 Hello friends and welcome to another session on problem-solving and we are doing congruent triangles now the question says PQRS is a quadrilateral and P and U are respectively points on P, S and RS such that PQ is equal to RQ. So this side is equal to this side given R PQT PQT is equal to RQU. So it's mentioned. So these two triangle this triangle is Equal to this triangle. So I'm sorry angle and these two angles are also given this one and this one Okay, these are equal. Okay, we have to prove that QT that is this QT is equal to QU Okay, so we know that if you have to equate or find two Different elements of two different triangles to be equal The best is prove that they are congruent part of congruent triangles, right? now QT if you see QT is a part of two triangles PQT or PQS and likewise QU is part of QRU and QUS correct. So either of them have to be congruent, right? Now if you see it will be a little easier to do the These two triangle parts, right? So if you see QTS and QUS if you consider that to be congruent So what will happen then then there is a common side clear There is a angle which is already given equal, right? So hence either we so, you know, obviously we can't have QT is equal to QU already given then the question is Not appropriate. That means we have to find something else and preferably this angle To be equal to this angle if this angle also becomes equal to this angle, then what will happen? This is one angle one pair of angles equal There's a common side which is equal and other pair of angle to be equal are equal then by ASA these two will be congruent and then you will be able to prove that QU is equal to QT correct now Out of these this is already given to be equal. There is a common side But how to prove that this angle is equal to this angle for that these two angles also should be Part of two different triangles which are congruent themselves So if you now see angle S is part of PQS and this angle S is part of QRS So somehow if I prove that these two bigger triangles PQS and RQS are congruent and automatically these two angle Let's say X and Y will become equal because of CPCT. So are these two triangles same? Let's check the Clearly this side is given to be equal. So PQ is equal to QR one side is given to be equal These two angles are independently or individually equal. So this one is equal to this one, right? So this plus this is going to be equal to this plus this So that's also there and there's a common side. So this gives a clear picture of How to go about the line? So hence the strategy would be to prove that so let me just remove all those marks So the strategy would be to prove that the two triangles PQS and RQS are congruent. Okay. So let me reiterate the strategy PQS RQS are congruent hence this angle X is equal to this angle X and hence These two triangles TQS and UQS are congruent and hence you can say QT is equal to QU. This is the line So before you start solving you should spend some time thinking thinking on how to go about the problem So let's solve it. So first of all as a customary, we will write what is given So given is PQ is equal to PQ is equal to RQ. This is given also Angle PQT. So let me write it here angle PQT is equal to angle RQU right and Also, PQS angle TQS is equal to angle UQS UQS right UQS is TQS. This is given right and what do we need to do? We have to establish to prove. What do we need to prove? We need to prove that PQ sorry QT or yeah QT is equal to QU. This is what we need to prove Okay, so let's begin the proof. So I will utilize the space here Okay, so how to go about the proof? So we'll say in triangle. So let's first prove that X is equal to X right in triangle PQP PQP and triangle RQS PQS, I'm sorry. This should be PQS PQ in triangles PQS Okay, PQS and RQS. They are corresponding So order of vertices is also right. So what do we know? We know that PQ is equal to how much QR already given So write given right so all the reasoning also have to be provided given. Okay. Now Angle PQT is equal to Angle RQU is it angle RQU and angle PQS Is equal to angle UQS. So if you add both Add both both of them LHS to LHS RHS to RHS you will get angle PQT plus angle TQS is equal to what angle RQU RQU plus angle UQS UQS Okay, so what is PQT plus TQS check PQP plus TQS is nothing but PQS So I can write this as PQS and in this case RQU plus UQS can be written as RQS So angle RQS So first thing this was established one Two and third one which we are going to establish. Anyways QS is equal to SQ Right. So using these three marks one two three. What can we say? We can say triangle triangle PQS Is congruent to triangle RQS Reason SAS, right? Check SAS S congruence criteria Okay, the moment this is true then let me again Make some divisions here. Okay, so the moment that is true Can we not say what can we say? We can say that Angle so this is what we establishing angle PQSQ will be equal to RSQ Okay, and why is this? CPCT congruent parts of congruent triangle now Once that is established We'll take the other two triangles in triangle which one PQS TQS and triangle UQS UQS, what do we know? We know that QS is equal to QS common side We need not write even common. So QS is common and we just found out that this new angles are equal and previously also PQS is equal to angle UQS This was given. Where was it given here? So it's given given and and we just prove that PSQ right or PSQ is equal to angle USQ U SQ by what by this here By this so it is PSQ, but P and P are on the same line So this this can easily we said that this is equal to angle PSQ is equal to angle USQ Correct. So there's no problem in that now. So PSQ is equal to USQ. Therefore therefore by ASA criteria angle side angle, we will be learning about this in the subsequent Videos that ASA criteria is also there many of you would also be already been knowing it. So by ASA congruence criteria What can we say? We can say that We can say that triangle EQS is congruent to triangle UQS UQS check the order of the vertices That's very important as I have been telling you again and again now the moment you establish this fact What do we now now learn? We learn that Angle Angle or other you can now say QT. That is what we have to find out QT is equal to QU Right QT is equal to QU and why is this C P C T Okay So because of the congruent part of congruent triangles again in this particular problem We had two triangles which were the two pairs of triangles Which were the two pairs of triangles which were congruent and that only helped us to prove this Given required Relationship right. I hope you understood this problem