 Hey friends, welcome again So I hope you're doing good with mathematics and you are able to understand the concepts through these video lectures Now going forward. We will be dealing with lots of other topics on geometry and you as you know, this is a series on similarity, so let's continue So in this session, we are going to understand what are conditions for similarity now as we know similarity we deal with Polygons, isn't it? So there are polygons polygons means There are geometric figures with more than three sides or even three side Geometric figure is called a polygon. So basically a closed figure. So let us say I have this kind of a closed figure And let me name this a b Cde right. This is how the polygon is and I have another polygon like this and its name is if Fgh I Let's say J. So you can see clearly here. It is a pentagon. Isn't it? This is a Pentagon Pentagon now I am talking about similarity of these two polygons first of all Symbol of symbol for similarity symbol or similarity How do we express similarity? So let us say if ABCdE is similar to Fgh IJ then we write ABCdE is similar to Fgh IJ Okay, so let us first understand what does this mean. So first of all, I am using a symbol here till design Right. So this is for similar. You might have known that for congruence if you remember for congruence What kind of symbol did we use? Like that right, but for similarity we use the symbol Now not only the symbol is important here the order in which you are writing the vertices So a then B then C then D then E The same order you have to write Fgh IJ. What is this order in this order is? We'll see when we see with the condition of similarity. So these are called Corresponding vertices. So a is corresponding to F B is current point corresponding to G C to H D to I and E to J. These are called corresponding vertices Please bear in the bear in mind. What is the importance of corresponding vertices? Okay, and similarly if you see AB is Corresponding to FG. So AB side is corresponding to FG side and so on and so forth. We'll come back to this little moment later now So what are the conditions for similarity? Under what circumstances we will say that this polygon or pentagon to be precise ABCDE is similar to Fgh IJ Right, so let us understand those conditions and the conditions are so now I'm writing the conditions one the condition is first condition is corresponding corresponding angles Corresponding angles must be equal Must be equal. This is criteria number one criteria number two says the lengths The the length Length of Their corresponding sides must be proportional Must be Okay, so let us take another Let us say another pentagon is like this or similar pentagon, which I drew above Let us say this is a B C D E and another one is like that Somewhat looking similar Is it now let us say this is F G H I and J so for the for the first criteria to be fulfilled We know angle a must be equal to angle F hence corresponding. So a vertex a is corresponding to Vertex F similarly angle B must be equal to angle G Angle C must be equal to angle H Angle B must be equal to angle I and angle E must be equal to angle J This is first criteria all the corresponding angles must be Equals and what does the second criteria mean? Lengths of their corresponding sides must be proportional means a B by F G so if you see these are two proportional these are two corresponding Sides length is it it similarly BC corresponds to G H CD corresponds to H I E D corresponds to J I and a E corresponds to F J Okay, so please take care of this corresponding sites and very very important now So hence, what is the rule or the criteria a B by F G must be equal to BC by G H Must be equal to CD by H I Must be equal to I E D or D E rather let's write in order D E by I J and E a upon J F So this is criteria number two in any two geometric Plane figures if these two criteria are met then we say a B C D E is similar to F G H I J Again the you cannot write like this or you cannot change the order you cannot write F J I H G this will be wrong to say why because in this case the corresponding angles are not same Why because a is equal to F. That's fine But B was equal to G here and here in place of G it is It is J which is coming and B is not angle B is not equal to angle J Right, so please mind that order is also important all the corresponding Vertices must be same all the corresponding sides should be proportional So hence if you see here the corresponding side is a B here it is F G, right? So these are the corresponding sides, but a B a B and F J are not the corresponding sides, right? So this one corresponds to this This one doesn't correspond to this. Okay, so please mind the order in which you're writing the vertices is very critical You must have the same order of a same sense of Now ordering right so for example E. This is the last angle here and J must be equal Second last angle is D. So D must be equal to second last here. I See must be equal to H and like that Okay, so order is important and hence what are the two criteria? All angles corresponding angles must be equal all corresponding sides must be proportional right now both of Both of them must be holding true Simultaneously if one is true and two is not then they are not similar Okay, and if two is true and one is not then also they are not similar But triangles is an exception for triangles. This is a this is for any this these two criteria So let me write it down. I'm writing that above two criteria above two criteria must be Simultaneously true that means both should both should hold true if one is true another is not true Then is the polygons are not similar and vice versa, but triangles are exception Triangles because of the virtue of their Geometry triangles are exception meaning what so in triangle in triangle any of The any of the above above to criteria is fulfilled fulfilled fulfilled then Then the second criteria is Then the second criteria is Automatically automatically fulfilled Okay, so you don't need to Check for both and it is only for triangles mind you not for any other polygons not even for Squires rectangles nothing only for triangles right triangles are exception that if you just prove any one of them either you You established this or you established this any one of them if you establish the other one is automatically Stablished this is by virtue of the properties of triangle right so hence for triangle you don't need to Prove or establish both the criteria anyone is established and The triangles are similar, but but for triangles any other polygon if you take you have to prove or establish Both independently right both independently allowed to establish only when both are established Then you can say the polygons are similar and what is what do we learn we learn that in Triangles it is not necessary only one criteria if you fulfill by the virtue of properties of triangles geometry The odd the other one is automatically fulfilled and hence the triangles are similar now in the next session will take up Similar triangles and their properties