 Okay, so let's begin. So I hope Okay, so most of you in in my my classes. I was doing Yeah, okay fair enough. So most of you are kind of acquainted with Concepts of quadrilateral the objective today is we are going to solve exemplar problems. Okay, and This is how it will be so first we'll just run through the main concepts and results which is in very Summarized form and then we will take up the questions one by one. Okay, so Let's begin. So the first concept in quadrilateral is size angles and diagonals of a quadrilateral different types of quadrilateral So we have studied trapezium parallelogram rectangle rhombus and square and on top of it. We have also studied kites Okay, so Just Yeah, so some of the angles of a quadrilateral is 360 degrees we discussed this last time angle some property of a quadrilateral Then a diagonal of a parallelogram divides it into this is where it all starts, right? So If you see we need to know the proofs of these terms as well. These are all theorem So a diagonal of a parallelogram divides it into two congruent triangles Please remember these properties in a parallelogram Opposite angles are equal opposite sides are equal and diagonals by set each other. So if you remember I had This thing in the group, I'll show you once again So this is what I shared. Oh, okay. You guys are not able to see my screen So this is what I shared in the group. I believe all of you have this card with you Um, yep. So basically the entire gist of quadrilateral is in this slide, okay, so if you see Trapezium only one pair of opposite sides are parallel two pairs of adjacent angles are supplementary. So what I have done is There are all different types of Quadrilaterals and their properties around their sides angles diagonals area and I've given you an example diagram as well So you can keep this slide in printed form in front of you whenever you are trying to solve problems Don't try to remember or memorize stuff. I Believe if you do enough practice on these questions, you'll be able to Buy hearted. So hence keep this in front of you all the time. So now Coming back to this a quadrilateral is a parallel gram if it's opposite angles are equal It's opposite sides are equal. It's diagonal bisect each other a pair of opposite side is equal and parallel So I believe you guys have already Gone through this Diagonals of a rectangle bisect each other and are equal and vice versa. They do not bisect at right angles So please remember in in a rectangle. They do not bisect at right angles But then definitely bisect each other On the contrary diagonals of a rhombus bisect each other at right angles, right? So, you know, you must remember so rhombus is a special type of parallel gram where the diagonals are not equal But they bisect at right angles. What is the difference between the tango and a rhombus? So rectangle diagonals bisect each other and are equal Rhombus Diagonals also bisect each other, but they're not equal, but they bisect at right angles. These are critical points here Now diagonals of a square bisect each other at right angles and are equal and vice versa So square is the most idealized form of a quadrilateral. You can think of Everything is good about square. All the sides are equal or diagonals are equal. That's diagonal bisect each other diagonal bisect each other at right angles So it's a good boy square. Now the line segment are well as well So the line segment joining the midpoints of any two sides of triangle is parallel to the third side and it's half of it This is called the midpoint theorem very very important theorem for all problem solving later on in geometry. Please Remember this. What is this the line segment joining the midpoints of any two sides of a triangle? parallel to the third side and Is half of it Okay, fair enough. Other Thing is other point is a line drawn through the midpoint of a side of a triangle parallel to another side bisects the third side This is a converse of the previous statement if you see The line joining the midpoints. So let me just go to my notebook so that I can explain this Okay, so here you go. So basically Yeah, so Now what midpoint theorem states is this So if there is a triangle, you know, this I'm just trying to revise it for you so that you Do not face. Let's say you do not have any shoes in this. So let me just redraw. Yeah, so let's say this is a triangle Okay, and let's say this is a BC and They're saying let's say point D is the midpoint of B is the midpoint of AB and Let's say E is the midpoint of AC. Then if you join D e So D e will be parallel to BC the third side as well as D e D e will be equal to half of BC so this is a Important theorem converse is also true. What is converse converse says that if you have a triangle again And you have one midpoint here and you draw a line parallel to the third side That's it. You do this now. You're deliberately drawing a parallel line from one of the midpoints Then the other midpoint e sorry the other point of intersection of this line will be the midpoint of AC right so this is midpoint So in the first case the Direct theorem you say, okay midpoints are already there. I join I get parallel line the converse is One I draw parallel line from one of the midpoint it intersects the other side on its midpoint So this is what is midpoint theorem now that you have any other doubt in this so before we start solving problems So we will go rapid fire. I believe you should be able to if in case you are not able to then I will take up the problem. You can message me privately as well. Okay, so let me begin Yeah, so this is the so start with MCQs guys Okay, so here's the question you can type in your answers and In the chat box so that We can I will keep a track of who's doing what and maybe I Would like to reward you guys. Yeah, so those who are performing So before you Say your answer, please check Okay, before you jump on Your answers you have to check. Yeah, go ahead guys. So first answer first answers are coming right instead of I would suggest instead of Doing it on the The thing, you know the common chat box you you send me send your answers privately to me. There's an option Where you can send your this thing. Can you see? I don't know if you're able to see this all of you must send your Answers privately to me so that you know, there's no common chat Okay, so most of you are saying 120 degrees. There are a few answers which are saying 90 as well So, okay, so there's no brainer in it. So 75 plus 75. So basically the first thing is Yep, first first thing is okay, let me just go back to yeah, so hence here. So Yes, I hope you are able to see the screen now. So hence I will be doing this Half here of there. Okay. Now. So question one is Tillice No, I till I say don't go to the next question because I will be keeping track of Your responses anyways, so we're going to count how many of you have Done it. Okay. So first question angle some property. There's no brainer in it angle some property angle some property and Yes, so hence most of you have done it correctly 75 plus 90 plus 75 and let this be X So 360 so X is nothing but 360 degrees minus 240 which is 120 degrees So first is correct 120. Go to the question number two A diagonal of rectangle is inclined to one side of the rectangle at 25 degrees. See the acute angle between the diagonal is Yeah, go ahead guys. So a diagonal of rectangle. So there has to be a rectangle rectangle and It says the diagonal is this the diagonal The diagonal of a rectangle is inclined to one side of the rectangle at 25 degrees Okay, the acute angle between the diagonals is Hmm. Oh, you at least let me know who's who otherwise you will miss. Yep. Go ahead guys. So I need Kate. Okay. I got the answer I need Kate the question read the question carefully. What does it say? It says a cute angle between the dial diagonal a cute angle Yeah, so please be very very careful. So I know you guys are tempted to write the answer and here you know here is the thing where you Make a lot of mistake. Yeah, so how to solve this. So it says 25 degrees. So how do I know? Which one is so let's say This angle is X and this angle is why I don't know which one is a cute So let it let me let me say this is X and this is why okay, so if it is 25, you know that You know, this angle will also be equal to 25 degrees Why is that? It's because ABCD if you see triangle ABC is congruent to triangle BCD You can see that. Yep. So when that is the case then this angle will also be 25 So hence, I know why will be equal to 180 degrees minus 25 degrees minus 25 degrees is equal to 130 degrees, but this is a Optus angle. So hence if this Y is 130 X must be 50 degrees. So answer is 50 degree Okay, I hope there is no problem in understanding why ABC is congruent to BCD Because sides are equal. There's a common side and there's a 90 degree So hence, so I will be doing it fast very good guys So I can see a lot of answers, but not are there are how many participants are there? There are more than 40 45 participants are there, but I'm not seeing all the responses guys Your responses are being tracked. So hence, you know, so those who are not participating if you do not share your problem Then we will think that it is not working for you. Okay. Next question number three. Go ahead ABC is a rhombus such that ACB is 40 degrees find ADB, okay, so ABCD is a rhombus ABCD is a rhombus ABCD is a rhombus Okay, so that ACB. Okay, ACB. So There's a diagonal which is joined to this thing. Okay, and they're saying ACB is What? ACB is 40 degrees. Yes You have to find out ADB ADB Answer is ADB to find out ADB how to find ADB So, you know, what is our property we're going to use in a rhombus diagonals bisect each other at 90 degrees So this angle is 90 so a we have to find out this X Okay, clearly if this is 90, this is 40. So this angle would be how much 180 minus 90 minus 40 so 50 Okay, so X is equal to 50. Why because these two lines are parallel So hence this and this angle are in alternate interior angle. So hence answer is 50 degrees Great. So all of you must participate in problem solving. Otherwise, you will think that the session is not good for you. Next and question number four The quadrilateral formed by joining the midpoints of the sides of the quadrilateral PQRS technique order is a rectangle if Don't send me all the answers. I don't require it right now. You just send me one by one Question two again. Question two again a diagonal of a rectangle. So there's a demand for question two So I'm doing question two once again. So it was given that there is a rectangle There is a rectangle and one of the diagonals. So this angle is given as 25 degrees. You have to find out the The acute angle between them Okay, so hence I'm saying I don't know which one is acute angle. There are two possibilities either X or Y. Oh Here you go. So, you know, so I don't know which one is acute. So X and Y. There are two possibilities I have to find out X or Y both So I'm saying A, B, C, D and I said triangle A, D, C is congruent to triangle B, C, D and why is that? Because A, D is equal to V, C rectangle opposite sides are equal DC is equal to DC common side and this angle is 90 degrees. So angle A, D, C is equal to angle A, C, D Both are 90. So hence why SAS? They are congruent. They're congruent. Then if this angle is X, sorry, not X if this is 25 So this also has to be 25 Now what is X? You can use external angle theorem as well. So X is nothing but if X is external angle It will be some of interior opposites 25 plus 25. Either you can use this That means 50 degrees or you can find this angle. Let's say this is Y So Y is 130. So X has to be 30 or 50 by linear pair. Okay, so I hope You understood how to get 50 degrees here. If not, then you can again reach out to me after the class No problem. So 3 is done. What about 4? The quadrilateral formed by joining the midpoints of the sides of a quadrilateral PQRS Taken in order is a rectangle if Okay, so let's draw a quadrilateral So this is a quadrilateral. Looks like a rectangle. Unfortunately, let me draw a little Different. Yeah, so this also is proper. Wait a minute guys Hmm, I'll draw a random so that there's no confusion. Okay, so one two three and four Yeah, this looks a little better. So let's say this is ABCD and they're saying That quadrilateral formed by joining the midpoints of the sides of quadrilateral PQRS. So Joining the midpoint. Okay, so this is these other midpoints. Okay, so midpoint midpoint Midpoint, okay, so don't go by the diagram assume that it is a midpoint or let it be hard. So let's say this is P Q or Yes, okay now See it will always be a parallelogram for sure. Why because if you apply midpoint theorem midpoint theorem which says that if you join two midpoints of a triangle, so let's consider this triangle ABD in this triangle SP is SMP are the midpoints of the sides where SP will be parallel to DV and In the second this triangle which one BCB in this triangle again qr are midpoints. So hence These two sides are again parallel so all the three like all the three sides are parallel So hence basically you will get from here PS is parallel to QR and similarly by the same logic SR will be parallel to PQ So that means without any doubt PQ are a Caesar Parallogram because opposite sides are parallel now. They're saying It's a rectangle that means this angle is 90 degree Where is that possible if this is like 90 degrees that that means it is a rectangle all are 90 degrees If that is 90 degree that means this is 90 degree So basically Yes, if all are Yeah, all is so it's a 90 degree. So basically you have to find out whether PQRS is a rectangle parallelogram Diagonals of PQRS are perpendicular diagonals of PQRS are equal. Okay Leveling is wrong. Where which question you are talking about Anithi? Oh Did I You guys taken in order, where is labeling wrong? Yeah, which one? Quadrilateral is PQRS and the midpoints aren't given a name according to the question The quadrilateral midpoint Oh, I'm sorry. Oh, okay. My bad. So basically there instead of PQRS you can take it as A, B, C, D Basically, they're talking about by trying with first one. So instead of okay fair enough So in this question replace PQRS by A, B, C, D. That's it. Okay. So that's the thing Okay, so I think most of you are confused. So let me rename it. So let me say this is not PQRS. Let this be PQRS is the original Quadrilateral. So that's what they're talking about. Okay, never mind. So let's say this is PQRS and R and S and this is not R So let me name this as A B, C, D D Okay, so midpoint theorem says that in a triangle if you have a triangle If you join the midpoint, what will happen? You will get a line parallel to the base Okay, so in this case consider P S Q or PQS triangle P Q S if you see QS is the base B and A are the midpoint. So if you join you'll get A, D parallel to QS or SQ Okay similarly similarly In the second case triangle R QS B, C is parallel to QS Right, that means combining these two you'll get A, D is parallel to B, C So A, D is parallel to B, C And similarly you will get C, D is parallel to A, B. What does this suggest? It suggests that One thing is very clear that A, B, C, D is a parallelogram A, B, C, D is a parallelogram. So A, B, C, D is a parallelogram, right? Now they're saying what are they saying? They're saying the quadrilateral is formed by joining the midpoints of the sides of a quadrilateral That is A, B, C, D in this case Is a rectangle. So when is this rectangle when one of the angle of A, B, C, D quadrilateral is 90 degrees? So let's say this is 90 degrees. A is 90 So A is 90 Right, if that is so then This is also 90 and all are 90 degrees So it's a rectangle, right? It's a rectangle. So when it is a rectangle, what can you say about PQ, RS is the question. Is PQ, RS is a rectangle? So is PQ, RS is a rectangle? So if you see the There are two methods of solving this. Either you eliminate your options Okay, most of you are struggling in this problem. It seems Again, I'm repeating PQ, RS is a quadrilateral given and A, B, C, D are respective points Okay, what we are going to what we are trying to do is we are trying to analyze what kind of a This thing is there. PQ, RS is what kind of a quadrilateral? Whether it's a rectangle, whether it is a parallelogram or whatever So, you know, how do we come to conclude that ABCD is a parallelogram? So ABCD, if you see A, B, C, D is a parallelogram. How? Because by adopting midpoint theorem, what does midpoint theorem say? It says that D and A are midpoints. If D and A are midpoints, then DA will be parallel to SQ in triangle PSQ. I hope this thing is clear to everyone Now similar logic, BC, this line BC will be parallel to SQ by midpoint theorem Now both BC is also parallel to SQ and DA is also parallel to SQ. So this is also parallel to SQ And CB or BC is also parallel to SQ. That means BC is parallel too. So these two lines are parallel So these two lines are parallel Similarly, these two lines are parallel I hope you got this. So when two opposite sides are parallel, then it is a parallelogram Correct. So that's what I am trying to explain. So hence we could prove that ABCD is a parallelogram If you didn't understand, no worries. After the class, you can ping me and then I'll explain again or I'll send you a solution So ABCD is a parallelogram. Now the question remains if it is a rectangle, they are saying ABCD is a What is that? Yeah. So PQRS, ABCD comes out to be a rectangle. The quadrilateral servo made is a rectangle You have to find out what properties PQRS Is having. So PQRS is a rectangle is a parallelogram or diagonals of PQRS are perpendicular diagonals of PQRS are equal. See if it is a if PQRS So I was talking about you can eliminate the options or you can do theoretically. Let's say no options are given. Then what can you say about this? Quadrilateral. So solve it like a theoretical problem. Let's say there is no option given. Then what will you do? How will you conclude that? How will you conclude about PQRS? now So if you see Yes So what will you conclude about PQRS guys? Okay, so this is 90 degrees given So if you see it is a rectangle guys Then uh What can you say about these two? Just a minute Yeah, I am just trying to say Uh, let's say you don't have these options. So forget omit these options. Now. Let us try to do it theoretically How do I let you know about the properties of PQRS? So never mind. So this is this is for sure. And if you see let me just name some more I'm you know, I will do the calculation I will just remove So let me just name them. Let me say this is uh PQRS And then let's say E and F So clearly E and EF So I'm writing here is the pay attention. So A, B, E and F is a rectangle again Okay, this is also rectangle Y A I'm highlighting it and then I'll remove the highlight. So this is a rectangle also Why because it's a parallelogram with one angle is 90 degrees. So this is a rectangle So they are This is 90 degrees. Okay. Now Now, so, uh, this is a parallelogram. So this also is 90 degrees So if you don't understand, I'll explain again. Don't worry. This is 90 This is also 90 Okay 90 degrees. Okay. Now what? So if these are, uh, 90 degrees, so this is perpendicular. Okay, and Yes So how do we do it without any? This thing. Okay, so this Okay, so fair enough This angle is equal to this angle. Generally, this angle is also equal to this angle. That's also there. Okay. Now what? Yes Force answer is C. How is C? Can you prove? The angles of PQRS are perpendicular. Can you prove that PQRS is the main the Yeah, from the uh, the outer quadrilateral is PQRS. I want to prove. I just don't want an option Anyone who has done this? Yeah, sir, if you draw the other diagonal also So what will happen in that case? Yeah, then one part of it if you prove that that'll be a parallelogram because the opposite sides are parallel So the opposite angles will be equal Who who's this? Shreya. Yeah, Shreya. You're saying join PQ Join PR. PR, PR. Okay, then what? Okay, now in Can you label the inside? Let's say this is point O Okay, so Yes, right direction. So hence what So hence if you see AB also is Parallel to CD and AB is half of PR Yeah, yeah, so hence this angle is also 90 degrees Yeah, so it's a it's a parallelogram. First you prove that it's a parallelogram The parallelogram already and it is intersecting at 90 degrees. That is also for sure. That's correct Yeah, so the opposite angles What are the options? So diagonals of PQRS are perpendicular. That's C Yeah, you can definitely say that. Can you also say anything about Why is PQRS is a rectangle not a correct option? Just try to deliver it more on this problem Can PQRS be a rectangle? Can that be an option? Think about it guys. Can PQRS What happens if PQRS is a rectangle? They're automatic. Will this AV CD will also be also a rectangle? No, sir. Why? It will form a diamond It will form a diamond. Okay, so let's see what I'm trying to say is this is a rectangle and you are joining the midpoints of These two it will it will actually lead to a another parallelogram, right? So it will lead to another parallelogram Can there be any possibility of PQRS being a rectangle and AV CD being a rectangle as well? Think about it. This is a very interesting problem Can both AV CD as well as PQRS be rectangle at the same time? Sir, if PQRS is a square then it will be possible. Yes, but PQRS I'm saying I'm not saying I'm not talking about PQRS being a square I'm talking about Think about it, okay Sir, but every square is a rectangle, right? I think not a proper square. I'm telling you not a square. Let's say PQRS is not a square then fair enough Let's not delegate more on that question. Go to question number five Quadrilateral formed by joining the midpoints of the sides of the quadrilateral PQRS taken in order is a rhombus if Yes, if so the same problem but now instead of Um What instead of rectangle is that there's a rhombus So ad is equal to ab It's a rhombus. So rhombus. What is what does rhombus say? So let me just create some space here So it's given that the same problem, but now The ab cd is a rhombus. That means ab ab is equal to cd is equal to d what ac Let me write in order a Ab is equal to bc is equal to cd is equal to da. This is given So what can you say about the quadrilateral option is d? Diagonals of PQRS are equal Hmm Diagonals of PQRS are equal. Yes. Why? because Ad is equal to half of qs by midpoint theorem. Similarly cd is equal to half of pr And ad is half of qs and ad is equal to cd. They are equal. That means Half qs is equal to half pr. That means qs is equal to Good, sir. We can't see. Oh, I'm sorry. Huh. This is what I was writing. Sorry. Yeah next I hope this is understood by all question number six to ten Fire Yeah question number six to sir. I have a doubt. Yes. Go ahead. Yeshika Question number four and five. Please go ahead and tell me unmute and tell me Yes, sir. I didn't understand the question number four and five properly Question number four and five meanwhile those who are those who want to solve more go ahead with six seven eight Nine and ten. Meanwhile, I'll explain What is question number four and five c? What they are talking about is simple what they're saying is There is a quadrilateral PQRS q r And s yeah, so now I hope this is visible Yeah, I hope I am I get So what they're saying is Find a midpoint. Let's say a Then midpoint. Let's say b then midpoint. Let's say let's say b at the midpoint and join a b Now they're saying a b cd is a rectangle After doing all this exercise a b cd is a rectangle So you have to basically Find the property of What can you say about pqrs? So they have given four options. It is a rectangle. It is a parallel whatever There are I can't see the options right now. So hence, how do I go about it? So I joined sq So if I join sq and pr So I joined sq and okay, so if I joined sq and pr, let's say this is o So it is a rectangle. So I know a b is parallel to sq by midpoint theorem Yes, sir. Okay bc is parallel to sq same logic So if two lines are parallel to the same line, the two lines are parallel to each other, isn't it? A b is parallel to cd I see in logic Correct. Yes, sir. That means a b cd is a parallelogram Now they're saying this parallelogram what you got is a rectangle. That means this angle is 90 Hmm if this is 90 this has all of them will be 90 Yes, sir. Right. Now if you see consider this This part this quadrilateral It means that parallelogram again It is a parallelogram. Why? Because these are parallel lines These are parallel lines. So this parallelogram Right. Yes, sir. This angle is 90 degree Hmm if this is 90 all of them are 90 90 90 so hence the diagonals are perpendicular to each other Okay Okay, okay. Yeah And similarly the sixth next question was if it is a rhombus If it is a rhombus Second question question number five was it's a rhombus a b is equal to Right Then a b was half of sq and bc was half of pr So sq is equal to pr Diagonals are equal Yes, sir. Okay. Yep question number six done guys Then a b cd is three is to seven is to six is to four So the question is You can mute now. Yep So question number six says a b cd is a quadrilateral a b a b cd. Yep. So a b c b is a quadrilateral such that the the ratio is Yeah, so all of you are able to do it So three is to seven is to six is to four. So three x then seven x then Six x and four x Okay, so a b cd is this isn't it so hence total angle some property. So three x plus seven x Plus six x plus four x is 360 degrees, right? So 10 20 x is 360 degrees so x is 18 degrees Right, so hence angles are 54 Then next angle is Yeah, so 126 degrees third angle is six x So it is one not eight and four x 72 18 seven 18 fours are 72 Yeah So you are saying They are Trapezium it's trapezium. Yes. Why because If you see These two angles a plus b Is equal to 180 degrees And c plus d also is 180 degrees. So hence they are supplementary adjacent angles being supplementary and only two pairs That means it is a Trapezium good guys question number seven seven seven done so If bisectors of angle a and b of quadrilateral abc intersect each other at p. So let me draw the diagram So there is a quadrilateral abcd Bisectors of A and b of a quadrilateral. So this is angle bisector So they meet at a point p. So this is x. This is also x. This is y This is also y At a b of b, uh each other at p of ba Of b and c at q and c and d at r and okay fair enough. So basically um You will get different different So this is another point Then This is another point. So like that. These are the final this thing. So ab cd and Let's say this is p Then b and c are q and r and s So it is p q r s. It's a rectangle How do I know? Because we know that What theorem will be applicable here? Theorem is the bisectors bisectors of two adjacent Adjacent sides Sorry not sides angles Off a quad relateral Meet at Right angles Okay, this is the theorem behind it. So if you see how how do I know that? So let's say consider aob triangle triangle aob. So this is not aob not apb apb So what is it? So if you see angle p plus x plus y is equal to 360 degrees Hey, hold on. Uh Yeah question question number seven Uh Okay quadrilateral angle Yeah, so no, we can't really see says it's a quadrilateral given guys. It's not a parallelogram. Just be What's your answer of say seven? Let me check your answers first seven. How many of you say a? Seven seven go go tell me the answer of seven. Let me see how many if you have said a Sir seventh one is a Sir seventh one is d Who's this Aditya Yeah, Aditya seventh one is d Yeah, then the janta is saying a how do you differ? And why I'm interested I'm interested in knowing this Good contest. Well, why it is or not a and why it is d or vice versa? Yes, sir So those who are saying a Uh, can anyone can come up with the proof please? homage you saying a why Can you come up with the proof, please? Hello guys Hello Yes, sir one minute. Yeah, tell me why it is a how did you come to come to conclusion that it is a Yes, go ahead. Hello. Hello It's not a Guys, it's not a parallelogram It's a parallelogram the bisectors of any two quantitative angles intersect at right angle that is for parallelogram, but it says bisectors of angle a and b of a quadrilateral abcd Yeah This is this is only true for parallelogram not for any Any just any odd Any odd quadrilateral it has to be a parallelogram only then right angles, but anyways, let's not get into this theorem Can we have some other methodology? So bisectors of a and b of a quadrilateral abc intersect each other at p of b and c at q And c and d at r and d and a s s e q r s is a rectangle. Whatever you have to find out How to do it So only only relation. I know is p plus x plus y is 360. So now let us Let us say this is z And this is you this I can definitely say Isn't it so hence I can say angle p Is equal to 360 degrees One one so it's 180 180, right? Okay. Thanks Is equal to 180 degrees minus x minus y that's what I can say about p nothing more Correct. Similarly. Can I say anything about r? So angle r is equal to 180 degrees minus u minus z Correct. So angle p plus Or angle p Yeah, so what can I say about this now if you see okay If you see x plus y plus u plus z Is nothing but 180 degrees itself isn't it Do you get this why it is x plus y plus u plus z 180 degrees because x is nothing but a by two y is nothing but b by two u is nothing but d by two and z is angle c by two right This is 180 degrees That's correct. Why because a plus v plus c plus d is 360. So this is true. So if you add Uh p plus r So from here if you add p plus angle r is 180 degree Plus 180 degrees minus x plus y plus z plus u which is again 180 degrees. So hence basically p and r angle p and angle r are supplementary So we can't see what you're writing Is that okay Similarly s and q can can be supplementary So two opposite Two opposite So opposite angles are supplementary. What is that case? Is there any case? so if you see It's very no so s plus q is 180 degrees And similarly s p plus r is 180. Sorry. I can't say anything about it. Yep So opposite angles are supplementary. What is that? Hmm opposite angles of a parallelogram are never supplementary Uh, they go yeah, you know option is there quadrilateral whose opposite angles are supplementary So d special type of So those who said a you know don't jump on to the conclusion So I know you were thinking that it's a parallelogram So if it were a parallelogram Then rectangle was the right answer But it was not So hence d is the right answer 7 d hence I stopped all of you Oh one second. They go Uh, yep, I hope this solution is clear to everyone why it is d So what did we do? Uh, I hope the diagram is clear to everyone So, you know the angles are equal and all that so angle bisectors are there So ap is the angle bisector of a so x equals to x So I hope this is clear to you y equals y z equals z and u equals u these are the four bisectors now what I did this p plus x plus y is 180 degrees by angle some property of a triangle Uh, and then similarly so p will be 180 minus x minus y Then similarly angle r plus u plus z is 180 degrees. So r is 180 minus u minus z I hope this is clear Then I simply added both of them before adding I just got this x plus y Plus u plus z is 180 y because a plus b plus c plus d is 180 Sorry 360 so a by 2 plus b by 2 plus c by 2 plus d by 2 will be 180 degrees, right Now what you do is you simply add them together So if I add them so p plus r in the left hand side will become 180 plus 180 minus x plus y plus u plus z all of these together Which is nothing but 180 degrees from here So hence I got 180 plus 180 minus 180 that means p plus r is all only 180 So p plus r is 180 means what right p plus r is 180 means p and r are supplementary. So hence option d I hope Now it is clear to everyone. Yep Is it clear to everyone good? So let's move on to question number eight. Let me see again. So you must Check your answers now. So you are making mistakes question number eight What is it a p b a p b And c q d are two parallel lines. Okay, so a p b are two parallel and c q d So then the bisector three participants raised and just a minute who's raising hands Yeah, Aditya Sree Vasishtha. Yeah, Aditya go ahead and say Tell me what do you want to say? Aditya, I can see your hand. We want to say anything. Anyways, let's continue. So This is the picture now. They're saying Now they're saying Then the bisector is off The angles at q Sanjana also is saying something. Okay, wait a minute Yeah, Sanjana go ahead and say what do you want to say? Say if you want to say just unmutancy. Oh, can I solve? Please go ahead. Please go ahead Okay, so um, if you draw a if you connect a p and q p and q let me connect it for you connected And so the bisector of ap q Is joined to the bisector of c q p C q p. So let me first join c q p So c q p angle c q p Oh angles. I'm sorry Yes, so uh come again. I I didn't read the question properly first So bisector of angle ap q Okay Is um, uh connected to the bisector of angle c q p. Okay Fair enough Yeah, and same way for the other side b p q and d q p They're joined. Okay fair enough like this So we know that ab is parallel to cd and um that ap q plus uh p q c is equal to 180 degrees. Okay. So this is 90 degree Yeah, very good. So this is That is also 90 degrees and then after that since um Uh, the two angles are Um either they're wait And tell me the names names names of the angles um So both the nine since both the 90 degrees are are equal. It makes um, uh angle p Wait, so can you mark the other two points? Let's say, uh, what point? So let's say this point. What point? What is your name? Let me say x and capital y So um angle x p y is also equal to 90 degrees. Why would that be? um, so so if you um add it, uh Uh, it'll be the same thing. No, since um, they're 90 degrees and also, um x p q is um also equal to um This one 90 degrees. Yes. Why? Because I'll tell you, uh, you have to articulate Small x this angle is also small x this angle is small y this angle is also small y Correct. So you know that 2x plus 2y is 180 degrees linear pair Yeah x plus y will be 90 and same way for the other side and since all angles are equal, it's a rectangle Very good. Awesome. Any problem in this understanding that Come again, who's this identify yourself? Oh, sir. Manir, could you please show the question? A solution or question solution you want? So the question sir, I want you want to see the question It's right here Not for the whole thing like it. I can only see half of it. How to do it. That's what you're saying solution Sir, I can see only half of the question. Could you please show the whole thing? Half of the question. Okay. I think it's uh, sir. Can we also say that y p q equals p q a due to alternate interior So uh, Angle slowly slowly y p q is equal to Angle a q p Why? Alternate interior. Sorry Why are they interior? It's not even that q b is parallel to p c No a p b is parallel to uh c q d, right? So p q is a transversal p q is a transversal perfect. So hence Yeah, go ahead. Go ahead. Then again next Sorry, sir. I got wrong. I'm sorry. Then then p b q would be equal to c q b a c q p No, no, he he got confused. Uh, say saying p y is parallel to q x. You already assume Sir, sir, sir, sir. Yeah um b p q is equal to c q p therefore half of c q p is equal to half of b p q Sanjana once again Yes, so then it will be parallel once again once again. I couldn't I couldn't pick it up Sir b p q is equal to c q p alternate interior. Oh, great. Awesome. Very good. Good. Good hold guys Therefore half of c q p is equal to half of b p q very very good. So this angle is also why that's what you're saying Yeah, yeah, perfect. So hence yes uh Is it be able to articulate mathematics make communication? Okay, good guys. So you have you know, it appears that you know, uh, you're having good So any other thoughts on this question number eight guys? Fairly dumb sir. Who's this? So it's me, Aditya Uh, sir, can we also show the congo congo? Can you also show congruency? Where so uh in this this question to prove that it is a rectangle Uh, congruency. See congruency is like going a still further step. You don't need to actually because Yeah, what I'm trying to say is before that stage So hence we don't need to you know, go for the kill overkill. You understand? Excuse me, sir. A result of this that you P x q y And all of them are nice. That's good enough Excuse me, sir. Yes. Who's this? Sir, it's me. Sir. I'm Sanjeev Yeah, Sanjeev go ahead So we can solve this even by co interior angles. No sir by the property Go ahead explain So since we know that uh, like we are told that angle p x or q is equals to 90 degree So like uh angle p x q plus angle x q t will be equal to 180 degree co interior angles q p x q p or x q x q y you were saying Hi, x q x q y Yes, sir. Yeah, so they are 90 90. So what I'm going to do is like that only we can prove all are 90 degree and like Yes, you can because once you know that lines are parallel and they are alternating angles are equal if they really do You know so many things but the thing is in the minimum possible steps is you just prove all the angles are 90 and you are done Right. Yes, sir. You can have multiple you know ways of uh, you know, that's the fun in geometry There's no one way there are multiple But all all roads lead to Rome. Can you just try ninth one? before we Ninth one so I want at least 10 questions to be you know Yes, it's good that we are analyzing on lots on these questions. Good. You should You know, you should spend time on the question. Just just don't be content with okay I got the solution and hence and we learned in problem number seven That you know, we need to brood over the problem. Yes. Go ahead and solve eight Oh eight is done. I believe this was the eight question. Oh question number nine Question number nine you have to do figure opt in by joining the midpoints of the science of rhombus taken in order is midpoints of a rhombus so This is our rhombus And you are taking joining the midpoints Very good. So if you join the rhombus Now midpoints of the rhombus so uh sites of the rhombus. What will you get squire? How and why So midpoints, let me name it a b c you must be ready with the proves as well. So let's say p q r s Oh Okay, so midpoints of the rhombus. Okay, so this is anyways 90 degrees And if it is 90 and uh, so this is parallel. So this is 90. So definitely it's a rectangle for sure Whether it is uh, it will not be a square by the way It will be a rectangle Yep 90 it will not be a square. Why because again if you see p q Is parallel to bd And hence p q is parallel to s r And similarly by the same logic. So p q r s is a p q r s is a Is a parallelogram for sure, but there is a 90 degree here Right if this is 90 because diagonals of uh rhombus bisect each other at 90 So this is 90 this is parallel. So this has to be 90. So this has to be 90 So all are 90 that means it's a rectangle One angle is 90. That means it's a rectangle and it's not a square for sure. Why because e q is equal to half bd And uh q r is equal to half Let's say ac and bd is not equal to ac guys because in rhombus diagonals are not equal Wrong option. What is the wrong option? 9 b is not correct rectangle Yeah, correct. Next last question and then good night to you one last question 10 d and e are the midpoints a b ac for triangle a b c and o is any point on side b c So don't get Righted by the language d and e are the midpoint a b and ac. So let me draw our triangle quick triangle So a b c d and e are the midpoints of the sides a b Okay Okay Uh, and o is any point on side bc. So let's say this is o So what now and o is joined to a okay. Let's join o Very good. If p and q are the midpoints of ob ob p and q so p p So this is p this is q p and q are the midpoints of ob and o c respectively then de q p. Okay interesting d e so so d e Yeah, what is it any thoughts any thoughts any thoughts? So It's a parallel gram. Yeah correct. Why? So since all sides are parallel to each other using a midpoint theorem Very good. So de is parallel to That's one Then since p is also midpoint and d is also midpoint. So pd is parallel to a o And similarly e q is parallel to a o. So hence my dear friends pd is parallel to e q So hence this these two sides are parallel And these two sides are equal equal. Yeah equal as well. Yep Ha so that you can use that also. So let's say who's this who said equal Sir Ashwin Ashwin said equal. So hence Ashwin basically what you can say pd is parallel to e q and pd is equal to e q either you use this So this is sufficient condition for The parallel again. Yes This is sufficient condition or You can also say one you say pd is parallel to e q and de is also parallel to e q. So these are two ways of sorry right So today we solved 10 questions in about 60 minutes. That means per question to around six minutes. Yes We did deliberate a little bit more on few questions. But yes, uh, the speed is not that great We should be able to solve minimum of 20 these kind of questions per Our so no problem. We'll slowly increase the speed So I believe you have the This thing with you next time. Well, we discussed will not discuss the mcqs will discuss these questions. Okay On the We sought answer. Okay, so I Please use this So it's me Aditya So in the eighth question The eighth question. Let me go back to the In the eighth question. Tell me what happened Question So, uh, if it is just given us a theoretical question without any options It is it important to do the congruency To show that it's a rectangle No, I I we didn't use congruence at all to show that it is a If you see what we do in this case, uh So we just said that he said So this is why why why So we said to it. Why is 180 degrees? Why? Because If x plus y Okay, yeah Similarly this angle becomes Yeah, okay, all the angles are 90 What is this? We didn't use I hope you guys also Sir Sir, sir, could you send this question this paper onto the group? Who's this? Yeshika? Yes, sir. Good guys. Thanks a lot and uh, sir So, where did you find this question from? These are exemplar questions your ncrt exemplar Oh, it's not in the textbook Okay Yeah, so thanks guys And the night to all of you So doubt Oh, who's this? So they're from nbsk and Yes, go ahead Oh, sir. I think I had a doubt in the last one Um, I think I think the ninth one. Yeah 9 So how is uh How is um Just one second. Just one second. Yeah, so how is bd equal to eq? Question number nine. Yeah Oh nine nine nine Yeah, this one can't see it Yeah, boy. I just Wait a minute. I'll show it again Can you see it? Yeah So, um No, so not not this question though Ninth one, right? Sorry the 10th one 10th one. I'm sorry. Oh a 10th one. Okay. So 10th one Yeah So in 10th one How is dp equal to eq? dp is equal to eq because They go by midpoint theorem consider this triangle this one first This one Oh This one. Yeah in this triangle. This is The line joining the midpoint. Yeah So can I say d What p is equal to half of a o a o Similarly in the other side Yeah Yeah, what can I say? Oh Half a o so both are So dp is equal to eq. Yeah, correct. Yes, anything else? No, that's it. Yeah, great Good night, dude. Thank you, sir. Bye. Bye