 hello friends welcome to another session on triangles and we are discussing now similarity criteria for two triangles to be similar so if you remember when we were discussing similarity we said that there are two criteria which when fulfilled can ensure that two geometric shapes are similar obviously the two geometric shapes have to be of the same category that is you can't really prove a triangle to be similar to a square if you take two triangles or two square two quadrilaterals not a square sorry quadrilateral or pentagons hexagons and things like that there are two conditions which have to be fulfilled for the two geometric shapes to be similar what are they first phase the corresponding angles must be equal each one of them so there are three corresponding angles if you see I have taken abc and defs two triangles so I'm giving you examples of triangles but even if it is any polygon the corresponding angles must be equal so in case of triangles what will happen a must be equal to d b must be equal to e and angle c must be equal to angle f right this is criteria number one all the three angles should be individually we say correspondingly equal correspondingly meaning what a corresponds to d so those two equals are corresponding angles so a is not corresponding to f if a were equal to f then we would have said a is corresponding to f in this case a corresponds to d or a is mapped to d so you can use whichever way you want d is corresponding to e and c is corresponding to f so this was criteria number one guys criteria number two was that the ratio of the three sides corresponding sides that is so for example in this case ab upon de dc upon ef and ca upon fd these three ratios must also be equal so if you look at these figures here ab by de i have written bc by ef and ca by fd now these values i have just calculated using a formula i have not mentioned the values over here so you can check if you want this is the formula i have written so you can see this is calculating the distance between the two points and calculating it so i have not you know uh mentioned any values over there it is calculating using the formula in the software now the idea for this session is uh we are going to prove or tell you that if in case of triangles if you want to prove that two triangles are similar then if you just match the three angles individually then you don't need to go for the second criteria right once again so for similarity to be established what do we need to know we need to know or between need to investigate that the two polygons you are considering their corresponding angles must be individually equal that is point number one or criteria number one criteria number two is their sides must be proportional corresponding sides must be proportional so ab by de in case of triangle bc by e f and ca by fd these three ratios must be equal which i am showing here right so you can see the angles are clearly equal so 65.77 65.77 78.69 78.69 35.54 and 35.54 is it so they are equal angles are equal now what i'm going to show you in this case you can see the ratios are also equal so hence you can say that the three are similar other two triangles are similar right so both the criteria that is corresponding angles being equal corresponding sides ratio being equal is fulfilled what i was trying to tell you is even if i change this configuration of the triangles but maintaining the angles to be equal they still are congruent so i wouldn't i'm not going to measure or i'm not going to care about the ratio of the sides but the best part about the triangle is if you somehow prove that the three angles are equal individually or correspondingly you don't need to go for the second criteria the second criteria is automatically fulfilled how we will now see so first we will validate it using this then we are going to prove that so what are we going to prove friends we are going to prove that if there are two triangles such that in corresponding angles of the two triangles are equal only that is given let's say if only that is given then the two triangles have to be similar meaning thereby the ratios of the corresponding sides would be proportional you don't need to prove anything beyond that so let me show you that it does happen so i'm going to change the location of the side df but i would maintain one constraint and that is all the three angles which are shown are not going to change so if they are not going to change how are the sides going to behave the ratios how are they going to be impacted let's evaluate that so i'm going to change this location so see i am now bringing this line df closer to e can you see guys the ratios are still the same so i have maintained e three angles equal the ratios are still the same and if i take it away see if i am now enlarging the triangle the angles are still the same but the side lengths you can check from here side lengths have been modified and the ratios also have been modified isn't it so you see both the change in value you see right so i'm changing df positions and see how the ratios at any given condition just same it is same right right so ratios are same if you maintain that the angles are same the ratios are always going to be same but in mathematics we know that we can't really you know just you know believe on what do we see we haven't seen all the cases any any which way so hence the best idea would be to give a generic proof to this so hence let's go to the next part of the session where we are going to prove this a a a similarity criteria okay so see you there in that video