 Hey hello friends welcome again to another session on triangles and we have been doing congruence criterion proving right so in the previous few sessions we saw different criteria what all SAS then ASA then AAS SSS now we are going to take up the last one which is RHS congruence criterion so again as we have been doing so what we'll do is we'll first explain the congruence criteria and then we will prove it here itself okay so first of all what's RHS so as the name suggests and most of you would be knowing it already you have done it in previous grades we're just trying to prove the theorem here right now so what's the statement and what we're going to prove and how so RHS stands for R for right angle H for hypotenuse in S for any other side other than the hypotenuse so clearly if you see here there are two triangles and if you see this is 90 degree guys so I have I have made two 90 degree right angle triangle okay so this is right so it's given what is given so let's start with the statement statement is if there are two right triangles right or two triangles where two angles are right yeah so two angles are right mean one and one triangle has one right angle and there is another triangle which is also right angle and if the hypotenuse are same that is this is AC here is equal to PR point number one point number two any of the two sides so let's say BC is equal to QR if that is so then these two triangles will always be congruent that's the statement right once again right angle do right angle triangles if hypotenuse are same and one of the other two sides are also same also equal then two triangles are congruent that is what we are going to prove and how are we going to prove is again if you see only one angle is there and that two non-included angle so hence SAS is out of question we can't do SAS is it so that means either we have to prove that this improved angle is also equal or if we somehow know that both the angles are equal whichever way so obviously the moment we get only one angle equal that is either of a being equal to p or in this case c being equal to r then in either of the cases we know that this these two triangles are going to be congruent right so hence our job is basically to find out such such condition or find out whether a is equal to p or c is equal to r so hence let us now do it formally and I'm writing given what's given so custom you have to follow to write this so what's given guys it's given that triangle abc sorry let me write here directly as angle two triangles are given triangle abc in triangle abc angle b is equal to angle q and this is equal to 90 degree both of them are equal to 90 degrees this is one then what is given hypotenuse ac the longest side is equal to pr this is also given okay and the third one is bc bc is equal to how much bq bc or let me write q r sorry q r bc is equal to q r I hope the given criteria is clear what is to prove to prove to prove what do we need to prove triangle abc abc is congruent to triangle pq r why are we doing it just for fun yeah so let's enjoy let's do it so abc is congruent to pq r and how do we do that um again we need some construction because until and unless the other angle is equal so that will be going to be tough right if the angles are not equal so how do I do it again as we did in sss criteria you can do the same here right so let's drop a line till this point here pq I'm extending such that q p dash is equal to pq okay that's the thing so hence I will just reflect the line pq around along q r so let's use a tool which is available with geo gibberish so let me just switch it off because it will not work otherwise so let's this reflection tool be there I'm going to reflect this line oh sorry if it just picked up the entire triangle I don't need that I just simply don't need that so let me do it once again and reflect about a line what I have to reflect this this line only about this line and fantastic I got p dash right now q1 dash is the points you know super posing q so let me just hide this q1 dash really will I be able to what do you think uh yeah but then q also disappeared so let me just yeah so I brought q back okay now what we are also going to join these two segments uh sorry points p dash to r okay oh yeah I think it got deviated okay so let me just redo it undo this one yeah so you need not deviate so hence p and right I don't need this g unnecessary confusion so let me just take this away okay now it's clear all of you are clear with the construction so let me also write it let me now switch it on and I'm saying construction what construction I am doing guys construction is this construction so pq so we did this construction pq extended so that it is equal to p dash q and what angle p a b c a b c is equal to angle p q r right and both are equal to 90 degrees 90 degrees okay so this is something uh which we did now and join what join p dash r p dash and r joint right so this is what we have done now we will again compare like we did in sss criteria a b c and p dash q r so I'm saying triangle in triangle a b c and which triangle and triangle p dash q and r let's observe these two triangles this one here and p dash q r there are few things which already know and that is a b is equal to p dash q without doubt why is this because I constructed boss so by construction I constructed it here where where did I mention it here I mentioned pq is um you know oh I'm sorry no sorry sorry sorry this is wrong why is this wrong because pq is equal to p dash q so will it help folks will it help uh let's see so okay so we have done this this is equal to this and this is coming yes it will help so forget about these two triangles what I'm I'm trying to say is um construction pq is not equal to p dash q actually this is what we need to do a change here pardon me what I'm doing is not pq is equal to p dash q actually intelligent thing will be to do a b is equal to p dash q a b is equal to p dash okay so this is the construction I am doing so correction guys correction some you know I was I might be in some other thought when I said that pq is equal to q p dash no a b is equal to p p dash q that's what we are doing okay now a b is equal to p dash q by construction what else I can say angle b is equal to angle p dash q and r perfect why both of them are 90 degrees check a b c is equal to p q r is equal to 90 degrees right I don't know why I did pq is equal to q p dash no so this is the thing right angle b is p dash angle b is p dash q r both are 90 degrees so it's fair enough if I made it also 90 degrees perfect and what else bc is equal to q r given bc is equal to q r this is given so I don't need to do anything over here where is it given here it is here bc is equal to q r so what can I say about these two triangles all of you are intelligent enough now you can figure out that a b c is congruent to triangle p dash q r right a b c is congruent to p dash q r the moment you say that what happens we can say a c hence hence all right therefore therefore therefore ac will be equal to p dash r so ac this is ac so this will become p dash r right but we have seen that or it was given what was given ac was equal to pr so can I not write here itself that this is equal to pr okay because ac was equal to pr it was given now if you look at these two triangles which one let's now consider triangle p q r and what else what else what else yes angle p dash triangle sorry triangle p dash q p dash q r okay p dash q r now in this case pr is equal to p dash r therefore can I not say guys what can I say let me draw a dot properly I don't know why it doesn't appear okay therefore here also I was struggling to make this dot okay so what does this mean this becomes pr pr p r p dash p r p dash is a nice also this triangle and hence I can say this angle is theta and this angle is also theta equal is it so hence I can say triangle q p r is equal to angle sorry not triangle q p dash r which is equal to theta both are equal wait a minute sorry for this glitch so I'll write theta right so q p r is equal to q p dash r equals to 0 or equal to theta now the moment I say that what do I mean now if if you see q p dash r guys q p dash r is actually equal to b ac also why because these two let me use this color here if you look at this very poor line I was trying to let's say yes so this is the line this is the attention pay attention here in this line so this will imply what angle which one b ac b ac is equal to angle which one b ac is q p dash r q p dash r right and now compare these two statements folks what do I get if you see q p r is q p dash r b ac is q p dash r that means I can prove that q p r is equal to b ac and that is what I was looking for understood so q p r is b ac q p r is b ac so hence I can write b ac let's say this is also theta right so q p r and b ac both are equal the moment I get this I am done so let me just make some space here to write okay so I think it will interfere with the right things let me see yes it is so let me not do that yes let me not do that so let me do undo undo yeah go ahead so what do we do so let me write in this small space running out of space here but anyway so now the moment these are so hence you can say triangle in triangle which one a b c and triangle p q r what do we know we know that a b uh sorry so angle c a or b ac angle b ac is equal to angle q p r right we just prove it where here see and angle b is equal to angle q 90 degrees both are 90 degrees and either you take ac or whichever way ac is equal to pr let's say so ac is equal to pr it's given isn't it so ac is equal to pr right so by this so by a a s congruence criteria I can say by a s these two are congruent triangle a b c is congruent to triangle p q r uh just in time or just in space okay so I could write that yeah so hence I declare the triangle a b c a b c is congruent to triangle p q m r fantastic understood right so what did we do we had two triangles we had drawn a b is equal to p q p dash correct an angle also b is equal to q we had maintained that prove that this particular triangle here is congruent to this one got what this angle theta uh sorry this side is equal to this side that is ac is equal to uh this p dash r which was in way in in the intern equal to pr so pr is equal to r p dash the moment we established that we got this theta is equal to this theta the moment these two angles are equal uh what happened uh you know and and this angle theta was equal to theta because these two triangles were proven to be congruent so hence a became equal to p and that is what we were chasing the moment a became equal to p another angle b is equal to q anyways given and to take ac is equal to pr so by a a s the two triangles are congruent and that's it that's how we achieve this congruence criteria right so in all these sessions what did we learn there are four basic criteria of congruence one is s as we learned in the first session then as a corollary to a as a was a as then we learned s s s and now r h s so four and one corollary so in a in a way five you can say but then basically there are four and with these four rules now our four set of uh criteria right we can prove triangles congruence right and we can solve problems related to that now this congruence criteria are going to part of your mathematical journey for a long long time so it's advisable to all of you that solve as many problems as possible on these concepts so that you ingrain them and hence never make or you should be you know perfecting the knowledge so that you should be able to solve any problem related to triangles and their congruences okay so see you in the next sessions with more problems and let's try to improve upon our knowledge thank you