 Hello and welcome to another session on triangles as we discussed in the previous session We are now going to prove the SSS similarity criterion in the last session You all observed how we validated this particular criterion which says that if the corresponding sites of two triangles are proportional Then they are similar right so we saw that in the GeoGibra demonstration in the previous session If you have not checked that out, I would request you to go there and have a look once and then probably come here and Understand the proof. Okay, so let's try and prove this Theorem, okay, so or this particular criteria. So how to go about it clearly first of all We need to draw a triangle. So let me draw a Triangle, so let's take two triangles ABC. So this is ABC and let me name it as DEF Okay, DEF. These are the two Triangles with us now. What's given? So let's write the given criteria first. So given is What is given guys? So SSS is already given right so AB upon DE is Equal to BC BC upon EF Which is further equal to CA Upon FD. So this is given. Okay, and we have to prove So we have to prove that they are similar. These two triangles are similar or in other words angle A is equal to angle D Angle B is equal to angle E and Angle C is equal to angle F Correct. This is what we need to establish Okay, so let's now establish it How to go about it? So first of all, we are going to do a construction again the ideas We already know that is of congruence and Let's say basic proportionality theorem, which we studied in this series. We are going to use them Okay, so the construction is something like this. So I'm marking a point E dash and F dash on DE and DF respectively Such that such that AB is equal to DE dash Okay, and AC is equal to DF dash Okay, this is the construction. I'm doing and then I'm joining E dash F dash is joint Okay, this is the construction. So this is what the Construction is about. Okay. Now what? Oh, let me just draw a straight line here. Yeah, so this is Okay now So how to go about the proof now, so let's write the proof Okay, so what's the first thing first thing guys? So let's first consider triangles ABC and and triangle DE dash and F dash Okay Okay, so clearly DE is equal to or DE dash is equal to AB and and D F dash is equal to AC Correct. Now in triangle DE F DE F DE F, can we not say that DE dash by DE dash by DE is equal to DF dash by DF Why why can we say that because the simple reason that DE dash is equal to AB and and DF dash is equal to AC given right and we had and we also had this relation that AB by DE is equal to AC by DF this was given Right, so I've simply removed this AV or replaced this AB by DE dash and this AC by DF dash in the given criteria so you can check this given criteria So here if you compare these two this was given and I have made use of that correct And I have simply replaced AB by DE dash and CA by DF dash because that's what I have constructed I hope this is clear. So let me also take this diagram Parallely wherever I am writing it so that becomes clear to you all the time So let me also copy this and paste again and so that I can show you The diagram every time I'm doing something in the proof Okay, so let me just so I'm just for the time being Reducing the size of it so that it is in the same frame. Okay, so I hope this is understandable Okay, now if that is so then what can I say I can say Ah that e dash f dash becomes parallel to EF This follows directly from here and why is that this is converse of Converse of DPT Isn't it converse of DPT so hence becomes parallel the moment it becomes parallel. What can I say I can say that Angle DE dash F dash will be equal to D E F and let me call this angle as X both of them, right? Fair enough. So this is equal because of corresponding angles These are corresponding angles Okay, now corresponding angles established next what is next? Okay, so the moment I say that now let's consider In triangle. So let us write here in triangles In triangles now notice the triangles carefully D E F. I'm saying D first E dash F dash and Triangle D E F Okay, so what's what's that D angle D is common? right common angle and And Angle D E dash F dash is equal to angle B E F which is equal to X just proved above therefore. What can I conclude by? by a a criteria criterion triangle D E dash F dash is similar to triangle D E F. I hope you noticed how did I write the similar sign similar right till the signs? Okay, these are similar the moment. This is similar. What do we conclude we conclude that two triangles are similar Then their corresponding sites will be proportional as well. So hence hence hence we can say What can we say D E dash upon D E is equal to E dash F dash upon E F Right, this is Corresponding parts of similar time triangle Is it so? Can I not now say that if this is so? Then I'm writing it here, but I'm writing but we know that D E dash is equal to a B by construction right by construction therefore, what can I say a B by D E Is equal to E dash F dash by E F Right, so just replace this D E dash by a B here right also a B by D E was given to be equal to BC by E F Check where was it given here see a B by D E is equal to BC by E F given it was given So I'm using that Right, it is given therefore that the triangles be at sight. Okay, therefore we can say E dash F dash by E F is equal to BC upon E F Isn't it if I can say that this means E dash F dash is equal to BC So what did we prove or what did we achieve? We achieved that this side BC is equal to this E dash F dash okay E dash F dash now if you see if we take in triangles now in triangle ABC and D E dash F dash What do we see a B is equal to D E dash by construction I'm writing by construction BC is equal to E dash F dash proved above here proved above and C a is equal to F dash D by construction again Okay, so what do we conclude then therefore? triangle ABC is congruent to triangle D E dash F dash right see and by what criteria by SSS Criteria congruence criteria, so we established that the two triangles are congruent therefore. What can we establish we can say? angle a is equal to angle D and Angle B is equal to angle D E dash F dash is equal to angle E Which is equal to X See again by corresponding parts of corresponding triangles CPCT Isn't it the moment we established that what do we conclude guys? What do we conclude that means the two corresponding angles are equal therefore? I'm just writing here for the want of space therefore conclusion conclusion is Angle A is equal to angle E. Sorry angle D Angle B is equal to angle E and Hence the third angle has to be same angle C is equal to angle F hence fruit I Hope you Understood this right so it was a little bigger proof So here here is the gist of you know the entire thing. This is a proof So basically what did we do we started with? The given criteria that the ratios are same then we drew two points E dash F dash on the other triangle making AB is equal to D dash and AC is equal to D F dash and Then we established the congruence between them and then eventually found out that angle A was equal to angle D and Angle B was equal to angle E and hence we could establish that the two Corresponding set of the angles are also equal and hence they are similar Triangles meaning thereby if SSS criteria is there if you have all the three sides proportional then automatically The angles would be equal and hence the two figures would be similar