Fundamental Geometry & Angle Properties | IGCSE/O-Level Mathematics | Class 25

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

Okay. Asalamuaikum. So for today's class we are going to start off from the very very basics of geometry. Everything about geometry. First we will see what do we have in geometry first of all and then we'll start off with this one. So, geometry is all about shapes, right? All about shapes. We learn about the different types of shapes and what we can do with those shapes. Find the area, find the perimeter, find the uh temp volume and everything. It's working on the geometry more prominently. It means that we're working on the shape. So the angle properties and the other geometrical properties include so shapes. So we have first of

all one D we have 2D we have 3D you people must have heard of these names 3D movie most of you have. So 3D movies, anyone has an idea what do you mean by 1D, 2D, 3D >> dimensions? One dimension, two dimension. What do you mean by this? 3D or 2D? 1D means one dimensional. One dimensional means that if there's something you just need. Dimension means what do you mean by dimension? Please go when you try to measure it you have dimension the length so you have a line simple line it is one dimensional only one thing that you can measure out of with one dimension so definitely this is your length for a 2D shape you have a square

you have a rectangle you have a circle any uh you can say shape which can be drawn on a page which does not have a third dimension. You you just have the x and the y-axis. When you work horizontally and vertically, x and the y- axis. So let's say you have a circle or you have a rectangle over here. So you have a dimension which is horizontal for a rectangle and you have a dimension which is vertical for a for uh this rectangle. So this 2D shapes, sorry, come again. Square triangle square triangle rhombus kite parallelogram or we'll discuss everything the 3D shapes mean now there would be three

dimensions three dimensions for for example we are living right now in a 3D world everything is 3D your room your room has a length It has a height and it has a width. Right? Three dimensions are needed to to work on it. You can say in simple words for example a simple cube or a cuboid something like this is a cuboid. So this has a length uh this has a height and this is it width width length and height. You can interchange them. You can there's not an issue if you call something else the height or the other one the length. It's okay. It's just about the perspective. The only thing that you need to

understand over here is that there are now three dimensions that are needed to fully say get the dimensions of this shape. So 3D shapes are like a cuboid. You have a sphere, a ball 3D. It's a 3 3D figure. You are living a 3D world. Your room take your example of the room. First of all, this is enough. So we would work on the now you know what do you mean by 3D 1D and 3 2D and now you would work more on the shapes. For shapes what do you work what do you work on? First of all the angle properties. We would be learning first the angle properties and then we would be moving on to the menuration. Menstruation is

you can say a separate topic but that is part of this geometry. Why I I usually take it the part of geometry because you are working on the shapes to get the uh area volume surface area and the parimeter and everything. This is again a property of the geometry finding these areas volume perimeter and surface area and everything total surface area. We would do all of this in detail. This is just an introductory class. Right now I'm just giving you an idea of what you would overall learn in this complete geometry. Just like in algebra I I gave you everything about algebra. It included the graphs.

It includes the the sketches, the coordinate geometry, everything was included. Similarly, geometry has all of the things combined. We would be studying in much much more detail separately the things as well menstruation and everything in detail. No worries. Okay. Okay, so now you have the overall idea of what this is geometry and now we can start off with the first thing and I told you right now we would start off with the angle properties the very simple angle properties and then we will slowly increase the difficulty even before angle property there are a few things that needed to be discussed

for example Okay, some more basic things. When you have two parallel lines, you have this arrow. The when you have the same number of arrows on any number of lines, you would say that those lines are par parallel to each other. What do you mean by parallel? Parallel means that they won't be intersecting ever. Even if you produce them as long as you want, these are parallel lines. Similarly, when you have something like this, when two lines are perpendicular to each other at an angle of 90, this is called perpendicular. Similarly, if there's a shape and we we know that two sides are same. So this is how you represent it over

here. They won't tell you, they won't write it down that in this shape these two sides are same. In this shape these two sides are same. What you what they would give you is this thing this small dashes on the sides. So the sides which are which have the same number of these small dashes they their size would be the same. So when this is given to me, I know that this length and this length is equal and this length and this length is equal because these two have the same number of small dashes and these two have the same number of one one small dash. So the these number of small dashes can tell you uh which side is

equal to which side. Okay. And another thing we know right now that these two lines are parallel as well. If I give the point the names A B C and D. So we know that AB line is parallel to CD line. Why? How? How do I know this? Because we have the same number of arrows. Look both of these lines have the single one arrow. So this means that this AB and this D C are is parallel. Then what about these? They would be something like this. So on these two line you can see again they are parallel. Why? Because these two have the same number of arrows. So two numbers on sorry two number of arrows on AD and two on BC.

Whenever you have they have given you a shape for example they give you something like this and they just have given you this thing. What can you know about this shape? Just by these arrows and these lines. I told you when the sides which have the same number of arrows. So you can see that this one and this one both have a single arrow going in the same direction. This means that these two lines are parallel. Okay, they have not written it down. They will not write it down. If they have shown this thing to you in the shape that they these two lines have the same arrow same number of arrows then we

would we can conclude that these two are parall to each other. Another thing that we can know from this thing is that both of these line this one and this one both of them have the same number of this dash. >> So trapez trapezium ah trapezium. Yes this is trapezium. We will learn more about everything. Please don't worry if someone doesn't know. A trapezium has one pair of parallel line. So this is what the arrow and the dash can tell you. Okay. This one. Yes. intersect BC. Yes, they would be at perpendicular means 90°. >> So A, B and BC are perpendicular. This is also perpendicular. This is also

perpendicular. So if they have shown it to you this sign on the shape in the exam, they won't. No, this one, the upper one, the rectangle. Mhm. points rectangle. It's a simple rectangle. Okay. >> G >> it's a simple rectangle and let's say that this is a shape and everything right now that I have drawn on it is given in the exam. This is the exact thing that is given in exam. So first of all we need to see what can we deduce with this with the only this diagram that they have given us. Okay. First of all you look for the sides which are equal to each other. So we look for these number of small dashes. So we can see that this one and

this one side this side BC and this side DC sorry A B and DC they have the same number of a small dash. So this means that this the size of this thing and the length of this thing they would be the same. Let's say this is five. This would also be five. Okay. Similarly this side A D and BC have the same number of dashes. Two two small dashes. So this means that if this is two, this would also be two. Even if they don't give you the value of two. If they give you the value of this thing as d a ds two by the by the this small two lines on both of them, we know that both of them would be would have

the same length. Similarly for the same number of arrows, you can see these two have the same a and bc have the same number of arrows. So these two are parallel. They won't ever intersect each other. Similarly, this AB and DC are parallel because they have the same number of arrows. So, this arrow and the small dash and this sign, they mean something. You should understand this. Now, if this comes in your exam, >> I got it. >> Okay. If someone wants to make notes, this is the very very basic. I can recall in my time, uh, no one taught me this. So, this thing I assessed myself, reduced myself when I was doing the

past. They give you all of these arrows and this small dashes and the pass. So I make sure that you people now know that these have some meaning and you can then deduce and then that can help you solve the exam questions. Okay, now we're moving forward with the basic angle properties. >> [snorts] >> Okay, the first one is simply angles on a straight line. Ice is a difficulty we would increase. Definitely many of you would have learned this thing uh very much in detail. This is a straight line. Okay. So let's assume this is the point O and on a straight line the sum of angle is is always 180. So sum

of angles on a straight line is 180. Now this is something you should learn. You should memorize. Why? because this property can help you solve the answers to the unknown. For example, let's say you have they have given you a simple question. Uh let's assume this is and this is another line. Let's say this is uh 32. This is 40 9. They're asking for this angle x. >> Sir 180 - 49 - 32. So now we know the angles on a straight line is is the sum of all these angles because 32 49 and x when these three angles combine we get a straight line. So the sum of these three would be 180. So 32 + 49 + x is equ= to 18.

>> Sorry >> 99 >> 99 okay 99. So this angle x is simply 99. So this is how you you have to learn this. There are some things that you need to memorize and there are some things that you need to understand and then apply them. This is one of the things that you need to memorize as well as understand it so that you can apply it properly and get the solutions. Okay, this is the most simple one. Now we'll move forwards with the second one. Angles on a straight line. We done with it. Angles on a single point. The sum of angles would be 360. For example, you have this point and you have these three angles A, B and C. You can

see when you combine all these three angles A, B and C, angle A, B and C, you have now the complete 360°. Yes. So the sum would be 360. So if you have any two of them, you can find a third one. You will see some possible questions, some example questions as well on this thing. Okay, let's assume let's find one. Let's make one and then let's say this is uh 165. This is 3x. This is 2x. They're asking you to find X. So what would we do? Simply we know that all three angles are on this center point around the center point 360. So this >> 39. >> Yeah. 2x + 3x + 165 is equals to 360. 39 x value. 2x + 3x would be 5x or

360 - 165 algebra should be good. Just be geometry at the end you will see yourself solving an equation. So if you if your algebra is good then your everything becomes easy. So 360us 165 is >> it will be 195 39 >> good. So now you got your value. So there are very few this simple but top everything is in everything. There is no topic right now that I can say can only come in P1 can only come in P2. They have the the CIE has never told us this thing P1 topic P2. So we should not segregate these things. practice properly. You should know everything how to do them manually. You should know everything how to do it

calculator with calculator so that you are prepared for the worst as well. Okay. So okay angle properties angles on a straight line angle properties I will share you these as well if you want otherwise if I don't forget thing. Okay. So now these are the two main things. Moving on forward. The third property you can say Z angle property or this is the you can say the layman word otherwise you would say this alternate angle property. Alternate angles. What is what is this actually? when you have a pair of parallel lines. [snorts] Okay, you can see this line and this line they are parallel and using this parallel line and if there's

a line which is crossing these two you can see a Z forming over here look like this sub Z. So whenever you see this Z forming the angles of the Z let's assume this is A. This would also be a the angles of the Z would always be equal. So in simple words we can say this is again something you need to learn. Angles of Z are equal. The Z can be also form in the other way. Okay. In any way that you can see the Z can form. This is also a type of Z. Sorry in this one in this Z the angles of the Z are yeah let's say this is B this would also be B okay so you if you see a Z forming in such a way that the that the lines of

the Z the first the two up up and down lines they are parallel this is the you can say uh the constraint these should be parallel. If they are not parallel then you cannot apply this uh property should be parallel. Sometimes your it looks like it is parallel. For example, this this is also a Z. You can see it's it's looking as like a Z. But you won't be able to apply over here this property that this is equals to this. We we cannot say that this a is not equal to b. Why? Because this is uh in fact z but the lines of the z they are not parallel. They have not told us. We don't know if this and this are

parallel. We don't know right now. The question has not given us. So the parallel the they should be parallel. If they are not parallel then you cannot apply the z property. Same is the case for the next one which is F property. F angles we call it corresponding. If you want to even if you have it parallel. Are these parallel? >> No. Neither the question has told you that they are parallel. If the question has not told you if they are parallel, you cannot apply the Z property because we don't know if this line and this line are parallel. Okay. >> Okay. >> Moving on to the next one. For the F property

or the more mathematical name is the corresponding angles. You have to learn both of the names or otherwise if you want to learn the alternative angle they will ask you which property are you applying over here. So for this angle property you would see if you have an F forming. So you can see an F forming like this. So when you have an F forming these is the inner angles of the F they would always be equal. Angles of F are equal or you can say because of the property of corresponding angles these two angles would be equal. We will do some practice questions as well. You will see how and this would be given to

you in the exam that these two lines are parallel. Okay, again they should be parallel so that you can apply the F property or the Z property. First you need to confirm if the two lines are parallel or not. Similarly you can have an F like this in the opposite direction. So this direction in whatever direction you can see the F is forming you can the inner angles of the uh of the F would be same. Similarly inner angles F. So if this is B this would also be B. Same is the example over here. This is also an F. But this line, let's say this is AB, this is CD. We don't know. We we can see right now

that AB is definitely not parallel to CD. So we cannot apply the F property right now. They should be parallel. The two lines of the F should be parallel. so that you can apply the f property. Okay. This is the first thing that you would check for both of the Z or the F property that you want to apply. First you check if if the lines are paral. Okay. The third one we call it the C property. Um sir I wanted to ask what's the difference between the Z angle and the F angle they look alike situation same the angles are different when you forming the Z this angle and this would be now you have to think which of the

property that you can apply over here so that you get the unknowns. Okay sir. Apply this property or this property. They would give you a situation very similar to this and you can apply any of the properties that you have learned to get the answer. So it doesn't matter if you get the answer using the F or the X or the other other properties. Right now we learned we will learn. So another one is the C angles which we call the interior angles of sorry. 180. Yes sir. Here we have also the same condition. Sir I did not get one thing. Mhm. angarable A or B alterpres I look for the F. So this is the F for

me. >> Okay. When you'll see the question, don't worry. Don't wait. Just now hold your horses. You have questions. Just wait. If I when when we will uh do all of the properties, then we would apply them on practice questions. Don't worry. Okay. >> Okay. Just learn right now for the get to know about all of the properties now and then we will apply them then you'll get a better of idea how to use them. [clears throat] So C property is uh when you can form a C categ form a So when you have the C property, what do you know that this angle and this angle the sum of these two would be 180. They both are the sum is 180°.

[snorts] Again the SH is the condition is that the two lines should be parallel. If they are not parallel, you cannot apply the C property. This is the third third rule. For example, if you have something like this, this is a line and this is line. You can see that this and this are parallel. So the C is forming like this as well. C is forming like this as well. So these two angles would have a sum of 180. Sorry. And this one and this one would have a sum of 180. C if you can see that the lines are parallel of the C then you can apply the C property and then you can you you will know the unknowns you will see some

questions we will apply these don't worry C FN X C FN Z these are just for your what do we say feasibility to learn in exact Z F and C Okay, the Z, F and C just to make it easy for you to learn. Alternate angles is Z, corresponding angles is is F and interior angles property is C property in your exam. Don't write the I'm applying the C C property. I'm the Z property or the F property. Okay. So for the last one X property or vertically opposite angles opposite vertically opposite angles. So in this one when two straight lines cross each other then the opposite angles are equal to each other. If this is A,

this is A. If this is B, this would be B. Simple. >> So are they perpendicular? perpendicular perpendicular then also I can apply this rule. When you have a perpendicular line you can see this is a again X forming right X plus sign X. So this angle the opposite angle would be equal right. If the upper one is 90° then this opposite one would also be 90° and similarly this end this one would also be 90°. So this is this can be applied in this perpendicular when two straight lines are crossing each other crossing each other not in this case look parallel they are straight crossing each other not parallel Example,

this is not a straight line. This line is a straight. AB A is straight. But this C to D, this is not a straight. You cannot, you can see there is a curve. You can see there's a change of direction. So here you cannot apply the vertically X property. The vertically opposite angle. This angle is not equal to this angle. A is not equal to B. These are different. The upper one is A. The lower one is B because the two lines which are crossing each other they are not straight. A B is a straight but the C to D it is not a straight thing. There's a change in direction slightly. Yeah. So you have to assess it in your exam.

you would get it the right way. [snorts] Complimentary supplementary is not as uh what do we say important? Complimentary angles simply means that the sum of anyone >> the sum is 90 >> compliment >> and the sub 180 >> supplementary compliment 180. I think compliment pair of angles. Pair of angles or sum sum is 90. Blue angles when you add them the sum is 90. So that is a the set of the complimentary angles. Supplementary supplementary pair of angles sum 180 >> 180 this is not as important right now just for definition we can understand this thing but you will see you won't need to apply this very Other

types of angles angle sa less than >> less than >> 90° >> uh less than 90 acute abuse or reflex >> reflex acute is less than 90 angle is less than 90°. Obtuse definitely is more than 90. >> So more than 90 or less than 180. >> H well more than 90 and reflex is more than 180. H more than 180. And if we have a straight line, the sum of the acute and the obuse would be equal to 180. Right? Acute plus obtuse line. So that is equal to 180. For example, this is a straight line. The angles on a straight line. If this is an acute, right? This is an acute. You can see it is less than 90 uh a cube and this one you can see this

is obvious. So the sum of these two you can see it is 180. You can also that these are supplementary angles. One more type of angle is the right angle. right angle when you have a 90° angle perpendicular. So right angle is neither acute nor obuse. It is we will call it a right angle perpendicular. Supplementary. Anglementary compliment. Sum compliment angles is a is a pair of angles. Just sum 90 acute angle. Acute angle definitely they would be over here. If you see this thing [snorts] What do you think about this angle? Is it an acute or abuse? >> This is acute. I think so. You can get an idea just by looking at

the graph at the figure. Just by looking at the figure. Okay. So simple you have known the properties now simple good example over here next question. So you can see this they're asking you to find D simple D and can I apply the X property over here X property? No sir property because the lines over here they are not straight. This is going like this and then similarly this line is coming straight and then bending a bit >> and then going downward. >> Uh yeah this is these are not the straight line and they are crossing each other then they would we can apply the property. So in this one we can apply the property of

angles on a single point. We know that the sum of all angles all of these would be 360. So we knew the identity the for the property and using the property we can form an equation. So we know that 6D + 139 + 5D + 7D + 23 sum 6D + 5D is 11D + 139 + 7D + 23 >> Sir answer is 11. The total answer for >> 18D is 360 - 23 - 139 18D is equals to 360 - 23 - 139 >> it's 11 198 on D would come out as yes 11 198 divid by 18 is 11. So this is how you get the now if I if I input if they ask you for each of the individual angles right now. So you can just input the value of D is 11. So 6 * 11 would be 66. So this is

66. This would be 55 and this would be 7 * 11 is 77 + 23 100. So just by looking at the value sus 139 this is greater than 90. Similarly this is man what this what is this 55 is it acute or obuse? Acute value simple or just by looking at it also we can see this is definitely 100 because it is greater than 90. Okay. Greater than 90 value. This is a good example. Now there can be more than one way that you can solve a question. Okay? But you will get the same answer at the end of the day. That's also possible. So it's okay if you're doing it the long way. It's okay if you do it the short way as well. Yeah. As long as you get

the same answer. So this is a simple question that they've given us. They're asking you to find X and Y. So what we can do over here is Can we apply X property here? >> X property. But before applying the X property, I need to see the straight lines. I know. >> This is a straight line. >> Sir, sir, what is X property? >> X property is this one. Bully. X property is when you have a straight line, two straight lines crossing each other, then the opposite angles are equal. Okay. So you look for straight lines. We will confirm the answer using your answer. Otherwise just wait for the others who are doing this for the first

time. Okay. X X [snorts] property. >> Don't take these down as well. Okay. >> Okay. >> Say you can see the X forming over here with the with the two straight line. So this means that this angle is 3 Y. This means that this one would also be 3 Y. the opposite angles should be equal. So this is also 3 y. And one more thing that we can find over here this angle this is 6x and this be equal to this complete thing. This angle 6x is equals to this complete thing. This is >> X property apply. Right now I have just applied the property. Now I know these values that this angle is equals to this angle. The red ones are equal

because of the 3 y + 6x + 34. >> No no no. You just write this thing. 6x is equ= to 34 + 186 - 4x you get this angle is over here. This angle is equals to this complete angle which is on the other side. Okay. Now this is the next part. The opposite angles are equal to this angle and this angles are opposite. Thank you. 6x = 34 x + 13 186 - 4x 122. This becomes 20 x= to 22. And now we want to find y. For that what we can do is we can apply the angles on a straight line. So we can we know that this is completely a straight line right the ab. So we know now that 6x + 3 y is equ= to 180 because it is a straight line. And now

we know the value of x. So 6 * 22 + 3 y by >> 48 you answer 48 48 >> uh 48 48 to 3 answer 48id 3 let me see 6 * 22 is 132 6 * 22 is 132 and then 180us 132 48 or 3 divide 48id 3 is 16. So you get the value of y as 16. Try to apply the first property. If that is not applicable, then apply the second property. Don't rush into think. This was a good example. We'll do just one more and then we'll end the class. Or I think I will have to end the class right now because I have another class. Okay, I think we can uh let's do one more question. Okay, this one. >> Sorry. >> I see you're using this. >> Okay. So now you can see they have given

you they're asking you for to find the D and C. >> So 7 C is >> you you find your answer then we'll confirm using your answer. Okay they are struggling in this topic a bit. So we have to consider for everyone. So they have now given in the question that this is you can see the arrow. So this is parallel to this. So when we have the parallel things now we can apply the identities [clears throat] the the rules. for the C you look for the Z any of them that you can apply over here you will apply and then get the relevant answer. So first of all I can see over here that you can apply the F this is the F right.

So you have the parallel >> [clears throat] >> uh the the what do you say the lines of the F. So this means that this is equals to 140. So 73 is equals to 140 and you will find the value of C which would come out as 20. So this means that this is also 140 sir. >> Now you can apply. Yes. ZB. Are you getting any unknown that you that they're asking you to find? anything else you can try applying everything that is possible you will get to the thing that you want to find actually so you can apply everything ZB C everything is applicable because you have the two parallel lines now for the D part we

know that this is a straight line and this is a straight line so we know the angles of the vertically opposite angles they equal. So this means that 2D is = 7 C because these two are the opposite angles. So 2D is equals to 140 and D is equals to 70. >> So this is how you get the balance. >> Okay. Uh copy this down. Take a screenshot. I think that is okay for me. Stop the recording.