 okay good afternoon everyone those who are watching I request all of them to please write their names in the chat box so that I can know who all are attending the class thank you so Varsha and Anchita are you guys studying together this is Varsha and Anchita if you guys have any doubt or any specific topic that you want me to cover you can post it in the chat box or on the group or you can send me the postman message on WhatsApp okay Tana is here Varsha, Anchita and Shraddha is here anyone else who is attending the class okay I'll wait for one more minute before I start the class what I do is okay Shraddha I take I will try to solve few questions from there too but before that I would like to solve few questions from quadrilaterals also because quadrilaterals is something which we haven't done that much so I have discussed many type of quadrilaterals with you so I will start taking questions best on quadrilaterals and I'm not solving any or I'm not proving any theorem as such I use the theorems directly I request all of you to please go through the book so that you can know how the theorems are proved okay welcome Aditi, Dhruti and great so is it fine guys I mean I'm not going to prove theorems because this is a revision class I don't think moving theorem except one or two like theorem like this the angle bisector of a parallelogram form a rectangle this can be directly asked in the question for a two marker or something like that so if anyone wants any particular theorem to be proved or if you want to understand how a theorem has been proved you can post that doubt in the chat box I'll take that up so I'm starting with quadrilaterals okay great so I'm starting with parallelogram now what are the properties of parallelogram so there are few theorems which has been given first theorem is suppose you have a parallelogram something like this if you draw a diagonal so suppose I mark it ABCD so this diagonal will divide the two halves into congruent triangle and how it can be proved this side is common to both these two sides are equal these two sides are equal so you can utilize any one of them or you can say that this angle because these two sides are parallel so this angle and this angle is equal and then you have two sides equal so as simple as that so what I am trying to say is that hi Aniru of okay Aniru there also joined good so one property is diagonal will divide now I'm starting in full flow diagonal will divide the parallelogram in two congruent triangles so this is the first theorem second theorem the more basic things which I have discussed in the class I'm not going through it that parallel sides are sorry opposite sides are parallel and equal and positive angles are equal and all these things and to adjacent angles at 180 degree this we have already done in the class so second is opposite sides are equal we all know it theorem 3 tells that opposite angles of the parallelogram so if this is a parallelogram so opposite angles would be equal so angle a would be equal to angle C suppose I name them ABCD and angle a would be equal to angle C and angle B would be equal to angle D okay Ritu has also joined good so we have almost everyone except Shreya and and Sanjana perhaps okay great now look at here now next one is the diagonals of the parallelogram bisect each other so till now if I have done theorems first was that diagonal creates congruent triangles opposite sides equal and parallel then opposite angles are equal then we have diagonals bisect each other now understand diagonals don't bisect each other at 90 degrees so please don't have that in mind so in parallelogram the angle is not 90 degree between diagonals so diagonals bisect each other this I have told you in the class also bisect each other also next theorem is so I prove one theorem out of this next theorem is in parallelogram bisector of any two consecutive angles intersect at right angle so how do you prove it this is a good question so in parallelogram these are just extension of what we have discussed in now and we have discussed in now that diagonals create two congruent triangles opposite sides are equal and parallel opposite angles are equal diagonals bisect each other as simple as that now in parallelogram bisectors of conjugate any two consecutive angles so I am taking angle a I am taking angle b and this is bisector of angle a and b now try to understand guys if a is equal to c so suppose this is c and this is d and let me make it capital for you and b is equal to be so try to understand angle a plus b plus c plus d is equal to 360 degrees now a and c are equal so I can write it to a and b and d are equal so I can write it to be so try to understand a 2 a plus 2 b is equal to 360 degrees so a plus b would be equal to 180 degrees as simple as that now try to understand what happens these are the bisectors of suppose a o so I am writing o a and o b they are bisectors of bisectors of bisectors of what angle a and angle b so what happens I take half a from here so half a plus half b which is this angle and plus this angle o should be equal to 180 degree now you look at here a plus b is equal to 180 here so half a plus half b would be equal to 180 divided by 2 which is 90 degree so half a plus half b is equal to 90 degree so I am writing 90 degree plus o that is equal to 180 degree so angle o will be equal to 180 minus 90 90 degree so what is the theorem theorem says that bisector of the any two consecutive angles or I'll say adjacent angles will meet each other at 90 degree so how this would be this would be 90 degree as this theorem suggests next theorem is if a diagonal of parallelogram bisect one of the angles of parallelogram it also bisects the second angle also prove that it is a rhombus so this you can go through this is not that difficult okay next one this is the last theorem of this theorem I want to prove the angular bisector of it off the angular bisector parallelogram form a rectangle so look at here what I am trying to do is that I am drawing a parallelogram for you this is a this is b this is c this is d now these are the angle angular bisector so let me draw a line so this is a so suppose ap then suppose this is angular bisector here now suppose this is angular bisector here and now suppose this is angular bisector here now guys try to understand it this question is very easy we just prove that this angle P would be 90 degree so what I am and this angle P would be 90 degree and if this angle is 90 degree this angle would also be 90 degree so let me make a bigger diagram for you so that you can understand it properly suppose this is a this is b this is c this is d so this is the angular bisector ap this is the angular bisector vr this is the angular bisector of angle c this is the angular bisector of angle d so I am putting it pq sorry this is s so this is s now we have just proved that this angle any two consecutive angular n angular bisector or angular bisector of any two consecutive angle will meet each other at 90 degree so angle P would be equal to 90 degree we have just proved that similarly angle r which is at the point where angular bisector of angle B and D are meeting so angle r would also be equal to 90 degree as simple now look at here now what would be this angle this angle will also be angle s is also 90 because consecutive angles a and b and their angular bisector is meeting at s now if s is 90 degree opposite angle to s would also be 90 and similarly q will be 90 degree so opposite angle to q will also be 90 degree so opposite angle to q and opposite to s that is also 90 degree so all the angles of of pq rs are 90 degrees hence it can be proved that pq rs is a rectangle at simple so now let me take a few questions for you guys and then I will move ahead so I am not taking easier questions I am taking difficult questions only now I am taking two three questions from level two of R.D. Sharma book so one question which I am taking is in this in a parallelogram I am just writing parallel so that you can understand that abcd the bisector of angle a also bisects bc at x prove that ad is equal to 2ab so now guys look at here so suppose I have a parallelogram so I have a parallelogram here and this is a this is b this is c this is d so bisector of a so bisector of a would be something like this and now bisector of a meets bc at x such that it bisects bc it means that bx is equal to cx it has been given also a x is angular bisector of angle a so how can you prove it so suppose this is angle one and this is angle two so angle one is equal to half a if it is an angular bisector now angle a plus b is equal to 180 degree if it is a parallelogram I am writing here because it's a parallelogram so angle b is equal to 180 degree minus angle a now in triangle abx angle one plus angle b plus angle two is equal to 180 degree so try to understand angle one can be written as angle a by two and writing angle one is equal to angle a by two already it has been given here so I am replacing this with angle a by two angle b can be written as 180 degree minus angle a plus angle two is equal to 180 degree now guys try to understand how to ad is equal to 2 ab what needs to be done I have to prove that this side is two times of this side now how this can be proved to prove it I need to so that try to understand if I am able to so that see what happened opposite sides are equal so if opposite sides are equal if ad is two times ab then bc will also be equal to two times ab so I am writing here if I have to prove ad is two times ab if I am able to prove bc is equal to two times ab are you understanding because ad and bc are equal in parallelogram and now bc is getting divided into two equal halves bx and cx so try to understand if I am able to prove that bx is equal to ab then my work is done so my objective over here is that what I need to do I need to prove that angle one is equal to angle two if I am able to prove angle one is equal to angle two in that scenario what happens I am able to prove that triangle abx would be an isocellous triangle and in an isocellous triangle what happens here is in isocellous triangles what what will happen is that sides opposite to equal angles would be equal that is why till now I have I have been trying to find out relationship between angle one and angle two how can I find out relationship between angle one and angle two so I have taken the reference of triangle abx if I take a reference of triangle abx so what happens in that scenario that angle one plus angle two plus angle b which I have written here is equal to 180 degree now angle one I know is a by two angle a and b I know the relationship so I can find out the value of angle b which I am finding out here in terms of angle b and that's how I will only have relationship between angle one and angle two so try to understand here what I write is this 180 and 180 is one now I have angle two and a by two minus angle a is minus angle a by two that is equal to zero so angle two is equal to angle a by two and angle one is also equal to angle a by two so see here I have proved that angle two is equal to angle one and if angle two is equal to angle one then I am writing here look at here then I can write that ab is equal to bx and if ab is equal to bx which is also equal to cx so bc is equal to which is equal to bc by two which is also equal to which is also equal to ad by two that's simple so I have proved here that this side ab is half of ad or ad is two times ab so in this question the question was all about proving that always remember they could have in this question they could have easily given ad should or bc should be equal to two ab but as soon as they could have given dc your mind could have got that prove that this bx and ab are equal because they have been divided into two parts so do you take it so just to confuse you they have given ad but you should know that in a parallelogram the opposite sides are equal so as soon as you see ad you should also look at the opposite side which is bc and now there has been a relationship so you can prove that this bx is equal to ab and you add bx plus cx so it means that bc by two is equal to ab or bc is equal to two ab or ad is equal to two ab so that's how you can solve the question next question which I would take is now let me erase let me make some space for you so that you can understand that okay now let me take next question of of parallelogram so next question is abcd is parallelogram l and n are midpoints of ab and dc respectively and our relationship has been given here which is al is equal to cm we have to prove that ln and bd bisect each other now how to solve this question so suppose first I draw a parallelogram abcd in this parallelogram what has been given l is on ab so let me mark l and m is on dc so let me mark m here so it has been given given that al is equal to cm now guys that's not something kind of we need to know we need to know that ab is equal to cd in a parallelogram opposite sides are parallel and equal now if l and m are midpoints of each other midpoints of ab and dc then obviously half of ab would also be equal to half of dc then half of it would also be equal to half of dc and what is half of ab half of ab al and what is half of bc that is equal to cm which has been given here which is a redundant data which doesn't need to be there so most probably if someone who knows this will not give you this relationship in the examination now what I have to prove I have to prove that ln and bd bisect each other suppose this point is equal now how do you prove it how do you prove that ln and bd bisect each other or so to prove that ln and bd bisect each other you have to prove that ob will be equal to od and om would be equal to om this is what you need to prove so if this relationship is proved these two relationships are proved then your question would be done now how do you prove it so try to understand see od and om are part of triangle odm so I am I am writing triangle odm so two sides are od and om are sides of it great and I am writing olb triangle and look at here ol and ob are part of it now I have two triangles and I have to prove that sides of this triangle is equal to sides of this triangle this can only happen when I prove either similarity or congruency between the two triangles now in our syllabus only congruency is there so your mind should directly go to the congruency that I have to prove congruency wherever you have to prove that this side is equal to that side at least once your mind should go to the word congruency that is there some kind of congruency which can be proved now try to understand see this angle one is equal to this angle two why so angle one is equal to angle two why because these two sides are parallel so I am writing writing alternate angles now this angle dom is equal to angle bol vertically opposite angles so if I have proven that two angles of a triangle are equal the third one has to be equal so I've proven that all the angles are equal as simple as that so what you can do is or to prove congruency you can also say that now you know this relationship so if you you also know that bl is equal to dm because they are half of bl is half of ab and this is half of dcm writing here this is half of ab and this is half of dc and if ab and dc are equal these two have to be equal so I have one side equal and the angles I have already proved that they are equal so I have ASA congruency here so angle side angle angle side angle congruency here or angle side angle so this angle is equal to this angle this angle is equal to this angle and this side is so angle side angle congruency here so if I have proved congruency so you can apply cpct what is cpct corresponding part of congruent triangles are equal so by cpct you can prove that ob is equal to od and om is equal to om okay so I hope you all understood it if I'm explaining something and you guys are unable to understand please ask questions next question next question is I'm taking this is the last question from parallelogram which I'm taking next and I'm taking because a construction is required extra construction is required here yes shardha AS congruency is also possible there a lot of congruences are possible because you can prove different sides and and because the other angle is also equal so you can say AS congruency was also equal so you are absolutely right about that okay now try to understand now suppose this is parallelogram abcd or no they are not saying this is a parallelogram they are just saying this is a quadrilateral so they are saying in a quadrilateral so I'm writing there if abcd is a quadrilateral in which ab is parallel to cd and ad is equal to vc prove angle a is equal to angle b now in this question try to understand all of you nobody is saying that abcd is a parallelogram so I'll never make the figure first of all which is which looks like a parallelogram so I am making a figure something like this so I'm just ensuring that ab and cd are parallel to each other and ad is equal to bc now try to understand in this question I have to prove that angle a and angle b are equal now how do you prove two angles are equal do we need to prove that I mean in this particular question I don't think there is a need to think about congruency or or something like else why because you know in a quadrilateral the angles would add up to 360 degrees so somehow you have to find relationship between different angles and how can we find that so try to understand it has been given that ad is equal to bc now what I do is that I draw a line parallel to ad here and I mark it e such that ab, ad is a parallelogram so in that scenario a plus e is equal to 180 degree now try to understand now ab, ad is a parallelogram then ad would be equal to b sorry once again have I made a diagram correctly yes I made a diagram correctly once again ad is equal to bd okay so ad would be equal to b okay and now try to understand what needs to be done here now this angle and this angle is equal sorry not equal 180 I mean the relationship between them is angle a is equal angle a plus angle e is equal to 180 degree sradha in this question we are not discussing that we are not saying that abcd is a parallelogram we are just saying that abcd is a quadrilateral in which ab is parallel to ab is parallel to cd so I am just saying this and this are parallel and these two are equal now what happens here is I can write the relationship ad is equal to be now ad is also equal to bc hence it has been given here this is because of the parallelogram and this is because it has it this has been given here so that's why I'm writing here so now I can write that now I can write that be is equal to bc so this and this are equal and what kind of angle they are making so if be and bc are equal so it means that let me take a triangle bec if be is equal to bc then angles opposite to be angle opposite to be is this angle c is equal to bc angle e so angle c and angle e are equal now try to understand so I can write this angle c I can write this as this is angle e is 180 degree minus angle a can I write it what is angle c guys what would be angle c can anyone druthi what are you asking tell me druthi what's your doubt you can ask me you can post it in the chat box okay how is angle a plus e is 180 degree one second oh I'm sorry angle a plus b is equal to 180 degree I'm sorry angle a is equal to angle e I'm just writing here oh I have corrected it druthi angle a plus b is 180 degree and angle a is equal to angle e so I've corrected it okay so guys try to understand now angle a and angle e are equal I have proved that angle c and angle I'm just removing it I have proved that angle c and angle e are also equal so if I can write it here angle a and angle c are equal now look at here if angle a and angle c are equal these two lines are parallel and angle b is equal to angle c also how angle b is equal to angle c because they are alternate angles so there we can prove that angle a is equal to angle b that is why I was getting confused because I have written a plus e is equal to 180 degree so when a is equal to e c and e I have proved that they are equal how c and e are equal because I have made this diagram so ad is equal to b and it has been given that ad is equal to bc so this is an isocellus triangle in isocellus triangle angles opposite to equal sides would be equal so angle c would be equal to angle e angle a and angle e are already equal so angle a c e are equal now these two are parallel lines a b and dc so a parallel line cutting sorry a transversal cutting two parallel lines so angle b and angle c on the other side would be equal so it means that angle a would be equal to angle b so that's how you have to solve these kind of questions now I'm taking different type of questions which are difficult in nature so I'm moving ahead do you have any doubts you can post it if you have any doubt you can post it in the chat box okay nobody is posting anything so I'm moving ahead then now next topic in quadrilaterals is sufficient conditions for quadrilateral to be a parallelogram okay Shreya is saying inequalities triangle Shreya is it triangular inequalities Anchita is also saying inequalities is it triangular inequalities that you guys are talking about okay so what I will do is okay what I will do is remind me at 5 30s sometime I'll start with triangular inequalities in between there was somebody who asked me this question of Euclid geometry so there was a line given where it has been given like this this this was point a this was point b this was point c and this was point d so it was given that ac is equal to bd you have to prove that ab is equal you have to prove that ab is equal to cd so try to understand ac can be written as ab plus bc and bd can be written as bc plus cd now both sides bc is cancelled so ab comes out to be equal to cd this is what you need to do so whosoever has done that whosoever has asked please note how the solution can be done to this question okay I am moving to other questions so now what happens in your book what has been given is that what has been given is that a converse of different theorems which we we have given has been proved in your book so I will just remind you of the theorems that we have done and then we will start taking up the questions so I will take two three questions from more from parallelogram then I will move to rectangle rhombus square and then a few questions from here and there and then I will move to triangular inequalities okay aditi I will just check yes aditi I I solved your doubt question six I have already solved ac can be written as ab plus bc and bd can be written as bc bc plus cd from both sides bc will get cancelled out and you will get ab is equal to cd so that's how you have to solve the question that's it nothing extra is required okay so now let me take different portion so we till now we have covered that in a parallelogram opposite sides are equal now converse of the theorem can be said as a quadrilateral can only be parallelogram if it's opposite side are equal so this is the converse and and and for everything you have been given proof so I'm not doing those proof of the theorems you just go through it I don't think they ask also but you just go through it the second is a quadrilateral is a parallelogram if opposite angles are equal third one is a quadrilateral is a parallelogram if diagonals bisect each other so it's converse of everything so what I'll do is I'll take two three questions from here and then I'll try to explain you guys so the question is abcd is a parallelogram I draw a parallelogram so here I am so I have a parallelogram like this so this is a this is b this is c this is b and suppose this is ac and this is bd and they are saying that x and y are two points on the diagonals bd so suppose this is point x and this is point y such that dx is equal to by so they are saying that dx is equal to by have to prove that have to prove two three things here prove that please make this diagram on your copy and write whatever I have said so that when I change the so that when I change the page you have the diagram and all the required information on your copy so I hope you all are sitting with your copy and just not listening to me so I have to prove that first one axcy is a parallelogram so let me first prove it for you so let me first make construct axcy this is something like this now if you have to prove that any particular quadrilateral is a parallelogram what will you do first thing first we need to prove that opposite sides are parallel and equal the second thing we need to prove is or we can prove that x y which is a diagonal is making two congruent triangles if it is making two congruent triangles so I can easily prove that two sides are I mean the first theorem that we that we discuss is that the diagonal of a parallelogram creates two congruent triangles on on on either side of the parallelogram so you can do that also so in proof one what I do is that suppose this is 0.0 now if abcd is a parallelogram I know that ob is equal to od how because parallelograms diagonals of the parallelogram bisect each other now try to understand if if this is true then ob minus by would also be equal to od minus dx why because by and dx are equal already it has been given there now what is ob minus by is it equal to ob and is it equal to o x so I have shown that in this particular a quadrilateral axcy I have shown that these two o x and o y are equal it means that this x y which is a diagonal is getting bisected by this 0.0 so this I have already shown now here o a is already equal to oc because this is part of parallelogram abcd so I am showing that diagonals of this quadrilateral are getting bisected I mean they they are bisecting each other so the converse theorem is that the converse theorem is that if the diagonal of any particular quadrilateral bisect each other what will happen that particular quadrilateral would be a parallelogram so I have shown here that the diagonals of axcy are bisecting each other if for any quadrilateral I find diagonals bisecting each other then what will happen then we can write that this is a parallelogram so this was the first proof what is the second proof let me just remove it the second proof is I have to prove that ax is equal to cy now as soon as you prove that axcy is as soon as you prove that axcy is a parallelogram ax would be equal to cy next I have to prove that a yb is congruent with cxd now how can you prove that now guys try to understand what is a yb so this and this are equal this and this are equal and this angle is equal to this angle so you can prove that this is a congruent triangle in in parallel what I'm trying to say is that a b is equal to dc and by is equal to dx angle a v y is equal to angle x dc so side angle side congruency can be proved now let me move to the third one now they are saying axd again the same thing they are asking so I'm not getting into it so this was one question which I thought which was important for you now let me move to another question which is also a also an ncrt question so I will solve three two more questions from here and then I'll move to rhombus and and rectangle and then perhaps I'll move to what you guys were demanding which is inequality if if I'm not able to complete I may extend the class by 15 20 minutes let's see what happens so let me remove it now next question is no we have done this question so I'm not taking this yes next this this question is very important all of you please please focus you have been given so I'll write the question so that you have idea what we are trying to do if triangle abc and triangle def such that such that abc are respectively equal to equal to and parallel to parallel to def and def then show that one abd is a parallelogram now let me make a diagram for you so what is happening I have two triangles so this is my one triangle abc and this is my second triangle def now I match it and just give me a moment suppose something like this so this goes from here and this goes from here and this is c now ab is equal to b and bc is equal to def now what happens guys guys is I have to prove that abd it means that which two sides abd so look at here ab and ad are equal it has been given and bc and ad def it has been given so what more do I need to prove to prove that abd is a parallelogram so let me write first of all ab is equal to de and ab is also parallel to b also try to understand abd now look at here now only abd are involved this bc and def is not involved here guys so if I am able to prove always remember this question is not difficult this question was just about remembering the converse theorem and what was the converse theorem that for any quadrilateral if opposite sides are parallel and equal it has to be a parallelogram so I am proving that ab which is part of this this side and this particular side which is opposite to it are parallel and equal already it has been given to you that they are parallel and equal so you just need to write that they have been given and you just need to write the converse theorem that as for this this particular quadrilateral opposite sides are parallel and equal hence this quadrilateral abd would be a parallelogram as simple as that now another quadrilateral has been given which has been given bcef so bcef again here also what you need to do bc is equal to ef bc is parallel to ef so you just write that two sides are equal and parallel you have been proven this so just because two triangles have been given in this question don't get discouraged just mix diagram I know this diagram is not proper so let me make a proper diagram like this now this looks like a proper diagram so this is a this is b this is c and this is d this is e this is f so abd I have already proved which is the front side bcef which is the base of it I have already proved it now if somebody is asking you to prove ac is equal to df you should write that how do you prove ac is equal to df this will also be ac df will also be a parallelogram so if ac df is also a parallelogram then what happens ac becomes equal to df so this is how you have to solve this these kind of questions I hope you understood this question at least now let me move to properties of and this is not very properties of rectangle rhombus and square so for all of them for each one of them the angles no I have I included rhombus removing rhombus for rectangle and square I'm sorry I'll just correct myself for rectangle and square angle each angle would be 90 degree now for both rectangle and square opposite sides would be parallel and equal so sides are equal in rectangle opposite sides are parallel and equal in rectangle the diagonals will bisect each other I'm not saying they are perpendicular bisector I'm just saying they'll bisect each other in square diagonals will do bisect each other but they'll also be perpendicular bisector in rectangle and square the length of the diagonals would be equal I'm not talking about rhombus till now so what what I have done I have till now said that in rectangle opposite sides parallel and equal diagonals are equal they bisect each other they do not they are not perpendicular bisector all the angles are 90 degrees till now I have said that square I'm saying all sides are equal diagonals are equal they are perpendicular bisectors all the angles are 90 degrees now let me move to rhombus so in rhombus opposite sides are parallel once again okay one most important parameter of rectangle or property of rectangle is diagonals bisect each other at 90 degree now each side I mean this is this is I mean from here only from rhombus only we construct square by making all sides equal so we need to understand that in rhombus all four sides would be equal opposite sides are parallel this is 90 degree now what is the difference between rhombus and square this angle is not 90 degree in rhombus so opposite sides opposite angles are equal rhombus all sides equal opposite angles equal then you have diagonals perpendicular bisector so these are the few properties of yes that rhombus that's what I have been saying that in rhombus we have diagonals at 90 degree I have told two times that in rectangle we don't direct the diagonal do not in rectangle the diagonals are just bisector they are not perpendicular I'm not saying that this angle is 90 degree so let me now solve a few questions from here and I'll solve only difficult questions as you know so let me take a question which question I should take so okay let me take a theorem only let me check how many of you can solve it okay please remind me guys at nearly 520 that I have to I have to go to triangular inequalities so anyone please put it in the chat box I'll move to triangular inequalities I'll try to solve difficult questions from there do you guys have any problem in heron's formula anyone you can just post it heron's formula okay now let me take it the question if if diagonals of a parallelogram of parallelogram are equal and intersect at right angle then the parallelogram is a square so what I am saying I'll just make a square first of all and I'll say this is abcd the information that has been given in the question is people are saying they don't have problem Shraddai saying she doesn't have problem in heron's formula all others are you guys listening to me at least right yes or no if the diagonals of a parallelogram are equal and intersect so diagonals of the parallelogram are equal so ab is sorry ac is equal to db that has been given to me now what more has been given to me suppose this point is oh what more has been given to me that they buy they they intersect each other at right angle so aob is equal to 90 degree that is also equal to boc and that is also equal to doc sorry doc and that is also equal to aob okay great nobody has problem in heron's formula so I will not teach heron's formula then great now how do you prove that this is a square what we need to do so we need to prove two things first we need need to prove that this side and this side are equal why I am proving ad and ab is equal or I will prove ab and bc is equal because it has already been given that it is a parallelogram so in parallelogram I will write what has been given to me ab is equal to bc it has been given to me and ad is equal to bc it has been given to me if we prove ab is equal to ad or bc then all four sides would be equal and then I have to prove that this angle and this angle is adjacent angles are 90 degree why because a and c are already equal because it is a parallelogram and b and d is already equal because that's a parallelogram so if I prove that a and b or c and d any two adjacent angles are 90 degree I will be able to prove it so now try to understand that how to first thing first if I have to prove that sides are equal what should go in my mind the first thing should be I have to prove congruency somehow so let me take triangle ab and triangle ad in triangle ab I am taking side a o in triangle ad I am taking side od and I am writing it equal how I am writing it equal because I have been given that that diagonals are equal if diagonals are equal half of the diagonals a o is equal to half of ab and od is equal to half of bc and ab is equal to bc that has been given to me and that is why I have written a o is equal to od now let's try to understand angle now see this is 90 degree and this is 90 degree so what I can prove I can write not prove I can write that a o b is equal to angle od which I have already written there this is 90 degree perpendicular it has been given to what else so I have proved I have written that this side and this side is equal a o is equal to no one second I can also write that a o is equal to a o because a o is common to both of them so I am proving the congruency ss congruency I am proving here so by ss congruency what I am what I am trying to prove that these two triangles are congruent as simple as that and if these two triangles are congruent then by cp ct sides opposite to this 90 degree angle would be equal so I can prove that ab is equal to sorry I'll write ab is equal to ad if I prove that ab is equal to ad automatically I prove that all four sides are equal now how do I prove that this is 90 degree so now look at here guys now see this side and this side is equal now let me make a new new diagram for you on the next page so this is like this this is abcd we have written and this is the diagonal and this is the diagonal now try to understand this is 90 and this is 90 and this is 90 now we have proved that this and this are equal and this and this are equal how we have proved just we proved that these two triangles are or we can say that half of the diagonals would be equal because diagonals are equal now I'll try to understand if I mark it O so triangle AOD don't you think it would be an isocellous triangle and if it is an isocellous triangle so suppose this is theta and this is theta so theta plus theta 2 theta and this is 90 degree so that is equal to 180 degree so theta comes out to be 45 degree and here also this and this are equal and this is 90 degree so this theta one and this is theta one so try to understand you can write that theta one is also 45 degree so you you what is angle a angle a is equal to theta plus theta one which is equal to 45 plus 45 which is 90 degree as simple so anyhow you can prove that that would be 90 degree or what you can prove so you take triangle ABD you take this triangle in triangle ABD this and this is equal so this and this is equal so this is how much this is 45 degree this is 45 degree so how much this complete angle would be 90 degree if I take I am making here perhaps you are not understanding I am taking only one diagonal so I have just proved that AD is equal to AB so AD AB equal so this is an isocellous triangle what happens in that scenario in that scenario you can prove that or you can prove from here you can prove that ABD is congruent with BSE so in in in that so any way you can prove this all the congruences can be proved and angle can be found out I will like this method so utilize this method or you can say that this this this and this so this is theta one and this is theta one similarly you can say this is theta and and and the other one is theta so theta theta one every angle can be found out and you can find out value of theta and theta one so anyway you can find out this this answer now let me take one good question which is an NCRT question so how many of you are prepared for math 2 examination properly I know none of you will reply for this question none of you will reply for this question so it has been told that abcd is a rectangle in which I have abcd which is a rectangle I am solving a question in which abcd is a rectangle in which diagonal ac bisects a and c both so ac Sraddha is saying okay okay sir what do you mean by okay okay preparation is ac bisects angle a and angle c so you have to show that or prove that that what is abcd abcd is a rectangle none of you should get less than 85 percentage Anchita is saying it's going on preparation always go goes on 85% is minimum for all of you that's the target so out of 30 it has to be more than 25 or if it is out of 50 it has to be more than 42 calculate how much you need to do so what do you have to prove we have to prove that Sanjana is asking how many marks it is for I don't know how many marks your examination is for you have to show that abcd is a square okay now look at here so how do you prove that abcd is a square what is the property of a square we have to prove that if if it is a rectangle so let me write ab is equal to cd and ad is equal to bc why because of rectangle all angles are 90 degrees so here I don't need to prove angles what I need to prove I just need to prove that ad is equal to ab if I prove oh it's 50 marks great so your target is forget about this question target is minimum 42 marks this is minimum that you have to achieve and maximum target is 50 so I'm just saying target is minimum 85% because last time many of you did not get 85% so this this time it has to be 85% now if I prove that prove ab is equal to ad then in that case I will be able to prove that abcd is a rectangle now how do I prove that ab is equal to ad now it's pretty simple that I have to go to some kind of congruency or something like that now guys try to understand what has been given to me it has been given to me that angle a or you can write that if this is angle one and this is angle two so anyway angle one is equal to angle two now let me draw a diagonal from here also and let me write down angle one is equal to angle two I have written now what is the another angle now this angle which I am assuming to be angle three would be equal to angle four here so angle three is equal to angle four here now try to understand guys angle one plus what can be done is that one second what I have to prove I have to prove that so look at here can I somehow prove that this angle and this angle is equal because what happens in this scenario if I prove that it is an isosceles triangle also if it is not a congruency question yes ac is common I am just proving I am not trying to prove any congruency here I am just trying to prove if are the angles equal like if I can prove angle three is equal to angle two somehow I will be able to prove and that should be the case see angle three is equal to angle two why angle three is equal to angle two because angle three is equal to a by two angle two is equal to c by two and angle a is has been given to given equal to equal so angle three is equal to angle two and what does that do that that tells me that triangle triangle acd is isosceles as simple so if it is isosceles then sides opposite to sides opposite to equal angles would be equal so what is the side opposite to angle three that is dc dc is also equal to ab and that is equal to ad and ad is also equal to bc so I have proven that all sides are equal angles are already 90 degree that can only happen in the in in in case of square so you prove that this particular in this particular case it is square now let me solve one more question from here and then I'll move to triangular inequality and at last I'll ask you guys whether you have any doubt so try to understand this question is that this is I made it so I have abcd and what does it look like yes Sanjana I'm just going to after this question um so that you don't need to prove that 90 degree in rectangle if it has been given to you that it is rectangle it has to be I mean angles has to be 90 degree so if somebody is saying that it's a parallelogram every time I'm not go and prove that opposite sides are equal and parallel and triangles would sorry diagonals would bisect each other if it has been given the theorems of that particular quadrilateral I'll take as it is and use it in my questions so rectangle if it has been given to me I will assume that this is 90 degree so it looks like trapezium so it is a trapezium now what is in trapezium ab would be parallel to dc so ab is parallel to dc now ad is equal to dc it has been given to me and I have to show that angle a is equal to angle b best way to show show that this is the same question we just did this question in another form make it up parallelogram I'll draw e here and what I'll do this ad if it is a parallelogram I'll make is such that adce is a parallelogram and ad is equal to c ad is also equal to bc it has been given to me so and d is equal to e so this angle and this angle would be equal so I can prove that this angle and this angle is equal or this angle is this angle and this and this angle is equal so this kind of question we have already done now let me move to triangular inequalities which is in demand today so guys give me just one minute I'll drink water and start so do you want me to go from very basic I mean theorems also though there are only two theorems three theorems how many of them three four theorems please reply quickly do you want me to do theorems or only questions I'm waiting for your answer okay Sanjana is saying theorems also great I'll do theorems also for you so I'm starting in one minute just give me one one minute I'll start in one minute yes I'll solve problems and theorems both there are only four theorems you can open the book if you have ad sarma or a survival anyone you can open the book there would be problem I'm using ad sarma only so I so if it is in front of you that would be beneficial so I'll explain most of the questions and then okay so what is the first theorem first theorem if is if two sides of a triangle of are unequal longer side has greater angle opposite to it theorem one longer side has greater angle opposite to it and this is very easy proof how do I prove it just look at here so try to understand this is my triangle and triangle is abc I am saying that two sides are unequal so I am saying that ac is greater than ab assumption this is so that I can confine myself to this particular statement now try to understand I have to prove that larger side has greater angle opposite to it so what is the angle opposite to ac angle b and that should be greater than angle c this is what I have to prove now how do I prove it simple construction if ac is bigger than ab then a portion of ac would be equal to ab again I am making this statement if ac is greater than ab then a portion of ac would be equal to ab so I am making this line point d on ac so construct bd such that ab is equal to ad now try to understand if this is angle one and this is angle two so how many of you think that angle two would be greater than angle c or let me write it acd and why I say this because angle two is exterior angle in triangular angle in triangle bcd now any exterior angle would be equal to summation of two other interior angles I am only taking one interior angle c so I am writing here angle two is equal to angle dbc plus angle acb so definitely if I delete this dbc angle two would be greater than acd now try to understand if according to this construction where I am saying ab is equal to ad so then triangle abd would be isosceles if triangle abd is isosceles then angle one would be equal to angle two because in isosceles triangle I have been using this theorem in today's class for longer time so in isosceles triangle angle opposite to equal size would be equal so it also mean that angle one is equal to angle acb now look at here this angle one is only part of this angle b so angle one plus this angle three is equal to angle b so angle one plus angle three is equal to b so if one only is greater than this c so one plus three would obviously be greater than this c so that's why I am saying that angle b would be greater than let me write here angle b would be greater than angle c so in this case what I did I revise it for you what I did was I assume that ac is greater than ab if ac is greater than ab in that particular scenario what happens is I construct a line bd which is online ac and this bd is constructed the objective of constructing this line bd but I can make an isosceles triangle and ab and ad I am making equal and that is isosceles triangle so I am putting angle one is equal to angle two I am knowing I know that angle two would be greater than this angle because angle two is equal to summation of both these angles and would be greater than any of the of any of angle three and suppose this is angle four or whatever it is now angle if angle one is greater than four angle one plus three which is complete part of b would obviously be greater than four so that's why angle b would be greater than angle c okay now let me move to another theorem theorem two what is theorem two so theorem two tells that it is converse that in a triangle theorem two in a triangle the greater angle has longer side opposite to it now always remember in converse kind of theorem I will utilize the first theorem so in this theorem which theorem I will utilize I will utilize the theorem that I just proved that the longer side has greater angle opposite to it now I have to prove that greater angle has longer side opposite to it so try to understand this theorem proof of this theorem is very easy so I have a triangle abc where it has been given that ac is greater than ab so I have to sorry I have to prove that ac is greater than ab and I have been given that b is greater than c so there are three scenarios that ac would be equal to ab if ac is equal to ab b and c both would be equal if ac is equal to ab so isocellus triangle b and c equal which goes opposite to what I have assumed here is it okay if ac is less than ab so what happens in that scenario so as soon as I write ac is less than ab so angle opposite to ac would be lesser than angle opposite to ab so here b would be lesser than c I already know that angle b is greater than c so there is only one scenario left out which is angle sorry which is ac is greater than ab so what in this particular proof what I am trying to do is that I am trying to say that ac is not equal to ab because if ac is equal to ab both b and angle b and c would be equal which is opposite to my condition given that b is greater than c I am saying that if ac is lesser than ab so angle b would be lesser than c by first theorem which I just proved so this is also opposite to what I have assumed earlier so there is only one case left out that ac is greater than ab so this is how you prove this now let me move to third theorem so third theorem is theorem 3 sum of two sides any two sides a triangle is greater than third side so suppose this is triangle abc just look at the proof this proof is a bit tedious because I have to do the construction so I have to prove that ab plus ac is greater than bc so this plus this is greater than this how do I do it so I draw a line from here and suppose this is point d such that this construction I am writing here ab is I am writing here I am making this ad equal to ac so ac is equal to ad so try to understand let me assume angles this is angle one this is angle two this is angle three this is angle four this is angle this internal angle five and this is angle six let me assume angles like this so if I am saying that ac is equal to ad in in that particular scenario what happens is that what happens is that what happens is that this becomes an isoceros triangle so angle opposite two equal sides would be equal so ac and ad are equal so opposite to ad angle five is coming and opposite to ac angle four is coming so look at here angle five is equal to angle four now okay guys let me just make a bigger diagram for you so that you can understand it properly and let me remove it also so a little bit of bigger diagram here will help our coach so this is a this is b this is c this is d I have already written here ac is equal to ad now this is angle suppose this is angle one this is two this is three this is four this is five I have changed the angles six so in triangle adc ac is equal to ad now if ac is equal to ad in that particular case angle opposite to ac angle five is there opposite to ad angle six is there so angle five is equal to angle six now try to understand now suppose I take this angle one second look at here guys now suppose I take this angle so what is this angle angle three and I take this angle angle six angle three plus six would be greater than angle five how do you say this this is the only part of the proof where students struggle to understand properly and they leave it here how do you prove it that angle three plus angle six is greater than this angle anyone can tell me can anyone tell me write something so what happens is look at this angle five what I'm trying to do is that I'm telling that angle three plus six is greater than angle six as simple this is just saying that angle three plus six is greater than angle six and why am I doing it let me discuss the concept first leave the question altogether I have to bring some kind of inequality how do I bring inequality so to bring inequality what I do in this particular question is that I am just saying that this angle plus this angle is greater than this angle so then why I'm going here I am going here because I don't need angle eight side ad here what I need I need side ac here and that is why I am going to angle five here as simple as that so from here I can say that angle three plus six which is this complete angle c so let me write it I will say that angle in bcd which is this complete angle six would be greater than angle bdc I mean three plus six which is this complete angle bcd will be greater than this complete angle bdc so it means that it is just telling me that what is the side opposite to bcd so what is the side opposite to this particular angle so side opposite to this particular angle is this complete bd and bd is greater than what bd is greater than side opposite to bdc now guys understand opposite to this there are two sides ac is also there and bc is also there now you have to look at the triangle which triangle I am using I am not using triangle adc that you will use ac I am using the complete triangle bdc so I will use bc so I will write bd is greater than bc now I am writing here what is bd bd is ab plus ad and that is greater than bc I know ad is equal to ac so I can write that ab plus ac is greater than bc so this is like this how many of you understood it please let me know how many of you understood it how many of you are there few of few of you have already left it those who are there how many of you understood it should I need to explain once more okay shraddha understood what about you sanjana did you understand it okay rithu is saying understood oh anshita is saying these guys didn't understand so I'll explain once more anshita you try to understand I am constructing ad which is equal to ac so I am going to first this triangle adc and I am saying that angle 5 is equal to angle 6 now what I why I am saying angle 5 is equal to angle 6 because I have constructed ad equal to ac so in the isosceles triangle what happens angles opposite to equal sides would be equal that is why I am saying that that is why I am saying that angle 5 is equal to angle 6 now try to understand this particular angle 3 plus 6 would be greater than angle 6 now angle 6 is equal to angle 5 so I am taking angle 5 what is 3 plus 6 so angle bcd is equal to angle bdc so what is 3 plus 6 3 plus 6 is this complete angle bcd so I have written bcd and angle 5 is angle bdc so I have taken angle bdc so opposite to bdcd this complete side bd is coming so bd would be greater than by theorem 1 greater angles have larger side opposite to it so greater angle is bcd opposite to it larger side would be there so I am writing bd greater than what is opposite to bdc opposite to bdc bc would be there so this is bc so bd is how much bd is ab plus ad so I am writing bd as ab plus ad and that is greater than bc now I know from here that ac is equal to ad so ab plus ac is greater than bc so this is how you have to do it now let me move to a few questions so I take difficult questions from here so that you can solve those questions in your exam I am I am solving this question if if if you are with your rd sharma book I am solving page 12.76 question 18 on that page so it is saying that pqr is a triangle I will solve some four five questions from here pqr is a triangle and s is any point in its interior so s is any point in its interior so I am assuming s to be here now it has been telling that prove that sq plus sr is greater than pq sorry one second is lesser than that is why I got confused in lesser than pq plus pr now how do we do it so in this particular question a construction is required so let me extend this sq you can do it on this side also so extend it to this side and suppose this is t so what happens I take triangle pqt and in triangle pqt pq plus pt would be greater than qt and what is this qt why I am keeping qt on this side why I am not keep keeping qt on other side I am not putting qt on other side because for sq I want it I want to show sq lesser than something so that is why sq is part of qt so I am keeping it on the lesser side so pq one second guys I did something I just closed it it got closed one minute okay so look at here I will write pq plus pt is greater than sq plus st because qt is equal to I am writing here sq plus st what else I write now I will go to this triangle because I want sr I go to this triangle so what I write in triangle in triangle srt st I want to cancel st so here st is on the lower side here I will put st on the other side so st plus tr and see pt is there here and tr I am keeping here so if I add pt plus tr I will get pr so st plus tr is greater than sr now this is my equation one this is my equation two I will add both the equations so add equation one and two so I write pq plus pt plus st plus tr is greater than sq plus st plus sr now st st is gone pt plus tr is pr so pq pt plus tr is pr so pr is greater than sq plus sr this is how you prove it did you all understand it now let me move to another question so I am taking difficult questions only I am not taking easier questions you can go through it questions are easy before so fast six seven questions and you will get this kind of questions only so nothing more than that let me move to this question on piece 12.75 I will solve two three more questions so if it is going beyond six o'clock please be signed into the session question is show that sum of three altitudes and remember this sometimes this is used directly like if you go to sum of three altitudes of triangle is less than sum of its three sides so I am making a triangle here and this is my triangle abc now this is a this is ad this is one altitude this is be this is second altitude and this is cf which is third so I have to show that ab plus bc plus ca is sorry ca is greater than ad plus be plus cf now how do you do it I to understand so there was one theorem which I didn't prove and that theorem which would be utilized here and what was that theorem so I am discussing that theorem and then I will remove it so suppose there is any point and this is line and this line is suppose x y so and this is point o so what is the minimum distance of point o from x y so minimum distance of any point from any particular line would be a perpendicular distance drawn from here so try to understand if this is om and this is o x and o y so minimum distance would be om not o x and o y y how to prove it so go to any triangle om y now this is 90 degree so in this triangle all these angles would be lesser than 90 degree hence greater angle will have larger side opposite to it so this is the greater angle 90 degree or both the angles are lesser than 90 degree so o y would be greater than om similarly you can prove that this o x is also greater than om or even if you don't know it by pythagoras theorem you can say that hypotenuse is greater than the base also and perpendicular also so look at here if this is 90 degree it can be easily said that ab is greater than ad and it can also be said that ac is greater than ad can i say that obviously i can say that because if you look at this if i go to in triangle adb ab is hypotenuse so hypotenuse would be greater than the perpendicular ab what is ab ab is hypotenuse and ad is what perpendicular so hypotenuse will always be greater than perpendicular because i am anyway not including base here ad is what perpendicular so what i write from here i include i add both of this so i am writing here ab plus ac just add it should be greater than equal to 2ab now let me just remove this similarly i can write that similarly i can write that in in in this bc would be greater than cf and ac would be greater than cf why because in triangle bcf and acf bc and ac are hypotenuse so let me write this again so ac plus bc is greater than 2cf similarly i can write in this triangle that bc is also greater than be and ab is also greater than be so i write that ab plus bc is greater than 2be now i have three equations this is my equation one this is my equation two and this is my equation three if i add all these three equations so ab is coming how many times two times here ac is coming how many times two times here bc is coming how many times two times here so it becomes 2ab plus bc plus ca is greater than two times ad plus be plus cf and you have already proved the inequality so to cancel your inequality is proved so this is how simple it is now let me move to another question which is on page 2.76 only so i'll solve two three more questions and then i'll wrap up the session let's solve another question so it has been given to we prove that first one sorry page 12.76 again question 17 so it has been given to the show that the difference of any two sides of triangle is less than the third side so suppose this is triangle abc and i have to show that ac minus ab is lesser than bc sorry this is ab is lesser than bc so what i do do so try to understand from here extend a line d here and what do you do with this this line so there are four angles getting formed suppose this is angle four on this side this is angle one this is suppose angle two and this is suppose angle three so how many of you think that and obviously here i forgot to write ab is equal to ad because i want to get this kind of question we have already done that when we did this kind of question to prove theorem which theorem the first theorem that greater angle has larger side has greater angle opposite to it now guys try to understand this angle three would be greater than angle one trying to understand let me make a bigger triangle for you from here i am drawing a line ad and this is my angle one and this is my angle three i am saying this angle three is greater than one why because exterior angle three would be greater than one of the interior angle and i am also saying that this angle two sorry this angle two and this is angle four this angle two would be greater than angle four why because exterior angle two would be greater than any other exterior angle four now try to understand i have constructed ab is equal to ad ab is equal to ad so if ab is equal to ad angle one would be equal to angle two so try to understand i can write from there that if angle three is greater than angle one if angle three is greater than angle one angle three would also be greater than angle two because one and two are now if angle if angle three is greater than angle two angle three will automatically become greater than angle four so if angle three is greater than angle four in that scenario what happens what is the side opposite to angle three side opposite to angle three is bc and what is the side opposite to angle four that is cd now try to understand bc and what is cd cd can be written as ac minus ad cd is equal to ac minus ad now this ad is equal to ab so i can write ac minus ab so ac minus ab if somebody is asking you that you don't have to prove ac minus ab you have to prove ab minus ac so where you will do the construction try to understand whichever side is coming here the tip of the construction should be here and whichever side is getting whichever side is negative whichever side is getting getting subtracted the initial point of the construction line should be there so see ab was getting subtracted so initial point is starting from b and it is going to ac so this is how we have to do this so now let me move to a question on page 12.77 and then one more question and then i wrap up the class so this is second last question of the class after that if you will have any doubt in one particular question i can solve otherwise now sraddha if you are there most of the questions of that sheet would be solved you go there only this kind of questions are there so most of the questions of the sheet i have given should be solved with the help of the concepts that we have discussed now now look at here we have done this kind of questions this kind of this this type of question too many in the class also this is any particular line segment l r on which i have three points so this is point q this is point p and this is point s now here is a point a so i am drawing a triangle a q and this is going to r so this is triangle a q r in triangle a q r a p is perpendicular so a p is perpendicular to l r and it has been given to me that this is given what has what what more has been given to me that p r is greater than p q this has been given to me it means that this is greater than this what i have to prove is show a r is greater than a q now how do you prove it so look at here i will draw any particular line s now why am i drawing any particular line s because it has been given to me that p r is greater than p q i am drawing any line s try to understand in this kind of question time and again what i am trying to do is wherever any particular side is greater than other side on the greater side i am taking a point and making portion of that greater side equal to the smaller side so here i have this i am drawing this point p s such that p s is equal to p q now i can show that look look at here this is 90 degree so i am not going into details of this you can show that this is also 90 degree so this angles are now this is equal to this and ap is common so you can show that this is s a s congruency in triangle ap q and triangle ap s again i will i will i will do this p q is equal to ps ap is equal to ap angles are 90 degree both angle p both side is equal so s s congruency if s s congruency is there what do i prove i prove that a q is equal to a s that's what i do and if a q is equal to a s then this angle suppose i am assuming this angle to be one and this angle to be two so i can prove that angle one is equal to angle two and if i prove that angle one is equal to two suppose this is angle three so angle two would be greater than angle three why because exterior angle would be greater than one of the interior angles as simple as that so if angle two is greater than angle three then angle one is also greater than angle three now if angle one is greater than angle three so larger side is always opposite to greater angle so opposite to angle a is what one is what a r opposite to angle three is what a q i can write a r is greater than thank you so this is the case now let me go to the last question and last question i am taking from ps 12.78 only question 21 which tells that in quadrilateral pq rs q rs and suppose these are the diagonals and this is point o i have to prove that pq plus qr plus sr plus sp is greater than pr plus qs and how do we do it so guys try to understand i take this triangle i take triangle pq r so pq plus qr will be greater than pr then i take this triangle in triangle sqr qr plus rs is greater than sq then i take i have taken this i have taken this then i take other two triangles i am not not moving this there are four triangles one triangle would be this one triangle would be this on this side and then on this side so four triangles would be created due to two you just keep on doing this add all four of them this would be two times this this would be two times this two will be get cancelled from here the next thing that i have to prove here is so i have taken this triangle now i can take this triangle which triangle i can take these smaller triangle triangles one two three four so if you take smaller triangles one one two three four pq would be lesser than op plus oq similarly qr would be lesser than or plus oq similarly sr would be lesser than os plus or and similarly ps would be lesser than os plus op so if you add four of them this would be psi summation of side of quadrilateral this would be two op plus two or plus two oq plus two os and and what is this try to understand op plus or is equal to pr so i get how much i get two pr here and from here i get two qr qs so this would be two times pr plus qs and this would be greater than this so this is what we needed to prove so guys i am wrapping wrapping up the session now we have done enough practice from quadrilaterals and from triangular inequalities these are the two important topics which you guys told me to revise last class you guys told me to revise quadrilateral so that's why it was there in my mind and i revised quadrilaterals triangular inequality i did it because though we have already done these things in the class but this is an important topic and a lot of people get confused so it's good that we have revised these things wish you all the best for your coming examination i shall be available on tomorrow uh after uh four o'clock i i shall be available for you guys so not not four o'clock let's make it five o'clock after five o'clock i don't have anything at such so whatever your doubt is there you can post it you can get it in touch with me because i know 29th is your examination and on saturdays and sunday's also i shall be available for you guys i'm always available but these are the days when i have not much work to be done and i shall be available to clarify your doubts so please feel free to get in touch with me and if you have any doubt get it clarified as quickly as possible what i want from you guys is that no doubt should be remaining once i mean before after i mean before you go for your examination so for that one thing is important that you get in touch wherever you wherever you are getting the doubt if you have any doubt now you can ask me otherwise we'll wrap up the question thank you so much for attending the session and wish you all the very best thank you