 Hey friends welcome again to another session on congruent triangles and in the previous session we had one criterion and we saw how we can prove that criteria right and that was a s a and also we saw a corollary to that criterion what was that a a congruence a s congruence criterion isn't it now in this session we are going to take up one more congruence criterion but this time we are going to use geogibra and try to simulate the process draw the diagrams here and at the same time we will try to prove the given criterion now this time we have taken what side side side congruence criterion so the session will be divided like this or it is structured like this in the first part we are going to explain what does this congruence criterion mean what is written away screen and subsequently we will prove that if this criterion whatever is the statement of this criterion is actually true so we will try and get a general proof for it right I hope you understood what we are going to do in this session so let's start and after this we'll also do one more session on the other remaining criterion and post which will be taking different types of questions on congruence other questions could be regular you know primary level stuff that is you know they are very primitive level questions we'll solve those as well so that we get familiarized with how to apply that and later on we'll also take up questions which are let's say olympic level where questions could be solved only using congruence and basic topics of triangle okay so I hope this is clear to everyone and let's begin so here what I have done is we are going to discuss the side side side congruence criteria now what does this criteria mean so first of all it means that if there are two triangles as you can see on your screen a b c and b q r there are two triangles and how and what is the speciality about these triangle is that this side a b here a b is equal to p q b c is equal to q r and c a is equal to r p okay so only sides are given to be equal so if you remember for congruence criteria what all things are needed first all the angles corresponding angles must be equal so a must be equal to p b must be equal to q c is equal to r in this case it doesn't mean that all the time a will be equal to p no depends on you know what is the configuration you have got so in this case clearly I have drawn the image like figures like that so hence a is going to be equal to p here so that's what the current setup is all about now um over enough of that it's given that a b is equal to p q it's given so yeah so let me mention somewhere what's given so here is what is given so given is let me name them so a b is equal to p q this is first criteria which is there then b c is equal to q r b c is equal to q r and third one is c a c a the third side is c a and that is equal to r v right so these are the three conditions given there is no mention of any angle to be equal and we have an objective objective is to prove to prove what do we need to prove we need to prove that triangle a b c a b c is congruent to triangle p q up okay so you must be wondering why do we need to understand the proof so we don't need to mug up anything it's just for understanding intellectual curiosity that we need to understand how proofings are done and later on yes if it is required in an example we will be able to do that and mostly it's a problem solving you know problem solving scenario so hence if we need to start from point a and go to point b what should be our approach so that's it okay so let's begin our proof for this particular proof since no mention of angle is there we need to bring angles somehow because what have we studied so far we have studied two congruence criteria one was s as in that also there is an angle included and the other one was asa so there are two angles and in this case there is no mention of any angle at all so what do we do so somehow we need to bring the concept of angle some you know somewhere so and if you see any of the angles if you prove are equal for example a is equal to p or b is equal to q or c is equal to r any of these angles are equal then our job is done why because anyway sides are equal so then it will fit it into our sas scheme two sides are anyways equal in fact three sides are equal all the three corresponding sides are equal so if we just need to prove only one angle to be equal corresponding angles to be equal so for this we need to actually draw or do some construction now if you have a question that how do we know that we have to do some construction probably at this stage i'll not be able to answer that why because this comes with a maturity in the subject matter right point number one and anyways as we are seeking an angle and straightforward it's not appearing that hey there is any angle equal then we need to think out of the box and that's how we get to this okay so what kind of construction but so this is what we are doing just you know see so what i'm going to do is i'm going to draw a line okay let me just switch it off for some time okay now so i'm going to draw a line what type of line i'm going to draw a line like this okay and just a moment i'm going to reflect about a line so let's say i'm going to reflect this around this what does this mean i'm sorry so i did wrong reflection just a minute what i need to do is need to reflect what reflect this line with respect to this length okay yeah so i got this reflection what does this mean reflection mean this pq here it appears to be pq dash or not that point is something else let me just take away that point as well so q dash is not required i don't require this i don't require this okay just a minute yeah so it's gone and then i have to also uh yeah so pq is there so i hope this is clear to everyone so this point is q and this is p dash so what is the specialty about this let me just say so what i'm going to do is i'm going to draw p dash q p dash q is equal to pq okay this is the construction i'm doing and ensuring that angle p q r is equal to angle p dash q r this is what i'm ensuring what is p q r and p p dash q r so let me just switch it off again and i'm i'm measuring the angles which angle so i'm measuring these two angles p i'm measuring angles for r q oh sorry the other angle i have to measure wait a minute so let me undo it yeah and now i'm measuring the angle which angle r q p 68.2 and p dash q dash r both are 68.2 so just to show you that they are equal i did this okay so once again um i will undo it i don't require all these angles i'm saying can you see this this is 68.2 68.2 just remember i or just you know uh keep in mind you don't need to measure the angle but they are same just you need to know and p q is equal to q p dash that's what i have done okay now let me take away these i don't want these either these also so let me take that away okay so now what i'm going to simply do is i'm going to join this segment which one so that the triangle is completed okay so this i did i don't need the name so i will hide the label not needed so now you can see okay now uh let's do what i'm first of all let me just so what i'm going to do is i'm going to take this here okay take this and reduce the size maybe i don't need this much space so let me take this here okay just so that i can write on the other side okay now so i'm writing here p q r is p dash q r and p dash q is p q so hope this is clear now let's take triangle so this is given i'm going in the opposite direction so please don't mind okay so usually i should write like this never mind so what i'm going to do is consider triangle which two triangle a b c and p dash q r triangle and triangle p dash q r right p dash q r and a b c do you not see that a b is equal to q p dash right or p dash q y because i constructed it so i can write this as by construction also angle b is equal to angle p dash q r and why is that again by construction by construction okay so uh that's what so earlier when i saw when you showed it to this actually i should have shown you this which angle are these angles actually it is this angle which was equal but since these are congruent triangles so hence for the construction sake i saw it showed it to you these two angles are equal let us say this is x so this is also x okay i have done it by construction okay in any ways it was a reflection so you can understand now uh when see reflection i did only to construct c p dash equal to length p q with the same angle let's say just for the sake of symmetry otherwise the construction is you have to draw q p dash such that p q is equal to q p dash and this angle and x and x are equal that's it okay i hope this is clear to you so i'm comparing these two triangles i got a b is equal to p dash q i got angle b is equal to angle p dash q r which is equal to x both of them are equal to x so this is clear i believe then what um after this i have been saying uh let's say what else q r b c if you see b c is equal to q r come equal sides anyways it was given right see b c is equal to q r here that means what this will ensure that triangle a b c is congruent to triangle p dash q r not t q r p dash q r right by which criteria it is nothing but s a s criteria i hope you're convinced the moment that is true what do i get i get these results from here what all one is that p q or a b is equal to a b is equal to p dash q right this side will be equal to this side but folks a b was equal to p q look at this point a b is equal to p q here right that means i can add simply p q so what do i get p dash q is equal to p q so this side is also equal to this side fantastic what else i will get p dash r is equal to a c and hence since a c was equal to p r so all these are equal right i'm writing it here so by c p c t i get a c is equal to p dash r but a c was also equal to see r p so i can write this or pr sorry um yeah a c was equal to pr so hence i can write pr here right so p dash r becomes equal to pr so these are two equal sides this is equal side now let me do this again so the moment there are equal sides there is mouth watering what is that so let me draw a line here fantastic now i don't need the name so i will hide the label okay now let's go back to the writing stuff yeah so do you see this sides are equal what does this indicate very clearly this angle let's see if this is theta this angle will be theta y isosceles triangle pr q is an isosceles triangle right so triangle i'm writing it here triangle p r q p r sorry pr p dash my bad pr p dash right in triangle pr p dash let me just draw a line here so in triangle pr p dash since pr is equal to p dash r right these two sides are equal so hence you can say um angle p angle r p p dash is equal to angle r p dash p i hope this is clear look carefully theta both are equal to theta fantastic similarly if you take this two triangle if you see here this is let's say this is alpha okay so this angle is also alpha same reason so let me write it here now so in triangle why don't i just turn off the axis so let me turn off the axis so that we get more space here okay i turned it off now so hence you will get what triangle p c p dash p c p dash what do i get i get p q is equal to q p dash so hence i'll directly write q p p dash right so angle q p p dash is equal to angle q p dash p no problem in that i hope right which is equal to alpha now that means what if you add this theta plus alpha here and theta plus alpha here both are same or not is it so hence can i not say angle which angle q pr so q pr is equal to angle q p dash r q p dash r and both are equal to what theta plus alpha right clear no problem both are equal so what do i now know right so now take up let's take up this triangle a p q r p q r is congruent to triangle p dash q r and why is that why is that because p q is equal to p dash q we had constructed angle p is equal to angle p dash we just prove that and that is equal to theta plus alpha both are equal to theta plus alpha and pr or r p is equal to r p dash proved it where here check here right so that means this is congruent the moment that is congruent what do i mean or what do i get i get um right what do i get i get angle p is equal to angle p dash anyways it was there and i don't i actually i did not prove it that um yeah but anyways so so these two angles are equal now if you remember angle p dash q r was also congruent to a b c where is it that written this is here check this condition let me use this color this condition triangle a b c was congruent to triangle p q r and p q r is congruent to p q p dash q r right so um um uh yeah sorry this was to be proven so no don't ignore this one so this is not the point i was trying to make here what i was saying is very high this one so triangle a b c is p dash q r congruent these two are congruent and we got p q r is congruent to p dash q r now one triangle is congruent to the second triangle second triangle is congruent to the third triangle that means the first triangle has to be congruent to the third triangle is it so hence what can i conclude with i can say triangle a b c okay is congruent to triangle p dash q or p q r i hope you got the logic logic is see a b c was congruent to p dash q r and from here we proved that p dash q r is congruent to p q r so with those two combined if one is congruent to the second one second one is the third one so first and third one have to be congruent and hence we achieved our end objective so this is what we needed to prove we needed to prove that the two triangles are congruent if the three corresponding sides are equal correct so hence what is the conclusion conclusion is if there are two triangles where each side is equal to the other side corresponding side then the two triangles are congruent right that means each angles also corresponding angles will also be equal i hope you understood this theorem now theorems are one thing but we need to now solve a little bit problem you know some more problems onto these concepts moreover only one more congruence criteria we are going to discuss further that is r h s congruence criteria and then we will take up problems related to all of this i hope you understood liked it thanks for watching and let's meet again in a different session thank you