 Hey hello friends, welcome to this session on triangles and today we are going to start an all new series on congruent triangles. So this topic is going to be very, very interesting. It is a very important topic as well because the knowledge of triangles is going to help us later in coordinate geometry vector algebra and other things and as the name suggests we are going to study congruent triangles. So it's all about triangles first and then we also need to understand what congruent triangles means, right? So the basic properties of triangles like angles and property, exterior angle property and all that we have already learned. Now today and henceforth in all the subsequent sessions in this series we are going to discuss properties related to congruent triangles. So we'll start with first what is meant by congruent triangles, the definition. We will see whether congruence occurs only for triangles or there are other geometric figures as well where congruency is valid. Then we will talk about what is the criteria for two geometric shapes to be congruent to each other. Then once they are congruent then what all properties can be studied around it. Subsequently we will be also taking up some amount you know some important theorems like Pythagoras theorem, Apollonius theorem, there are many such theorems related to triangles. In fact triangles is a very enriched geometric shape or figure I would say because lots of properties related to triangles are there which we need to study. Not only interesting but also gives a lot of help or you know they come very handy when we are dealing with other mathematical subjects as well. So let's begin. Our approach will be we will be trying to validate all those concepts and theories on GeoGibra software. I would encourage you also to download the app or you can also use the online version of or the browser or you can just go through your browsers you know this thing you can do it on a browser itself. So that helps in understanding and geometry until unless you do it you construct it becomes very boring at times. And secondly you will not get the real insight of you know geometrical properties until unless you do it yourself. So as far as possible we will try to do it on this software okay. So here is the first thing. So let's first understand what congruent triangles are. Triangles we already know. So let's first draw a triangle. We have been working with triangles for a long time now. So this is a triangle it is named as ABC. And now I'm going to show you how to create a congruent triangle of this one okay. So I'm just going to copy it and make a simple. So here we have taken two triangles triangle ABC and triangle A1 B1 C1 okay. So what is what will be meant by the congruent triangles is this that if you have a close look on both of these they look exactly the same is it now the best part is if I take this one and try to superpose this on ABC it will sit on exactly like that can you see it's exactly if it's in so there is no difference whatsoever between the two triangles that is they are superposing each other right. So this is not as an example of example of congruent triangles. So hence if you again if you have to let's say enumerate the criteria of congruent that is when will they superimpose each other that is they're exactly a replica of the same you know so one is the replica of the other what do we infer that it exactly fits in in terms of size and shape right so that will be possible our intuition also says that will happen only when let's say angle A this angle A is equal to this A1 then B has to be equal to B1 and C has to be equal to C1 moreover these sides also have to be correspondingly equal now mark my words I'm saying correspondingly equal meaning what so AB has to be equal to A1 B1 it's not that AB will be equal to any of the three sides no only one side AB is equal to equal to only A1 B1 right this is so hence we say AB is corresponding to A1 B1 similarly BC will be corresponding to B1 C1 and CA side is corresponding to C1 A1 right if we go for angles angle A is corresponding to A1 B corresponds to B1 and C corresponds to C1 let's also try and measure these angles and see whether they are actually true so if I measure these so I'm getting alpha is 74.29 so this also must be 74.29 and indeed so you can see both the angles are 74.29 let's now measure angle B okay so angle B is 75.96 and here B1 is 75.96 amazing and what about these two so this is 29.74 and this one and this one again 29.74 so you can see both are or all the corresponding angles are equal right there is absolutely no problem right so this is what we learn about congruency the criteria are point number one the angles must be correspondingly equal A must be equal to A1 B must be equal to B1 C must be equal to C1 right point number two the sides must be correspondingly equal AB is equal to A1 B1 BC is equal to B1 C1 and CA is equal to C1 A1 right alternatively you can also say that the ratio of the two corresponding sides is one is to one right so AB upon A1 B1 is one upon one BC upon B1 C1 is one upon one and CA upon C1 A1 is again one upon one why did I you know highlight this ratio thing is later on when we are taking up similarity in the same ratios will be equal then right so what is similarity what is what are the properties of similar triangles and other things we will be taking up when we are studying similar triangles for the time being please understand that the criteria for two triangles to be congruent is one that their angles must be correspondingly equal and their sides must be correspondingly equal that's it these two so for you are you know for to convince you let us try one length so BC is 8 you can see here B1 C1 is also 8 what about C and A so length is 8.06 here is also 8.06 A and B 4.12 and A1 B1 is 4.12 now in this case you could see that the figures also appear to be same exactly same but there could be cases where they don't appear to be exactly same but yet they could be congruent what do I mean so let me just draw a random line here okay so I'm drawing a random line and I'm going to reflect this triangle about this line okay can you see that I got a reflection of the given triangle so for the time being if I just remove it so here is a thing that is that disappears this also I don't want so let me remove this and now just talk about these two triangles okay so these two triangles ABC and EFG no sorry FGH if you see they are again they don't look to be similar but they are exactly same shape but they actually are and how do I know so we can measure let's say angle first so between FG and GH let's measure the angle first so this angle is so you can see 75.96 it corresponds to point to be B and G are corresponding points okay let's measure the angle this one 74.29 so this A and F are corresponding and clearly the other H and C are corresponding okay they're same same angles now if I measure the distance between F and G so it will exactly match ABC AB and FG are same similarly F and H is 8.06 which is AC and H and G is 8 which is BC so hence side GH is equal to BC side FG is equal to AB and side FH is equal to AC right so hence triangle ABC is congruent to triangle F FGH right so FGH ABC is congruent to FGH so this is how you have to write congruent triangles and going forward in the subsequent sessions we are going to study more about properties related to congruent triangles and as we mentioned there are two criteria of proving or establishing that two triangles are congruent but triangle is such a beautiful geometric shape that we don't really need to prove each of the equalities so we can eliminate a few yet the triangles will be congruent right so one simple example could be you know that the sum of three triangles of a triangle is 180 degrees so if I have to match one by one all these angles that is if F is equal to A and G is equal to B then automatically H will be equal to C I don't need to prove that H is equal to C if F and G are equal to A and B respectively right or the otherwise B and A respectively so point being in triangles we can eliminate few equality you know we don't need to establish all the six correspondence that is the three angles and three sides in fact with three correspondence also the entire you know we can prove that the two triangles are congruent so this is what we are going to learn in subsequent session guys