 how are you and I welcome all of you to this new session where we are going to discuss another congruence criteria so let's carry on with our work and in the previous few sessions we studied about congruence isn't it and there were two criteria and we studied that if those two criteria are met then two planar polygons are congruent what are those criteria first one was that each of the corresponding angles have to be equal right so there is let's say two triangles are there so as I keep on saying you have to imagine that in your minds that two triangles are there so the base angle left hand side base angle is equal to left hand side of the base angle of the second triangle right hand side of the base angle of the first triangle is equal to right hand side base angle of the second angle and two vertices here top right they are equal so each of the three angles are equal point number one and the ratio of course not ratio sorry the sides AB let's say this one side here and this one here they are equal this one here this one here is equal this one here and this one here is equal right so without pen and paper also you can conceive this idea in your mind then understand what is meant by congruent triangles or polygons if there is there are two polygons then again each angle has to be equal corresponding angle has to be equal and what each corresponding side also has to be equal okay in the previous sessions we discussed about one criteria which was s as criteria and we also learned that in case of triangles you don't need to prove all the six elements to be equal when I say six elements meaning what there are three angles you have to equate is it A to B and B to Q and C to R right you remember and the AB is equal to PQ BC is equal to QR and CA is equal to RP right so these are three of there are six equalities equations if you notice for any polygon but triangle I'm re-emphasizing or any polygons quadrilateral pentagon hexagons for any such polygon you have to establish both all six of them or in fact the angles in equality and the sides equality have to be separately established but in case of triangles we learned it's not necessary and out of six if we prove three to be equal out of six elements three angles and three sides and if we prove that three elements out of these six are equal into triangles then the triangles are congruent but not any three of these six there are a particular combination of these three right out of those six if a particular combination we have studied those in previous grades in these sequence of videos we are establishing their proves and convincing ourselves that yes it is true right so what I'm trying to say is that there are few rules for triangles and triangles being a lovely geometrical entities a beautiful creature I call them geometric creatures very very interesting lots of good properties so for them only few conditions are there one condition sas we studied last time which was side angle and this side so side included angle and this side are equal correspondingly for one triangle and the second triangle then you know the two triangles we are now going to study another one and that's mentioned on your screen just have a look it's called angle side angle congruence criterion what is that so imagine two triangles you can now you are good at imagining two triangles two congruent triangles you have to imagine and let's say you have the base angle here left hand side base angle is equal to the other base angle here of two triangles the second base angle is also equal to second base angle here and the included sides are equal then they are congruent but don't mistake me when I say base angles you know you can have any two angles out of these two triangles equal and the included side that is the side which is common to both the angles are equal correspondingly of one the other then the two triangles are congruent to my friends and that's what we are going to establish today we're going to prove it and you know convince ourselves that yes if these two conditions are these conditions are met then the two triangles are congruent but what is the end result of congruence wow how and when do we know the triangles are congruent and I'm saying we know that when again all those six inequality equalities are established three anyways we are providing we have to just prove the other three are equal and hence the congruence will establish so let's begin what are we waiting for okay so just give me a moment so that I can I can just set up my this every time disturbs at times when you know when when you close the window it just gets changed anyways so let me yeah perfect okay so just give me one more okay so let me start with drawing the triangle so here is the tool and this will give me a triangle so let me draw a triangle here is one triangle okay let me confirm this and let me copy this one so I have to copy this triangle which is here so I'm copying it okay so so that I get exact copy of it so duplicate and let me take it here duplicate confirm confirm oh I didn't get a triangle basically okay let me try once again at times it does give some trouble but don't worry okay wait a minute here is the duplicate yeah so I'm selecting once again so duplicate yeah now this works so let me duplicate it confirm yes now let me name the triangles what are the name guys so this happens to be here okay this is b and c p q r two triangles right what's given so let us start with mentioning what's given so given is this what angle b is equal to angle q so angle b is equal to angle q and angle c is equal to angle r okay so second given condition is angle c is equal to angle r and third given condition is b c is equal to q r these are given and what do we need to prove we need to prove that triangle a b c a b c is congruent to triangle b q r okay just be careful with the order of the points guys that's very important fair enough now that we have a job at hand how do we approach and what do we do so what do we do here we already know that we have one just a minute yeah one criteria we already know and what is that criteria let me write it over here that criteria is s a s s a s right so if two corresponding sides and then included angles of two triangles are equal they are congruent you can learn that so can we use that here but then in this case we have been given two angles are equal and the included side only is equal the other side there is no mention about it so okay let us see what happens if we try to you know get one more side equal that is already b c is given to be equal to q r here if we establish think about it if we establish what if we establish a b is equal to p q somehow if we establish this then what will happen then we'll get in these two triangles a b is equal to p q because we established that angle b is already given to be equal to angle q and b c is given to be equal to q r then the job is done the two triangles are congruent by what criteria by s s criteria because we have learned that before but is a b equal to p q i don't know then what are the possibilities let's talk about it so one condition is that yes indeed they are equal if they are equal a b is equal to p q is there let's me take it as case one then what will happen in these two triangles there is nothing much to prove now so a b is equal to p q and angle b is equal to angle q and b c is equal to what q r so by what by s s by s s we can say triangle a b c is congruent to triangle p q r isn't it no big deal but then you can ask a question sir how do you know that a b is equal to p q i would say yes i don't know so let's see those criteria as well or those condition so what if a b is not equal to p q then euclid uncle said that if something is not equal in geometry if two line segments are not equal then one of them will be smaller than the other it's common sense also mathematical common sense says that so let us say what if a b is less than or let us first take greater than okay so a b is greater than a b is greater than p q guys okay so if a b is greater than p q don't you think we can find out a point on on this point let's say here here i can find a point here okay what kind of point it is let's let me call this as a dash now a b was greater than p q remember a b is greater than p q i'm assuming it's not true but i'm assuming because for the sake of argument i have to prove that you know a b is indeed equal to p q and i'm saying okay you are not convinced let me say for the time being that yes a b is greater than p q okay no problem so if a b is greater than p q friends then then what can i do i can definitely point find out a point on a b here i have mentioned as a dash such that what a dash b is equal to p q that's it i can do that if two lines you're saying are not equal then you can always find on the longer line a point such that the segment the other segment the smaller segment completely fits into the longer one isn't it so that point is a dash such that a dash b is equal to p q okay now the moment we start with this assumption where does the problem arise so let's start taking these two triangles into consideration triangle a dash b c and triangle q r let's take these two now now by assumption we are saying a dash b is equal to p q let's say by construction we are assuming a b is not equal to p q so a dash b is equal to p q okay let it be and my dear friends it's already given angle b is equal to angle q there should not be any doubt here and my friend it's also given that b c is equal to q r now if these three conditions are met then can't we say by s as what can we say by s as triangle a dash b c okay let me write triangle a dash b c is congruent to triangle p q r that means by c p c the corresponding parts of congruent triangles i can say but oh i have not drawn this so let me just yeah okay so don't you think now since a dash b c a dash b c it's congruent to p q r then this angle a dash c b angle a dash c b will be equal to angle r simple angle a dash c b is equal to angle r c p c t but angle r was equal to angle b guys a dash c b is equal to angle b oh sorry angle c not b my bad my bad so let me just do a correction so i'm saying angle c okay a dash c b is equal to angle c now this is not possible why can you say angle a dash c b that means this angle how can the part be equal to whole so this is x and this whole is y okay so do you think x is equal to y can't can't be possible it happens it is possible it's possible it's possible when when see we have a common side b c and both the other uncommon side is on the same side of b c even if you had one common side like that one uncommon side here and one common side here then we can we could have still said that x is equal to y these two angle could be same but the uncommon sides are on the same side of the common side so hence we can't say only possibility is what that it's possible only when then only when a dash c coincides coincides ac and the moment this is there then automatically we say then a dash coincides a hence hence what will happen then ba dash which was equal to pq will be now equal to ba right only only when ba dash and ba are same and you had started with the assumption that ba dash is equal to pq and only then this is possible yep so only criteria only condition left is that a dash and a are not two but only one point right so hence we can conclude that in that case ba will be equal to pq hence now now y is now y ss also again what will happen triangle abc now a and a dash are coincident points so abc is congruent to pq and we could establish for this condition is the case number two case number two now after proving this there is one more case left over and which is third case and that's that's what is that that is when the the other side or let's say here it was ab is greater than pq now ab is less than pq if ab is less than pq can't we repeat the same process so instead of taking a point on a dash you will take a point on pq the only difference will be what i'm saying is let me write over here let us say the third case case number three okay when let me draw these lines to differentiate now you're saying let's say ab is now less than pq guys this is the third case only case left okay this is not a greater than saying ab is less than pq so if ab is less than pq then what will happen if you see pq so you can find here a point p dash which will be equal to ab and don't you think the process will just repeat so instead of calling abc as abc and pq are as pq are you can swap the names the concept still remains the same it is not going to change right so you can repeat the process or you can say similarly similarly right it's not going to change much because only names have changed so it will not you know change the properties of the triangle similarly what we can say is here also in this case also triangle abc will be congruent to triangle pq so in all the three cases whether ab is equal to pq or ab is less than pq or ab is greater than pq in all the cases if the given conditions have to be have to be ensured that b is equal to qc is equal to r and bc is equal to qr then the only possibility left is ab will have to be pq and the moment ab is equal to pq with the given condition that angle b is equal to angle q and bc is equal to qr by sas congruence criteria which we proved in the previous sessions we can say that yes the two triangles are congruent and hence the criteria asa is established understood so if there are two triangles two angles are equal corresponding two angles are equal and the incubate sides are also equal guys then the two triangles are congruent congruent means you can just propose one on the other and it will exactly match so this is the second criteria which we learned now we will be solving some problems on these and then we'll take up some other criteria the two criteria are still left still left okay and just to mention here because it is aas a corollary direct corollary would be aas what i mean is this let me show you how so let me show you this what i'm saying is aas a criteria also leads to what aas criteria right any two angle and any side aas criteria okay what is this why is this also what i'm saying is if there are two triangles a bc a b c and another one let's say pqr pqr and any two angles are given x is equal to x let's say and here is y equals to y and any side let's say this this is not included side okay bc is equal to qr if these three conditions are met then also the triangles are congruent then again we don't need to check anything else why by angle some property you know by angle some property of a triangle if two angles are equal the third angle has to be equal because they sum up to 180 degrees isn't it so that means this angle z will automatically be equal to this angle z and hence again you see as a criteria is fulfilled so this is a this is side s and z is another a so as a gets fulfilled once again if two any two angles of two triangles are equal and any side is equal to the corresponding side of the second triangle then also the congruence criteria is achieved why plainly simply because of asp angle some property of a triangle since two angles are correspondingly equal to two angles of the second triangle third angle has to be equal and why is that angle some property because all three of them have to sum up to 180 degrees two are equal then 180 minus these two and 180 minus these two is the third angle so third angles have to be equal so hence any two angle and a side any side may not be included sides but in s as if you remember that s a that a was included angle not just any angle there okay but here as a or a as are just like you know two sides of the same coin so both are equally valid for congruence I hope you understood it but yes and unless we solve some problems it doesn't get ingrained in our you know blood stream so what I'm trying trying to tell you here is just after this session we'll be having some problem solving sessions where we'll be taking up some problems and solving them one by one what you need to do is understand the approach and understand how this particular theorem is being deployed over here over there to solve problems and then after let's say 50 not sums we'll be doing you will be good to go with this theorem and can apply wherever required I hope you like this session so let's meet again in the next session thank you and have a nice day bye bye