 Let's begin. So what's a quadrilateral guys? So we'll just do a quick Definition part and then we'll move on to problem solving as quickly as possible So let me start with a quick definition and few properties around quadrilateral. So quadri Quadrilaterals, okay, so the name is very you know Self-explanatory so hence quad quad means for and Laterals means sides So hence we say any closed geometric figure with four sides and we name it a B C D This is called a quadrilateral or quadree angle Quadree angle that both both names are there now So if you see There are four sides. So let me just write four sides are there or four laterals are there four edges are there whichever name four vertices are there a B and C and D and Four angles are there. Okay, and there are there are two things two diagonals are also there. So these are two Diagonals opposite vertices joined together in case of quadrilateral diagonals, right? So these two diagonals are there moreover, you know that in a quadrilateral We have four sets of Four sets of adjacent sides as well as four sets of adjacent Angles and you can see a B B C Is one set then BC CD is another one CD AD another one and AD AB these are Four sets of adjacent sides similarly angle a and angle B They are pair of adjacent angles Angle B angle C angle C angle D and angle D and angle a these are four pairs of four pairs of adjacent Angles, isn't it? This is the information about a quadrilateral apart from that there are two opposite sides What are two opposite sides here? Clearly it is a B and CD and the other one a D and BC there are two pairs of opposite sides similarly two pairs of Opposite angles and that is angle a and angle C and angle B and angle D these are You know the vital information about the Quadrilateral, why am I discussing all of that because later on you are now going to you know study the properties of a quadrilateral Now you asked me why do we study the quadrilateral in the first place? now, you know Sense of geometry is very very important. Let's say in the field of engineering Then those who want to let's say take up architecture Architecture and therefore or later on let's say design Right you want to design any particular you know Product or let's say you want to design a building or you are into engineering so you must be very very thorough with the knowledge of geometry so hence and Quadrilateral being one of the integral part of geometry dot or you know, we must learn Okay, next first property of a triangle, which is very very vital you all of you know already and that is angle some property of Angle some property Property of our quadrilateral. Okay, so this you can guess already We had an angle some property of a triangle and that was some of the three angles in a triangle is equal to 180 here What we are going to do you would use to repeat the same process and here is let's say the AB So this is the quadrilateral and we are saying angle a plus angle B plus angle C Plus angle B is 360 degrees It is very very easy to prove and we will use something that we have already learned in the previous sessions or classes And that is we will be using angle some property angle some Property of Triangle right how to do that construction join any of the two diagonals Let's say I'm joining DB. So now name them. Let's say this is angle one. This is angle two. This is angle three This one being four five and six. So hence What can I say in triangle a? DB Angle one plus angle two plus angle six is 180 degrees. Why angle some property for triangle Similarly angle three plus angle four plus angle five is again 180 degrees and this is in triangle D C Okay, now what name this as equation one name this as equation two and simply add both of them. So one Plus two LHS plus LHS RHS plus RHS. So hence you get angle one plus angle two plus angle three plus angle four Plus angle five plus angle six is 180 plus 180 on the right-hand side is 360 degrees Fair enough. So hence now you club two and three. What do you see two and three and five and six? The proof is kind of done. So hence angle one is angle a angle two plus angle three If you can see this is angle two plus angle three, so it's angle D Angle D and angle four is nothing but angle C and five and six put together here is angle B correct So 360 degrees you can rearrange and you have just got the desired Result, okay, so some off The four sides. Oh, sorry four angles of a quadrilateral 360 time to solve problems now. So let's solve problems Okay, so first problem set So R&R people you would have done it in the morning, but never mind you can do it again So let's start. So start guys. You can post your answers in the chat So first question is angle for quadrilateral are respectively in a hundred ninety two find the fourth angle quick do it Yes, calculation error dude Don't do calculation mistake. Okay, very fairly simple question. All of you would have Done it by now. Now second question in a quadrilateral ABCD the angle ABC and DR in the ratio One is to two is to three is to four find the measures of each angle of the quadrilateral Okay, first answer is 70. First answer is 70 why we will be using angle some property of the triangle No, sorry of the Party lateral. So first time I think I don't need to solve the first one for you. But anyways Let me just share Yeah, the whiteboard with this So that I can finally solve it as well. Okay, so here is the next question Yep, I hope you're able to see it. Yeah, keep solving if you are able to solve one Go to the next one. Yeah, so the first one is First one is Yes, so the first one first question so it is given the angles are hundred then 100 degrees and you have 98 degrees and third is 92 degrees and let's say the fourth one was X Okay, so how to solve this? So I'll tell you what in in the exam if it happens, you should start like this So this is just the information. So you should you should start like this. Let the for Angle V X okay Let the fourth angle be X now By Angle some property property of a quadrilateral a Quadrilateral we get what do we get hundred degrees plus 98 degrees plus 92 degrees plus X is equal to 360 degrees Okay, so hence, what will you write you will write hundred plus 98 plus 92 is 190 so 290 degrees Plus X is equal to 360 degrees. So X is equal to 360 degrees minus 290 degrees, which is equal to 70 This is the answer. So the fourth angle is very good. Yep, let's solve question number two I hope all of you understood the first one. This is Snow brain are very easy one. So they will give you some certain such It could be you know asked for not more than one mark In your exam. Okay, next question Next question was a quadrilateral ABCD angle ABC and the are in the ratio one is to two is to three to four Find the measure of each angle of the quadrilateral Okay, so in such cases whenever the ratios are given what to do you have to do Let Let the angles of Let the angles let the angles of the quadrilateral or the given quadrilateral Lateral B What? X Then 2x then 3x and 4x. So whatever was the ratio if you see x 2x 3x 4x all are in the ratio One is to two is to three is to four Okay, now what Therefore by angle some property of Quadrilateral Quadrilateral we get what x plus 2x plus 3x plus 4x is equal to 360 degrees So that means 10x if you see x plus 2x plus 3x plus 4x is 10x so 10x is 360 degrees. So what is x guys simply 360 degrees by 10 equal to 36 degrees Okay, so it x is 36 and 2x the second angle will be equal to 2 into 36 degrees, which is equal to 72 degrees 3x is 3 into 36 degrees I'm sorry. Yeah, 36 degrees is equal to 108 and 4x is equal to 4 into 36 is Equal to 1 4 4 degree and to check if you add all of them you'll get 360 degrees. Okay, this is Question number two all clear anyone has any doubt, please you can talk to me to chat. Okay next Yep So now let's say this question would have come in MCQ then you know You don't get this much time to you know write all the steps So hence in such cases, what will you do you simply add the ratios? What is this 1 plus 2 plus 3 plus 4 so 10? Yeah, so divide the total by 10 so 36 So automatically the angles are 36 times 1 36 times 2 36 times 3 and 36 times 4 Simple so these questions could be good candidates for MCQ questions. Okay now third one third one says Sides B and DC of a quadrilateral ABCD are produced as shown Prove that a plus B is equal to X plus Y That's mean why those who are solving question number five. I there's a correction in question number five and It it says prove that so I'm Yeah, so the question number five just a correction question number five I'm writing here So the correction is in the question number five you have to prove that prove that angle P Plus angle Q is equal to half Angle ABC plus angle ADC half angle ABC plus angle ADC you have to prove this Okay, so it was missing in the sheet never mind. So let's go back to question number three. It says There is a quadrilateral. So this is the quadrilateral. I'm drawing the figure So this is the quadrilateral Okay This is a quadrilateral. Okay. So now what is given? It's given these are the points a B C and B and It's given that It's you know DC is produced. So okay and BA is also produced You have to prove that a plus B. So this angle happens to be B this happens to be a and Join them This whole happens to be why and this hole happens to be X So you have to prove what you have to prove that a plus B is equal to X plus Y in such questions How do we start? Sir, what is the message by Neha from Neha? What is it? Could you please crawl down to the fifth question? I can't see the entire figure. Okay So here is the fifth question guys. I hope now it fits in. Yeah Now in this how to approach such questions, you can start from C They're asking about if you see a plus B are angles and they're asking about the sum of angles So whenever angles are being added upon You must have this thing in your in the back of your mind that it has got something to do with angles some properties whether it is triangle angle Triangle, quadrilateral, whatever right? So a plus B any of if you see A and B happens to be exterior angles to these to this figure, right? And if you consider only the triangle This shaded triangle, right? So this triangle for this triangle B is the exterior exterior angle So can I write and let me some let me put some other names as well So for example, let me put this angle as one and let's say this angle be two Let's say this angle is three and let's say this angle is four So can I say that? Angle B or let's say not angle B B is equal to angle one plus angle three by what theorem it is Exterior angle theorem for a triangle exterior angle theorem fun. I'm just writing in abbreviation I'm not writing in full in exam. You should write in full. So B is angle one plus three similarly A is equal to angle two plus angle four Now always keep in mind you want a plus B. So hence it is always advisable now to add these two So what will you get? You'll get a plus B is equal to angle one plus angle two plus angle three plus angle four in the RHS now Yeah, someone is saying something just a minute. What is it? Okay, someone is giving the solution never mind. So a plus B is this so what do we Get we get angle one plus angle two is nothing but If you see angle one plus angle two is why and angle three plus angle four is x so hence prove This was again a one-liner sum. So no brainer in this again a plus B is equal to x plus y So the third question is sorry. Yeah, go to question number four Here is where the Complication starts. Okay. I think how many of you could solve question number four Question number four sorry. Yeah, good. So most of you would have done it Never mind Aditi raised hand Aditi. Yes, Aditi. You can write here Aditi Yeah, so basically, okay fair enough. No problem. No problem. You can mention here four and find out four and five done Good. No problem. So let us, you know, I the The point is we need to learn how to express your you know, most of your answer scripts Which I have seen there were lots of problems in communication guys even if you know the solution You will you were not able to reproduce it on the piece of paper and there were lots of Mistakes regarding that so hence use these sessions to understand how to present your answers as well Now let's say question number four. What does it say? It says Yeah, question number four question number four says in a quadrilateral ABCD. So let me draw a neat quadrilateral. So Quadrilateral I am drawing ABCD always try to draw asymmetric quadrilateral to avoid confusion because many a times You draw symmetric quadrilateral symmetric figures and what happens is your mind gets confused. So don't draw You know square. Let's say if there is a there's a quadrilateral question. Many people start with drawing squares So hence what happens is they you know, your mind gets confused You're solving our expressions to draw, you know asymmetric figures now now. So this is a B C and D. Okay. Now they're saying A and B are our bisectors of A and B. So let's say this is a oh A oh and this is the oh, sorry Try to make The diagram as neat as possible I'll give you a mantra here and I always keep on saying in my classes. The more you make The life of the examiner easy Easier he she will make your life Okay, always remember think about Let's say you have to correct anyone's paper. How would you expect them to be so there is where maximum You know, so presentation does really matter. Anyways, so this is just a game. I'm cropping this up and shifting it somewhere else. Anyways now Come back to the question question says Um, what does the question say question says that a oh A oh is up by sector of angle a and B. Oh is my sector of angle B. So clearly do not hesitate to write this as X and X Why and why Okay, so you should present like this what you should write given I know many of you despise this but then Guys, you know, if you are not very, very prepared for methodical approach of question solving later on in your MCQ based test also you will struggle. So hence this is very, very important. This is just from experience. I'm trying to tell you So given what is given given is always let your mind know and let the mind take a full control of the entire problem So hence I'm saying given what is given a oh and B. Oh are By sectors, there is no problem in writing right as much as possible even you are by sectors of Angle a and angle b respectively Okay, the more you write the more clarity your mind gets and lesser the number of careless mistakes now to prove What is to prove you have to prove that Angle a OB angle a OB is equal to half Angle C plus angle D Okay, many a times we also adopt our Check what is the check so we can we can check whether the given problem is really correct or not how do we check So you can take some special cases. For example, let this ABC DB square now Let's say now let me show you how to you know, keep this checks in mind. So let's say Let's say this was the case. Let me draw This let's say ABCD is a If it is true for one quadrilateral, it will be true for any quadrilateral. Isn't it? Hello, are you able to see here be properly? Hello Guys are you able to hear me? Yes, sir. Yes, sir. So now so what I'm trying to say is see how to check. Okay, check how to check such kind of problems If it is true for one quadrilateral, it should be true for any quadrilateral, right? So let me take that square and you know, squares are right angles. So all the all the angles are right angles So hence if the bisector would be there, how much will be this angle 45 degrees 45 degrees and how much will be this angle 45 degrees? So hence how much will this this angle be 90 degrees? So let's check whether this fits in into our problem. So angle AOB. So this was a this is B and this is O So angle AOB so much angle AOB in this case is 90 degrees and let's check what is half of C plus D Now see this is C and this is D. This is also 90 is also 90 so half of C plus D again is 90 So hence it works. It works for a square. That means that means the problem is kind of correct. You know might might not be so hence we can approach and then we can generalize now what to prove this. So how to prove. So let's jot down the proof right again the what should come in your mind many a times this question is being asked to me How do you go about any problem? How do you think whether this is a you know correct step or what should be the first step? Many people struggle in the first step itself. Now there is no direct answer to that but yes over a period of time if you are conscious about problem solving You can get those trends and traits you know as you solve more and more. For example here again if you see there are some of angles given Some is being talked about. So hence the moment there is some of angle and there is a geometric figure automatically the mind goes to some property of any quadrilateral or any polygon for that matter So hence that we could definitely try that and if let's say that doesn't work then we'll have to also go to some other method. So this is what you know with experience again now angle AOB is half angle C plus D. So what is angle AOB by the way where is angle AOB. Let's look at it. Angle AOB is here now clearly in terms of I can express AOB in terms of X and Y this question is automatically leading me to do that why No we can't prove it stress because you know the question being asked here by stress is when we did this choir method checking thing Can we not use that method itself as a proof. No one specific case cannot be proof for you know all the cases right so it just works only for one case. So something which works for one case doesn't mean that it will be you know useful or let's say you know you can prove it or it will work for all the cases. For example let's say in one particular test out of 100 you get 80 that doesn't mean you will get 80 always in the next exam you have to get more than 90 or 95 right. So hence it cannot be generalized so by one test I cannot generalize whether you are really really smart or dumb or whatever right. So hence one specific case cannot be taken to generalize the proof so hence we have to take that that approach in geometry or any such proof so that no one can question our approach and it has to be generalized. I hope you understood. No no no the question next is you said that if it works for one question one quadrilateral it will work for others to know I didn't say that I'm saying. If it is true for all quadrilateral then it will be true definitely for a square as well why because square is a special type of a quadrilateral. So hence that's the thing so and if it is true for any given quadrilateral it must be true for a square as well but the vice versa is not true. That means if it is true for square it need not be true for all the quadrilateral. So hence that's the you know relationship please understand if it is true for a quadrilateral it will be true for any square for that matter okay not the vice versa. Now let's see the proof I hope you got it okay great. Now angle A O B is half so how do I express this in terms of X and Y so clearly if you remember angle A O B can be written as 180 degrees minus X and minus Y. This is a direct fault you know result of let's say angle some property of a triangle isn't it. So angle some property of a triangle this is a direct fallout of that so hence so wait a minute I'll just make some space here. Yes now what's next so you can again write this as 180 degrees minus X plus Y okay where again you can write this as can you not write X as a by two so hence I'm writing angle A by two plus angle B by two isn't it. So what is this 180 degrees minus half of angle A plus angle B okay now I need C and D so hence I have to find out I have to somehow replace a B by C and D so how do I do it. I know one more relationship that is angle A plus angle B plus angle C plus angle B is 360 degrees. So my dear friends angle A plus angle B will automatically be 360 degrees minus angle B minus angle C minus angle C minus angle D correct. And then if I multiply the whole equation by half I will get 180 degrees minus C C plus D by two now the problem is solved so now what you can do is you can replace this item here. This one here can you see by this item here so hence it becomes angle AOB is equal to what is this 181st so right 180 degrees minus the second thing can be written as 180 degrees minus angle C plus angle D by two I just substituted this and hence you got angle C plus angle D by two hence proved. Okay, I hope you understood this. See more than I know most of you would be knowing how to solve it. I'm just trying to say you have to develop a construct of problem solving and it is you know very, you know, it's lacking in most of you but when I see your answer scripts and you know the way you approach, you have to, you didn't understand the last three steps. Once again, see. So A plus C plus C plus D A plus B plus C plus D is 360 degrees. Hope you understood this, then angle A plus B can be written as 360 degrees minus angle C minus D. Just take these two on the right hand side here. Okay, then what angle A plus B is this so if I multiply this by half I have to multiply this whole by half. So if you open the brackets what you what will you get you'll get 180 degrees minus this. Yep. Okay now. Did you all understand now. So, hence, now what I will come back to a OB so a OB was what this one. 180 degrees minus half angle A plus B so and minus half angle A plus B was equal to 180 minus this. Simply replace this one, this item here I replace this one by this one, this one. Yep. And hence, if you replace it, you will get the desired so 180 minus 180 gets cancelled and this minus and this minus one minus gets multiplied becomes plus one and you get C plus D by two. Now did you understand all of you. Is that fine. Clear area we are just trying to understand the methodology most of you are grossly missing on the methods to adopt and how to present your answers. Okay, so hence please pay attention even if you know how to solve it, but I know when you reproduce in the paper. 180 minus 180 becomes 360 that's what the kind of mistakes you have done in your last midterm exam so please be attentive here. Okay, now before we proceed further. There could be multiple methodologies yesh it's just a way of expressing it properly that's it if you if you know there is one way works better than the other please go about it but then. No, no, no, please suggest don't worry you know you are very very welcome to suggest anything. I'm just trying to hint upon is one thing which is you know which I observe in your all your papers is the way you present your answers is little you know shaky you know the approach is not that correct. So hence that has to be taken care of okay in in bit of all this I'll give you one more generic theorem and that theorem is angle some property of any any polygon. Angle some this is an additional information for you. Property angle some property of any polygon you must be knowing few of you angle some property of any polygon. Okay, so what is it if you have if you have a polygon or polygon or polygon with n sides with n sides for example n is equal to three in case of triangle. I'll just give you who who's who's Eminem you know I know I need to know your name please tell me your name don't have some random names in my class I need to know each one of you please tell me your name. Yeah so a polygon with n sides so angle some angle some property property angle some property of a polygon. polygon of n sides is nothing but some of some of all angles of the polygon polygon polygon with n sides and sides is equal to how much is it anyone knows n minus two times 180 degrees. Okay, please remember again I'm writing n minus two times 180 degrees this is what the generalization of this particular theorem is you can use it for any purpose you want now you can check when n is equal to three what is a triangle right. Three sides triangle CC when n equals to three and minus two into 180 is 180. Right, when n is equal to four. What will happen four minus two into 180 is equal to 360 degrees. So this is for triangle this is for quadrilateral this is for quadrilateral file and say we have a pentagon pentagon pentagon will have five sides n equals to five. Right, so hence it will be five minus two into 180 degrees is equal to 540 degrees. Okay, so this is how it goes so hence you can you can now predict for hexagon for hexagon angle some property hexagon means n equals to six. So it will be nothing but six minus two times 180 degrees which is nothing but 720 degrees okay now can you tell me guys what will be the angle some property or the sum of angles of our polygon with 12 sides 12 sides polygon. One thousand eight. Yes. One thousand nine eighty. Nine eighty why? 180. 180 awesome guys you guys. Very nice. 180. 12 minus two times 180 degrees. Yeah. 180. 180 degrees. Right. 1,800 degrees. Very good. Thanks a lot. Okay, all these things you will learn in organic chemistry later on the angles of a polygon becomes very, very important later on. Okay, so please keep these things in your mind. Okay, now what about 360 divided by n? What is 360 divided by n? What is it? Come again. Exterior angle. What is this? No, I come again. What is what are you talking about? Sir, what about 360 divided by n? That formula. 360 divided by n formula. What is it? Exterior angle. Exterior angle. Yeah, who's this? Sir, Shadli. Shadli. Yes, Shadli. What is what? Who's the person who's asking the question? I didn't recognize you. Sir, I'm assure them, sir. Naful. Oh, yes. Tell me what is this? Shadli, you're saying something about this. Sir, 360 divided by n is for exterior angle. State the sum. Sorry, state the theorem. Sir, if n is the number of sides in a regular polygon, then each angle is going to measure 360 by n degrees. That is, that is the, that's not external angle, no? Oh, each external angle is going to be like that. That's what you're saying. Okay. Did you understand harsh, harsh, right? Yes, sir. I understood. Yeah. So regular polygon might mind you guys. I'm not talking about regular polygons here. What is an 11 sided polygon called though? If I don't know, see best way is to go to Google and you learn how to study simultaneously. Okay. Open. So you don't need to know everything. So let's say undecagone is my prediction. Let's see. 11 sided polygon. 11 sided polygon. Yeah. Name. So someone has asked all these things. Dual. What is it? Um, hendecagone. Okay. The person who asked this question, Aditya, hendecagone. I don't know how do, how do you pronounce it? But yes, keep Google friendly. You know, with you, keep it with you all the time. Okay. If possible. Yeah. Now, uh, don't take again is for 12 coming back to our. What? Where did it go? Yes. So now the next question. Okay. This, that's what I mentioned in the very beginning in the question number five, something was missing. It's not given what is to be proved. So don't worry. I'll give you, I'll give you enough things to prove now. Okay. So question number five, guys, how many of you could solve this question to prove, to prove in this question is angle P plus angle Q is half angle ABC plus angle ADC. This is what is to be proven. All of you could do this question. Anyone who could do it? Yes. No, don't know. Maybe come on guys. How many could do this? I did it. Yeah. Okay. Great. Great. Let me tell you how to approach such kind of problems. So hence again, you need to understand the approach more than the real, you know, solution, solution. You guys, I know you guys are pretty smart. You will be able to do it, but approach as well as presentation is something which is very, very crucial. Now, the question is bisectors of angle A and D quadrilateral ABCD. Now, many of you, the first mistake you will do is ABCD appears to be a parallelogram, isn't it? In the figure, it appears to be a parallelogram and almost, you know, a good chunk of people would start the problem solution by considering or assuming that ABCD is a parallelogram and hence that is first stage of, you know, mistake, right? So hence do not get confused or let's say guided by whatever the figure is given. Please read the question thoroughly at least twice. So it says in figure bisectors of angle B and angle D of quadrilateral ABCD. So hence what I used to do during my time was I would skew the image to that extent that I don't need to follow whatever is there. So, you know, unfortunately here it is not getting done. No, it's fine. Again, this is also, so I just try to distort the, you know, try to generalize the sum, you know, by drawing this kind of a diagram now. Now name it according to what the diagram is ABCD. And now it says angle B and angle D. So angle D bisector is going here and you produce it. So this point is Q and angle B bisector is coming here and you produce it. Wow. The software is intelligent enough to predict good. So hence this is Q and this is P. Now you have to prove, prove what angle P plus Q. So this angle plus this angle is equal to is equal to half of ABC and ADC again P and Q. Okay. So how to solve again P and Q some and you know, normally you're seeing that we're trying to solve similar type of problem. So hence somewhere the angles some property must be there, but you know P D Q B is a quadrilateral. So either you can use the angles some property of a quadrilateral itself or you can try and use angles some property of a triangle. There could be two approaches here, right? So let me say where all is angle P. So clearly. So given and all. So you have to start with this given. What is given BP and DQ are bisectors, bisectors of angle B and angle D respectively. This the question itself will help you but sparing 10 seconds to write such given statements will help you. Okay. So right and AB and CD are produced produced to Q and P respectively. So this is a respectable and always try to number the given items. Why? Because the more you number the information, the more you get a clarity. Okay. So those are the two facts which are given now to prove to prove what is to be proven. So angle P plus angle Q is nothing but what is it? Half of, half of angle ABC, ABC and ADC. This is what is to be proven. Okay. Great. Now what? So now once I have taken down the question. Now I'll use the entire space. Now clear how to go about it. Now if you see this is one triangle guys. So I will use this information. P is lying in one. This thing and then let's use the information which is given VP and DQ are bisectors. That means if I coin it as X, this will be X. And if this is why this will be why is it? Now let's find the proof out. So how to go about it. Let's try to work on angle P first. So if you see angle P plus angle C plus X in this triangle, this triangle, angle P plus C plus X. What is it? It is nothing but it is equal to 180 degrees. Okay. Is it fine? Now angle P plus angle C. So hence if you see the left hand side comprises of P. So hence I am getting the P here. P thing is here. So keep that in mind. P is here. Now let's talk about Q. Right. Let's talk about Q. Q angle Q. If you see where is angle Q here? So this falls in this triangle. Correct. So angle Q plus angle A plus Y is 180 degrees again. And what is the basis? Angle some property. Okay. Now what? So I want to get angle P plus angle Q. So either you can represent P in form of all other angles and Q in terms of all and all other angles and add, or you can add these two equations right away. So for making your life easier, I'm expressing it like this. Angle P is equal to 180 degrees minus angle C. And what can I say about X? X is half of B, isn't it? So I can write this as 180 degrees minus angle C minus angle B by two. I hope it's clear to all. Next is angle Q is equal to 180 degrees minus angle A. And Y can be written as angle D by two C angle here. And X is B by two and Y is D by two, isn't it? Okay. Now what I am naming this as one and naming this as two and add one and two. One and two. See, can you not see that I get the LHS? I get the LHS, right? LHS is sorted. I just need to arrive at the given RHS. So let's see. So if you add, what will you get? 180 degrees minus angle C minus angle B by two. And plus 180 degrees minus angle A minus angle D by two. Okay. Which is equal to 360 degrees. 180 plus 180 is 360 degrees minus angle A plus angle C within brackets. I'm clubbing them together. And this is nothing but angle B plus angle D by two. Clubbing minus sign together. Now you'll ask why I'm clubbing this together because I am always having this desired result in my mind. In the desired result, you see only B and D are talked about. So hence I have to somehow get rid of this item. And again, to get rid of that item we'll again use. So this is quite, you know, intuitive that 360 degrees also there, A and C is also there. So can I not write this as angle B plus angle B. This item is this. Why? Because A plus C plus, I'm writing here A plus B plus C plus D was 360 degrees by angles and property of this triangle quadrilateral. So hence A plus B will be 360 degrees minus C minus D. I'm not putting the angle sign. You can understand from here. Okay. So what is it? So hence clearly this item is B plus D and minus again angle B plus angle D upon two. So take the LCM you'll get two and it is two times angle B plus angle D minus one time angle B plus angle D. And hence you get the desired result which is angle B plus angle B by two. Okay. Hence, probably right. Hence prove. Okay. I hope you all understood how to solve such kind of problems. Now the question is, can we consider BD? No, you're again. How can you consider BQDP as a quadrilateral? The question is, can we consider PBQB as a parallelogram? For parallelogram guys, you must have PB parallel to DQ and sorry. So there is no information regarding whether they are parallel or not. This information is not there. So hence guys, we can't do that. Okay. So don't assume things until unless there are enough information to prove what you're assuming. Okay. I think there is an easier method. Go ahead. Please, please, you know, enlightened. No problem. There could be one geometry. Some can have multiple ways of solving it. You can, I, as I told you, you could have taken this approach that PDQB is a quadrilateral and PNQ being two of the opposite angles. So you can start from there as well. Okay. So one sum can have multiple ways of solution. Again, mind you, my intention to work out problems with you is not to find out solution only, but also to train you on how, how you should approach a problem and what, how should you express yourself? Okay. Because this is just one, one such sum in every sum, you have to be diligent about your expression. Okay. Now moving on. So this is, I think you can do the exercise and let's now again go further. Okay. Where in now we are talking about this table. So hence this table could be useful for you. You can use this table. Okay. The table is the properties, all the different types of properties or different properties of different types of quadrilateral. And whenever you are trying to solve any sum, it's always an advisable thing to do is that you have an A4 sheet in front of you and keep the properties of parallelogram as we are expressing right now in front of you and then try to solve the problem that is always, this is one, one thing you can do. Another thing that I will recommend is make a list of, list of all theorems, all theorems in one place, one place. So you can take an A4 sheet or a dial or a notebook wherever and you keep on writing theorem one, theorem two, theorem three, only the theorems, not proofs. Keep writing all the theorems at one place. Okay. And then what you can do is you can keep it in front of you whenever you are solving any geometry problem anytime. Okay. So now let me give you this table and what I'm talking, just give me a second guys. Okay. Now what I was talking about is but our what some proof cannot be used. See, Aditya here, what I am talking about is see, forget about what's happening around. I'm just trying to say this is the right approach of learning mathematics. So, you know, scores will follow. Don't worry, you know, if you practice and you have enough, you know, knowledge and right approaches, you will definitely get the marks. So never mind. So what I'm trying to say is please maintain a list of this and the table which I'm now going to give you. So hence all of you can draw it as well. So hence I'm writing it as type. Type of body lateral. Then we'll talk about the sides. Then we'll talk about the angles. Then we'll talk about the diagonals. And then we'll talk about the area. All information at one place. Okay. So if you want, you can draw the table with me. Or you can, you know, your, your call. So this is what I'm going to do. So now, okay. So never mind. Let me first start with a trapezium. So hence first information is about trapezium trapezium. Okay. What about the sides? Sides, you can write one pair of opposite. One pair of opposite sides are parallel. Okay. Only this. Angles, nothing such specific thing. Diagonals. Nothing specific property. Area is given by half into some of parallel sides into height. Height is what is height? Height is this. So if I have to draw the triangle diagram. So this is, let's say this is a B and this height is H. So it will be half into a plus B times H. Clear. This is all information about trapezium. Now there is a specific special type of trapezium as well. And that is isosceles trapezium. So I'm writing it here again. So let's say isosceles isosceles trapezium. What about this? So same as above plus nonparallel sides. Nonparallel sides are equal. Okay. Here base angles are equal. Nothing specific about diagonals. Actually diagonals are also equal. Diagonals are equal and area remains the same. So this is the information about isosceles trapezium. What is an isosceles trapezium? You should see this is an isosceles trapezium where this side is equal to this side. Okay. This side is equal to this side. These are parallel lines and this angle will be equal to this angle. Isosceles trapezium. Now next is parallelogram. Parallelogram. What about the sides? Opposite sides are parallel and equal. What about angles? Opposite angles. Angles are equal and adjacent angles are supplementary. Supplementary means some of them is 180 degrees. What about diagonals? Equal and bisecting each other. These all will have to prove one by one. But I'm just giving you the crux of the info. What is the question sir in trapezium? Adjusting angles add up to 180 degrees. Yes. There you can call it. What Shardul is trying to say is this angle plus this angle is 180 degrees. Adjusting angles but not all adjacent angles. Two pairs of adjacent angles are supplementary. Equal and bisecting each other area is given by base into height. Base into height. What is that base into height thingy? This is parallelogram. This is base and the distance between any two sides. Any two parallel is H. Area will be into H. If you know this base and let's say this height then that's also is okay. Let's say this is A and this is P. Area is also equal to A into P. Whichever both ways are okay. I will redraw so that you guys do not get confused. Okay. So what is it? I'm saying so this is a parallelogram. Let me draw a little parallelogram. Let's say this is A or other B. This is A and let's say this height B H. And let's say distance between these two parallel sites B P. Okay. So the area is area is equal to either B into H or A into P. Both are same. Okay. So this is about parallelogram. Okay folks. So I'll just quickly complete the table so that I can leave you for there. You guys must be tired. So just give me one two minutes and I'll complete this table. Then we have something called rhombus. Rhombus. Where again everything of parallelogram so everything stays the same. This is same. Opposite sides are parallel and equal. But side all sides are equal. Okay. What about angles? Opposite angles are again same. Adjusting angles of supplementary opposite angles are equal. Yeah. This this remains as it is. And this one they are not equal but bisecting each other at right angles. Okay. And let's say this is the rhombus. So so hence hence let's say this is my D1 and D2 apart from the above two formula. The area formula let's say this is D1 and this is D2. So area is also given as area of a rhombus is half into D1 into D2. Okay. Okay. After this so diagonals bisect the angles where which one? Yeah. There are multiple features. Aniket. There are many other things which will be there. But right now we'll focus on something which is basic comparable to all. Now. So next is kite kite adjacent adjacent sites. Two pairs rather two pairs of adjacent sites are equal. Angles one pair one pair of opposite angles are equal. And diagonals larger diagonal larger diagonal bisect the smaller one smaller one at right angles. Okay. And area again is the same half into D1 into D2. What is the kite guys? See this is a kite. So little bit more. I'll try to make it again. It takes a diamond shape. But anyways what I'm trying to say is these two sites are longer. Yep. Again, every time I do it takes a never mind. So let's understand like this. Yeah. So this is the longer diagonal. This is the smaller diagonal. So hence it is half into. So this is D1. And this is D2. D1 into D2 is the area. Now this is right angle. This side is equal to this side. This side is equal to this side. And this side gets bisected. This one. So the longer diagonal is bisecting the right. Okay. So these are few of them are, you know, all sir. Can you go up? Okay. I'll definitely go up. I hope you know now. So let me just, you know, yeah, this is the table, which I was talking about. What you can, what you guys can do is anyways, you have the, you know, book right now with you. So it will always be a good practice to, you know, tabulate everything and do one thing. Let's do one thing here. All of you can draw your own. Send me a snapshot of your table with all the, all the, you know, body laterals and their properties written down neatly. With some diagrams and all, and the best one gets a reward from me. Is that fine? So the best such one, I will publish that in our forums and everywhere. And then I'll, you know, give the reward to the person who is going to send me a very good diagrammatic representation of all these information tabulated at one place. And I'm going to give you a good reward from my site. So that's the thing. So hence you jot down everything. And next class, when we'll meet, we'll see how in the same diagram, each property can be proved and then utilize for further mathematics. I hope the thing is clear to you. So there are a few queries for kite. The opposite sides are parallel. No. It's not parallel. Okay. Opposite sides are not parallel for kite. Okay. Is rhombus or kite? Yeah. You can say it is a special case of a kite. You can say so. For that matter. It's as good as saying is a square a kite is a, you know, a square could also be a kite. So hence, yes. You can say so. But then in a strictly theoretical or definition point of view, since kite is a diagram where only two pairs of adjusted sites are equal. So hence in that sense, you know, school of thought says that rhombus is not a kite. Our diagonals equal in parallelogram. That depends. Maybe may not be. For example, in a, you know, you can, you can all such questions. How do you handle what you can do is draw schematic diagram. So let's say for example, this is a parallelogram here. Right. Do you think the diagonals are equal in this case, or you can, you can, you can go ahead and say, you know, you know, one case is good enough to prove that, you know, it's not, you know, they are not equal. So hence not necessary. Diagonals need not be equal in parallelogram. Anything else? Guys. Any other question? Yeah. Any other questions? I think I could answer all. Okay. Very good. Okay. Okay. So I think I could answer all. Okay. Very good. By when you should send as quick as possible, I will wait for two days and within two days, if you send, I will keep on posting it in all the groups that this is what I have got till so far. This is the best. If I get better than that, let other audience also decide which is the best one. And okay. A decision will be mine. I'll be giving you reward. If you send me a good diagrammatic representation, the way I have done it, you can improvise upon it and you can come up with your own diagram and let me know as quickly as you're done. Okay. So hope the class was useful and we'll meet again. So thanks for attending it. Goodbye and good night. Thank you. Thank you, sir. Thank you, sir. Thank you, sir. Thank you, sir. Thank you, sir. Thank you, sir. Thank you, sir. Thank you, sir.