Triangles & Their Angle Properties | IGCSE/O-Level Mathematics | Class 26

Generated automatically from the lecture captions, so spoken maths may be transcribed imperfectly.

Okay. So we're continuing with uh geometry topic that we started previously. So for today's class we had previously discussed all of the basic geometric angular triangle properties and for now we moving on to the uh angle properties of triangles. Geometric properties angle properties of triangles. Sorry. First of all, we need to understand how many types of triangles are there. So you people might also know about this right angle triangle. This is very common right angle triangle and then we have obtuse angle triangle. We have acute angle triangle. If anyone knows about this [snorts] right angle, I expect that

everyone must know about the right angle triangle. Uh what is it? If anyone can tell me [snorts] right then we would discuss bit more. Right angle triangle is when you have a right angle in it. Regardless of any other angles that we have in the triangle, we should have one 90° angle. Perpendicular 90° angle. Whenever you see a 90° angle, we call it a right angle. Simple. Obtuse angle triangle. We know what is an obuse angle. It is an angle which is greater than 18. Sorry, greater than 90. So if we have a triangle in which any one of the angle is greater than 90 you can see this is greater than 90. Let's say this

is 110. So this is an example of a obtuse angle triangle. For acute angle triangle this is very simple. All of the angles would be less than 90. So you can see this is less than 90. This seems to be greater than 90. Let me make up a more simpler one. H this you can see all of these may be equal or not be equal but all of these you can see are definitely less than 90. You can obs with simple observation you can see no no angle is greater than 90. [snorts] Then we have a little bit more about the triangles. We have equilateral triangles. equal. We have scaline and we have uh what do we call it? Isoselus

equal little triangle. To understand this a simple example is when you have everything equal for all three sides are equal to each other and all three angles are equal to each other. How much is the angle of an equal equal little triangle? Who will tell me? [snorts] Zenob what is the angle of? >> Every angle is 60. >> H yes 60 because 60 60 and 60 180 divided by 3. There are three angles in the triangle. Uh total and each of them is equal. So the total sum of angles is 180. So 180 divid by 3 is simply 60. And all of the sides will also be equal. If this is five then all three of them would be five=

a formula. Yes, you can generate a formula but there is no specific. You need to just simply understand what is it. Yes, you can make up a formula for it. A scaline triangle is when you have everything different. The angles are different. All three angles are different. And all three sides are different. No side is equal to the other. Everything is different. For isocles triangle you have two sides and two angles equal. So this and this angle you can see they are looking uh like equal and this is different. Similarly the two sides are equal. So if this is five, this is five and the other side may be different. It won't be five.

Similarly, if this is 30, this is 30 and the other angle definitely won't be 30. So you have two angles equal and two sides equal. This is called an isocles triangle. There is nothing much over here to uh what do we say? First you have to understand all of these names and according to how how they look like and then you work on it. Further more we can study a little bit more on the isosis triangle. We have some special >> sos special properties of isosis. Let's make up a big triangle. And if you understand this thing a lot of things would be would become easier then okay so in isocus triangle there [clears throat] are two angles

which are equal. So you can see this angle and this angle. Let's say these are equal. I told you how do we show equal angles. you have same number of arcs on the angle. You can see both I I drew two arcs on both angles. So let's say this is point A, this is point B, this is point C. So I can say now that angle A B C is equals to angle A C B. We have to understand how do you write or show which angle that you are referring to. >> [snorts] >> There are two three parts of this. You have this thing. When you write this uh you can say something like this symbol. When you draw this symbol, this is showing you that you're talking about

the angles. Okay. And another thing, another symbol is this one. This which I drew over the B. This symbol this is telling me that the angle lies on point point B. So look you can see A B C means that it is starting from A. The angle actually is at B and then it is going to C. So look you can see the angle is on B. That's why I have this sign over the B not on A. The angle is not on A. So that's why I drew like this A B C. The angle is on B. Similarly, on the other one, I wrote A B C. The starting point of the angle is A, then A to C, then C to B and the angle is on C. So, you should understand what how do

you write this in your exam to show the examiner if they if they're asking you to prove or show something the angles or if they write something that this angle is equal to this angle. So, you should understand what which angle are they talking about. So, there are two things over here which you need to understand. This symbol this is referring that they're talking about to the about the uh lines sorry the angle and this symbol is telling you on which point the angle is lying. Okay then so this is one of the properties that two angles are equal and the sides that are opposite to that ang to those angles they are also equal. So

this means the side which is opposite to this angle and this this angle these two sides would be equal. So I can say that that a b is equals to a c. This is how you show that the sides are equal. [snorts] So you write like this that length of a is equals to the length of a c. Okay, this point the other point the other angle point which is different from the other two angles we call this the vertex. Vertex of isosis triangle the other angle which is not equal to the two other angles. We call this angle the vertex of the triangle. Vertex of the assoc triangle. And now one more important thing is when you draw a

perpendicular from this vertex to the base. Look what I'm drawing. This is seeming not to be equal. But the property is let me make you Let me make some corrections. This and then this is okay. The green line you can see the green line is a line which is from the vertex and dropped to the base of the isocosis triangle and it is crossing the isosis triangle [snorts] at the base in such a way that this would be perpendicular and uh the base is bicted. Now it is in is divided into equal two halves. So if I write this down perpendicular from vertex to the base biseects it. So if I if I name this point x, this

would mean that bx is equals to cx. bx is equ= to cx. And you can also say that a x b is equals to a x c is equals to 90 perpendicular. You have to understand what how do you write this? What does it mean on the shape and the [snorts] bisect you divide it into two equal halves? >> H from vertex perpendicular. This is a symbol for to show perpendicular. >> So I can write let me write down like this. P E R P perpendicular from vertex to the base bisects the base. You can say you won't see this property a lot being used. Okay, but this is one of the uh advanced properties. You should have this in your mind. So if there there is

some uh place where this is being applied, so you can use this. Okay. >> [snorts] >> This can make your life easier sometimes. We will use this property later as well in circle properties as well. So this is how you can write this is how you explain it. [snorts] This is the very very basics of this angle. So you're learning everything from the basics. [snorts] Definitely if you're seeing this for first time there are some things that [snorts] you need to learn that you need to memorize and when you will apply them in your practice questions note down [snorts] okay now we moving forward some angle

properties of triangles The basic one everyone knows sum of angles is 180 of any triangle. Sum of interior angle let me write sum of interior angles. When you call interior angles, this means the angles inside the triangle. So if you have a triangle something like this triangle, let's say this is 10. Sorry, not 10. Let's do it 70. This is uh 60. They ask you for this angle x. So you know that the sum of these three angles x + 70 + 60 is 180 because this is one of the properties that you have been using from very early on. So sum is 180. So if your algebra is good enough you form this equation then you solve for x.

So 180 - 60 - 70. So that would be 50 50°. [snorts] Okay. The second one, exterior angle is sum of opposite interior angle. For this one, a good example would be something like this. But first of all you need to understand what is an exterior angle. [snorts] Exterior angle means that if you if you name this triangle A B C and if you produce a single side of the triangle if I produce if I continue I produce this further. So this angle would be your exterior angle the angle outside the triangle and this one would be the sum of the inside. This one opposite insides. Okay. So if I name this uh C and this is

A and B or you can something say like this. So this means that uh a of outside exterior is equals to c + b for this triangle. Or in the other way this uh you get the same answer. So logically if you don't even remember if you don't even memorize this one you can get the answer for the exterior angle very simply you can find this angle you know the sum of angles is 180. So if you have this one you have this one you subtracted 180 you will get this answer and when you get this answer you will again subtracted 180 you will get the angles on a straight line property you can get the outside one [snorts]

we'll find the Look uh let me give an example. Let's say this is 20. This is um 70. I'm just giving random angle values. They're asking you to find this one X. [snorts] So you can see that this X angle is simply produced after you produce the base further. So I told you if you apply the property we know that this exterior angle would be the sum is equals to the opposite interior. Opposite interior you can see 20 and 70. This doesn't seem like [snorts] seems like 90 when I when I make questions of my own sometimes. So let's make it 30. So now you can see >> [snorts] >> The x would be the sum of 70 + 30. So

you can see it comes out as 100°. Or if you find it the other way around. If you can find this angle, this would be how much? 180 minus 100. This would be 80. And then you can see this is angles on a straight line. So again 180 180 minus 80 you will get this 100. So even if you apply this property or you find it some other way around you can you can get the same answer. Okay. So there is no hard and fast rule when you are doing angles. Yeah there's no hard and fast rule. You can get the same answer using different properties. So it's up to you how how it depends on you. Okay. I think these are

properties of uh triangles and let's do some questions now. These are simple ones simpler ones. [sighs and gasps] So you can you can see they have given you the interior angles and they're asking you to find unknowns. So if I do this one part D. So the three angles are uh d angle plus 4 D angle plus 3D angle and we know the sum of all three of these would be 180. So the rest of it is algebra. So now you solve it. This would be 8d is equals to 180 and d would come out as 180 minus sorry 180 divid by 8 which would be how much? 22.5 >> 22.5 >> 3 + 4 + 1 8 [snorts] >> similarly if they're asking you for this

one E E is a better you can see in E they have shown you that two angle two sides are equal this one and this one is equal this means that the angles of these sides the opposite angles to the sides they would also be equal so if This is 62. This would also be 62. They are not telling you. We have applied the properties of isocus angle and we found this to be 62. We applied the property. And now when you know the two angles 62 + 62 plus e, this would be 180. You formed the equation and now you use your simple algebra. So 180 - 62 - 62 140 124 and 56 56 56 is the answer. This is your part E. This was your part

D. Who will tell me what would be F? Anyone? >> 60. uh equilateral equilateral. You can see that they have shown that these three are equal or you can try start doing these this one and D2 D2 D3. So now you have you have used up sorry you have learned all of the previous ones also. Okay. So uh okay that you only apply the rules of the triangle or if you only apply the Z F or properties everything you can the question will be combined all together. So whichever property that you can use now you can and then you can find the final answer. Okay don't worry about it. If you're using the properties correctly

then it doesn't matter where you're using it. You can use it in a question with the circles properties as well. So things will be combined all together. So it's up to you. Okay. So in this one I told you the best method whenever you have a very complex shape and everything is jumbled up. Find anything that is possible that you can find out which be properly whatever the proper rules that you can apply and you can get the answer of the angle you should find them out and eventually you will you would uh find the angles with that which they're asking you to find. Okay. So I won't get the angle A and

angle B directly just just by looking at this question. I'm not as expert. [snorts] So what do you do? you find slowly in this one I can see that I can find this angle I I know the other one it is 1 115 so I subtract this with 115 I get sorry from 180 so 180 minus 115 is uh 55 65 they had not asked me to get this angle okay you should understand similarly I can also see this angle I can find out because this is 90 this is 32 so I add the 90 + 32 I will get uh 122 and 180 this would be 180 minus 90 minus 32 now use your calculator 58 Okay, I can apply this angle this triangle triangle right angle triangle.

So right angle triangle we know the sum of the three angles is 180. So 65 plus this is 90. Look this is 90 and this one is a plus a is equals to 180. systematic number you will get the right answer. And then uh this would be 155 and a would come out as this would be uh 55 plus a is equ= to 180 and a would be 180 minus 155 which would be >> 25 a 25. So steps 25 and I'm not ashamed of it. Okay, you can do the entire page for your working you are getting the right answer. It is much better that you get the wrong answer in the first time but get the right answer. This is the challenge and don't hurry by don't hurry.

[snorts] Okay. So now what can I apply over here the angles on a straight line? You can see these three angles A, 58 and B. All three of them are on a straight line. So sum of A, 58 and B is 180. I have already found my A. Look 25. I can replace this with 25. So B would be 180 minus 58 minus 25. This was how much? 97 97 good 97 okay moving on to this one I'll explain this one and then I'll give the rest of the three as your homework okay on your group and then you will share with me after the class after half an hour more over here what I can apply I I'm just assessing any angle that I can find it

out find out I can I will use later find. So I can see I can fit this one because you can see this is a straight line. So we know that this angle would be 45 + 64 plus the angle of B hi look how am I writing is 180. BHI is telling me that B till H then I and the angle is at H and I have written like that BHI. [snorts] So what would be my BHI with 180 - 45 - 64? Fast fast fast fast fast fast fast fast fast fast fast fast fast fast fast fast fast fast fast fast fast fast quickly >> 71 >> - 64 - 45 71 yeah so this is 71 uh I can use the Z property look I can see a Z over here because you can see this is a par line par

UF F or Z. So I know that this angle would be equal to this angle complete. [snorts] So this means that 71 is equals to C + 32. The complete angle of the Z. This angle this this and this. So this angle is of the Z and this complete angle of the Z. 71 is on the one side of Z and the other side is 3 + 30 >> 38. So 38 38 38 [snorts] 39 >> 39 39 97 >> 37 9 39 >> 39 >> 71 - 32. [snorts] You got this? So write this down. Now you can find simply this D because you can see this is a triangle and you know the sum of angles in the triangle is 180. 64 + 39 plus uh D is 180. How much would be D? >> 77. >> 77. Yeah. The math simple calculator 77.

Okay. All good. Any questions? Yes sir. All good. you will get to the point that you are that they're asking you to find. Okay, definitely. This is it for today's class.