primary school.

[00:35:33]How much degrees does a circle have? Like a full circle.

[00:35:38]We all know like a triangle is 180° or all three angles. A circle, isn't it 360°?

[00:35:47]All circles have 360°.

[00:35:52]Do do you all remember this? Right?

[00:35:56]Um you could take a circle and break it up into different angles. Like for example, instead of looking at the yellow angle, how about this angle here?

[00:36:06]Isn't there an angle behind this one? If I combine the red angle and the yellow angle, won't that be a full circle? Think about it.

[00:36:18]Let me erase this clutter here. Isn't this a full circle here?

[00:36:23]And if I take this big angle and convert it to an eye, isn't he looking upwards to look at that situation? I know they didn't tell us nothing about this red angle, but you got to use a you got to think outside the box. Behind this yellow angle is the rest of the 360° to make a full circle. Once again, the red angle plus the yellow angle is a full circle. And that total would be how much? 360°.

[00:36:56]But but the more amazing part, but the more convenient part is the green eye and the red eye are now looking in the same direction. Now I know they ask you about the yellow eye. They want to know the value of the yellow angle. But we need to find that red angle first.

[00:37:17]Right? The one touching the center is the red angle and the one touching the circumference is the green angle. And the green and red are facing the same direction. So rule two is based on the red and the green.

[00:37:37]Anybody wants to tell me the size of the red angle now? Not the yellow. Don't focus on that. If the green one is 100° using rule two, because the red one is touching the center and looking the same general direction as the green one, won't the red one be 200°?

[00:37:57]Won't it be twice as large as the green one? Look at it. The one at the center looking in the same direction as the green will be twice as large.

[00:38:09]You think you all could figure out the size of the yellow angle? Now, what would be the size of the yellow angle?

[00:38:15]If we just reason out using rule two that the red angle is 200 and a full circle is 200°. It come like this. You have a full circle which is 200°. If this piece here, sorry.

[00:38:33]If this piece, if you have a full circle and this piece is 200, how much is the yellow part underneath?

[00:38:43]A full circle is 360. You have 200 already. How much is the yellow portion?

[00:38:52]Easy question, guys. Everyone's supposed to get a car from here. I giving him a best here. Giving him my best here.

[00:38:58]Right.

[00:39:01]Right. If you're in form one and form two and remember this class is mainly for form for people going to form three and above, right? If you're in form two, form one or form two or you're going into form one, don't feel too bad if you don't catch everything, right? But my piece is is more to suit the form three and above today, right? You don't have to pick up on everything. You just have to pick up on some of the things, right? It's how much?

[00:39:26]60°. All you had to do was take 360 and minus the red portion. 360 minus 200 is 160. So the yellow angle is what? Oh, that's a terrible color there.

[00:39:40]Let me get better color. Is 160°.

[00:39:44]Everyone take down everything on the screen there for me. Right.

[00:39:53]You don't have to take down this note here. Right? They that is just irrelevant right now.

[00:40:00]Draw the diagram and put in the answers in it. Right? The main thing here is to understand the reasoning right now.

[00:40:07]Right? Is to understand the reasoning.

[00:40:10]It's this is me focusing on the fact that not because something looks like rule two because something touching the center and something touching the circumference means you could automatically say one is twice the size of the other or one is half the size of the other. You have to keep remembering that part that was that I was so annoying about. I kept reminding you for it to be ruled too. The eyes must also be looking in the same direction. Must, must, must, must must. Right? And this here I would say is the most popular trick when it comes to rule two in the exams. The most popular trick in the exam when they ask you to find an angle, they they don't already point and say, "Hey, find this angle or find that angle." They just always put three [snorts] letters to state which angle they want you to find. And you know that that it could be a little confusing at first. If they want you to find angle A, D, B, right, and you're kind of trying like you're not too sure which angle they want you to find, always look at the middle letter. That that's the best way to do this. The middle letter.

[00:41:27]You're probably studying, but why did they give you the middle letter? Why did they give me three letters and I have to know it's the middle letter? Ah, that's life. All right. So, in the first one, technically they're asking you to find angle D, which is this angle here. All right. And in similarly, in the next one, the angle they want you to find is angle C, which is this one. And in the next one, they want you to find angle B. Now, I I I know I said I was going to work out this first one for you, but you think you all could give this one a try?

[00:42:05]I I just give it a try, right?

[00:42:09]I want you your best to try to figure out all three angles. It's either rule one or rule two, or you have to use both of them. Remember, we only did two rules so far. So, using rule one and rule two, don't guess.

[00:42:22]Try to reason on which one is which one and send your answers on the chat.

[00:42:28]Right? Once again, I don't care if you're wrong. I only care about honest attempts like your like a real attempt even if it's wrong. Right? For those who know me, you know that that's my policy.

[00:42:42]Right? I have no problem with wrong answers.

[00:42:46]Good luck. And I give you three minutes, one for each one, right?

[00:42:57]go to the next slide. Right. So, I only concentrated on part A, but I colorcoded for my explanations. I colorcoded it. In the exam, it would just be black and white and you ain't getting no colors in the exam. Right? So, each pair of how you say um associated lines I put in different colors. You'll see the blue, you'll see a green, you'll see a kind of purpley and a orange. [snorts] So the first one they ask us to find what? Angle A D B. So that's this angle here, D. They want us to find I'll color code it. They want us to find this blue angle here. That's angle D. Now, I have a good feeling this is rule number two.

[00:43:49]If you if you forgot what rule number two is, you can just quickly turn back.

[00:43:52]Obviously, rule number two is the one that talks about one going to the center and one going to where the circumference.

[00:44:04]And the one going to the center, the center one is the one that's twice as large. Right now, I know you're not seeing the um the arc. The examiner won't actually highlight the arc for you. But if I take point A and connect it to point B like that, isn't this the ark here? The ark starts at A and it finishes at B. And look at the blue lines here.

[00:44:35]One pair of legs starts at A. the beginning of the ark, it hits the circumference and bounces back to B. So that's the blue pair of legs. Now look at the purple set of legs.

[00:44:53]One starts at A, the beginning of the ark. It hits the center center and goes back to B. But are the eyes facing the same direction? If I draw a eye here, isn't he facing kind of looking kind of upwards? And if I draw a eye here for the 144, is he isn't he looking kind of upwards? Both of them are looking in the same general direction. They're both looking upwards.

[00:45:23]And one touching the center and one touching the circumference. Like I said, that's a clear case of rule two. Which one is always be the larger one? The center. the circumference will be half the value. The one at the center is twice the value. So by default, the one at the center the the circumference is half. So if you take 144 and divide it by two, you're going to get 72°.

[00:45:50]So this is 72°.

[00:45:52]How much people got that correct? I wasn't able to check all the answers.

[00:45:55]You can just raise your hands. 72 degrees. And you have to state the rule that you use. All right.

[00:46:03]Now, this is how you're going to write it for the examiner in the exam. You can't just say rule two. Let me explain why.

[00:46:13]If you're going different textbooks, there are seven rules you're supposed to learn off in all. Do you really think the textbook has if you go to 10 different textbooks, do you really think they have the the rules in the same order?

[00:46:28]No. Right? They will have it in different order. The examiner won't know what rule two is for you. I mean we I just calling it rule two so we could communicate in class. In the exam you have to do what? Give a full reason.

[00:46:44]Yes. How? Yes. I know you're probably rolling your eyes. You have to write a small paragraph a statement of how you arrived at the angle A DB being 72.

[00:46:59]Right? And when you're quoting angles, you have to quote three letters. You can't call this angle O. A O B. So this angle is A O B because the angle starts at A, the middle is O, the 144, and the angle extends out to B. So when you're naming angles, you have to use three [snorts] letters in the exam. You just can't say well a angle D is half of angle O.

[00:47:35]Angle O official name is A O B like this green angle here. His name his official name is not seen. What what two letters C extending out to? A and B. So you put C in the middle and you put A here and B here. Or you could put C here and put B and A here. When you're naming angles, you'll get accustomed to in a little while. You have to know how to name it for the examiner. So everyone, if you notice, I wrote back rule two here and I explained to the examiner my thought process.

[00:48:10]Read through this. Make sure you understand. It's the same thing we just did. It's rule two. I just applied rule two, but you have to write it out for the examiner. When you're finished reading and taking it down, raise your hands.

[00:48:23]Like I said, this is the hardest topic I I think it have in CXC months because you have to be able to word out things, reason out things, see patterns.

[00:48:34]It's not just calculations, it's pattern recognition.

[00:48:40]Right?

[00:48:42]We still have part B to do. So for part B, they're asking us to find angle A C B. Now remember, we just figured out this angle here down here was 72°. We now figured that out. So update your diagram and make sure you have 72 down there. They want us to find angle AC. A C B. That's this green angle here. They want us to find. You all realize this is rule one. I'm not too sure if you all saw that. Let me put back in the ark.

[00:49:27]Okay, look at this. This is clearly rule one.

[00:49:32]Forget the 144°, right? Look at this. The ark starts at A.

[00:49:40]It hits the circumerence and bounces back to B, the end of the ark.

[00:49:48]And the green one starts at A. It also hits the circumerence and bounces back to B. And both the eyes, the green eye and the 72, the 144 have nothing to do with this. He have not to even look there. This blue eye, well, the 72 is looking upwards and this green one is also looking upwards. This is clearly rule number one. Both of them touch the circumference. Both of them start both legs start at A, finish at B and looking in the same direction. So that means the green one is also 72. Remember rule one said both of them would be the same exact angle. Right? But how you put that in writing for the examiner?

[00:50:39]Remember you can't say hey examiner I use rule one. He won't know what rule one is. You have to write out what rule one is. So you have to memorize the in words word for word what is root one.

[00:50:52]Rule one and then you write a little paragraph showing you how showing the examiner how you arrive at A C B is also 72°.

[00:51:04]That is what you have to write or something very very similar to it to get full marks in the exam. Write it down and when you finish raise your hands.

[00:51:16]Right.

[00:51:18]Can't remember what part C was. Part C is angle O D A.

[00:51:28]Now this one is a little hard to if you trace the three letters O B A. You see if I if you look at the middle letter here, you don't know whether B is this angle, this angle, this angle, or the whole thing, right? It could be a little confusing which one is B by looking at the center letter. So in a case like this, you got to be a little more accurate. O B A. Trace the three letters. Start at O, go to B, and then go to A. And let's see where the center of that is. You start at O, go to B, and then go to A. It's this angle they want you to find. Right?

[00:52:11]So, let let me draw that in a little bit neater for you all. They want you to find this angle here between the pink well purple line and the orange line.

[00:52:23]Right now, we got we're going to have to review something very very quickly here with sea maths. You all remember something called an isosles triangle.

[00:52:33]Remember an isosles triangle two sides have the same length and two angles are the same.

[00:52:43]Tell me something. Are these two purple lines the same length?

[00:52:49]Isn't this purple line a radius ra?

[00:52:53]Isn't this purple line another radius?

[00:52:55]Because both of them starting at the center and going towards the circumference.

[00:53:00]then both of those would have the same length cuz two radius it's the same circle right so if here's 10 cm if then here's also 10 cm so that means that this here is an equilateral triangle this length and this length is the same remember the symbol for same length the Apollo's stroke in um Apollo's stroke like that in sea maths okay so those two lengths are the same which means means this angle and this angle is the same.

[00:53:34]These two angles are the same. It's an isosles triangle.

[00:53:40]Remember how you find they call these the base angles. The two angles that are the same are called the base angles. You think you can finish off that question now using maths. If in in the maths exam they gave you an equilateral triangle sorry isoclesles triangle like this and they told you here was 144 and they're asking you for this angle here.

[00:54:03]How would you find that angle? Remember all the angles in a triangle add up to how much? 180°.

[00:54:11]And these two angles here this one and this one have the same exact value.

[00:54:18]You can more than finish off that question for me. Right? You just have to remember if two radius have the same length and they create a triangle, it's going to be an isosles triangle.

[00:54:32]What's the value of angle peak?

[00:54:39]You can use a somebody asked you. Yeah, you can use your calculator.

[00:54:43]Definitely use that calculator. Whenever you can't use a calculator, I'll make it super clear. Right.

[00:54:57]Right. So, answers are coming up. It's just you take the 180° because a full triangle is 180. You minus the 144°. Let me do that. 180 minus 144 is 36°.

[00:55:15]So the 36 remaining 36 has to split up evenly between these two cuz these are the base angles in the isocles triangle.

[00:55:24]So you divide it by two so you get 18°.

[00:55:28]So this angle will be 18° and like I said that's based on sea maths right once you've realized wait now. So this question really the third part had nothing to do with rule one or rule two.

[00:55:41]It came back to what? Sea maths, right? But let me put this in words. You have to be able to express this for the examiner. You just can't drop 18 on the screen there on your exam paper and expected to get it correct. Right? Look at here. I have the ex well something very similar to what to say to get full marks. Now, you got to keep in mind that for the exam, everybody's going to write these things slightly different. But once you convey the same message, you're going to get the full marks. Read it, take it down, raise your hands when you finish. We have a few things