 Hello and welcome to another session on triangles in this session We are going to take up another theorem and the theorem says that angles opposite to two equal sides of a triangle are equal Okay, so I have drawn a triangle a bc you can see a bc is a triangle drawn here and I have measured the sides ac and bc you can take this right and Both of them are equal to ten point one two I have also measured the angle a and b alpha and beta fifty seven point zero nine fifty seven point zero nine degrees Okay, so what we're going to do in this session is First we'll try and validate this in terms of if I change the position of a b and c So what happens? So does this theorem hold for all the configuration and shapes of a bc or it is just a specific case Okay, so we will try and validate with different different Locations and orientations of point a b and c that's point number one After that what we are going to do is we are going to prove this theorem right so proving is also important Just validation will not help validation is just to understand that yes It works so I'm constructing a triangle and seeing for myself that is the theorem does Work and and if it works then it is our you know our next step to find out The way to establish it generally that means We have to give up general proof of it. So let's start with the validation So what I'm going to do is I'm going to change the position of c and you have to observe the angles on so now This is the new position of c and you can see what is the value of alpha forty point nine nine forty point nine nine The sides remain the same if I change the location of b Let's say now I'm changing the location of b So you can see the value of a c and b c has changed, but Yeah Angle remains the same angle Angles all both the angles always are equal. Okay, even if I change the location of a let's say Okay, and here is point c sites are equal angles are same So you can take it to this side as well see sites are always equal and as sites are equal the angles are same So this is taking a reflex angle in this case because it has been measured like that. So let me go back to this position and Come back here. Okay, so you can see at any location of a and b or any configuration of the triangle The theorem does appear to be holding good, right? So all the cases if the sides are Sides are equal the opposite angles are equal now you'll ask how to prove this theorem So that is interesting. So let's try and prove this How do I prove this so what I'm going to do is I'm going to bisect angle a cb and definitely I'm going to use the concept of Convrew and triangle. So I'm going to bisect a angle c basically. So you can see there are two Two this thing I don't want this bisector. So let me take it away. So yes, so I am bisecting it and let's say this point Your intersection point here is B. Okay, and let me take away this bisector and simply let over here Segments stay over there. So what I'm going to do is now I'm going to add a segment so Cbm join right. So what is the construction guys construction is so let me also write now So I am going to write So what I've done is So let me first take this Diagram and let me sideways where we get some space like okay So let me write this through on the right side. So hence what I've done is angle if you see so What do I the construction is CD? Is Bisector of angle a CD Okay right now in Triangle, let's say a CD and triangle BCD BCD What do we know we know that AC is equal to BC it's given What else you also know that angle ACB ACB is equal to angle BCD Why because CD is the bisector CD bisects angle C And what else you also know CD is equal to CD in both the cases common They are common Therefore, what can I say therefore? if you see therefore CD Therefore triangle ACB is congruent to triangle BCD, right? BCD by what criteria by? SAS Criteria right by SAS. Therefore, what is this? You don't get confused so it is by s a Okay, therefore, and I'm writing here therefore therefore You can always say angle a is equal to angle B Okay, and this is nothing but corresponding parts of corresponding Triangles right so hence prove So we prove that angles opposite to two equal sides of the triangle are already equal And in fact the vice versa is also true That is the converse is also true if the angles are equal Then the sides have to be opposite sides also are equal right once again if the opposite angles are equal of a triangle Or two angles of a triangle are equal and opposite sides of the two Angles are also equal right so that converse is also true So I hope you understood the term and let's now solve some problems based on this term