 Hello and welcome to another session on triangles. As you know, we are going through a series of sessions where we are dealing with Different similarity criteria for two given triangles So in this session, we are going to take up the third similarity criterion and that is SAS similarity criterion Now what does the statement say? It says if in two triangles one pair of corresponding sides are Proportional and the included angle. So let me highlight this word included angle Angles are equal then the two triangles are similar. Okay, so let me first try and make you understand What this theorem is and then we'll go for the proof Say we have a triangle ABC and let me draw exactly similar triangle Okay, so here is the triangle I will paste it so I'll get another one another triangle So let me just position them in such a way That becomes easier to grasp the points. Yeah, so let's say this is one triangle and This one is another one. So let me shift it to this side and this one is Yes, so these are the two triangles, let me name them and the names are a a b see and This one is D E NF They are similar triangles. Okay. Now. What's given given is given is this that To one pair of corresponding sides are proportional pair of sides are proportional meaning AB by D E is equal to AC upon D F it's given and Included angle now that hence I highlighted this word included Included means the angle between the the two sides which are proportional. Okay, so hence in this case This angle is the included angle. So AB and AC. I'm taking see AB and AC are the Sides which are proportional to the corresponding sides of the other triangles. So hence this angle is equal to this angle Okay, so this is given and what is to prove we have to prove that they are similar So we have to prove to prove that triangle ABC ABC is similar to triangle D E F Okay, this is what we intend to prove how to go about it. So obviously whatever we have learned so far Especially the two other similarity criteria or rather three wherein triple a double a and SSS Similarity criteria we have already understood and we have also seen basic proportionality theorem and its application So probably any of these combinations of theorems will be helpful in achieving this result as well so like we have been doing we will do another construction here as well and The construction will be exactly similar What I'm going to do is I'm going to take up points E dash and F dash such that AB is equal to DE dash and AC is equal to D F dash and E dash F dash are joined Okay, our Joint so let me join E dash F dash. Okay, the picture is clear Fair enough. Now what? so if you join them and we now very clearly know that triangles ABC and Triangle, let's say D E dash F dash. Let's pick up these two Triangles, we know what do we know? AB is equal to DE by construction by Construction is it similarly angle A is equal to angle D. This is given and Dear friends, we also have AC is equal to D F dash. This is again by construction So then very clear it is that these two triangles are congruent So therefore I'm writing it here. Therefore Triangle ABC is congruent to triangle D E dash F dash the moment I say that What do I get? I get BC is equal to E dash F dash first of all that and also also we can see we can we can We can say that since AB by DE was equal to AC by D F Therefore we can say I can replace AB here by DE dash because that's what I constructed and I can replace AC by So I can replace AB by DE dash and Here I can replace AC by D F dash Right. So what do I get? I will get and I'm writing it here. Okay, so let us Conclude here. So the moment I say that I can say that E dash F dash is parallel to EF This is by converse of Bpt no doubts about it, right? So that's done by converse of Bpt. So what do we get? So we get that angle DE dash F dash that is this angle X is going to be equal to this angle X is equal to angle E let me write that okay and Already it was given that angle B or we had established that angle B was equal to angle DE dash F dash Y congruent parts of congruent triangles, right? Therefore we can conclude angle B is equal to angle E Angle B is equal to angle E also Angle A was angle B given already given Therefore folks we can establish that triangle ABC is similar to triangle DEF and that's exactly what We wanted to establish, right? So this is another similarity criterion that the two triangle two sides and Included angle again, please pay attention if the sides. Let me write it over here if the sides are AB let's say this was given AB by DE is equal to BC by Is equal to BC by EF is given and Angle B is equal to angle E will be the included angle in this case S AS criteria will hold SAS criteria will hold but but let's say AB by DE is given to be equal to BC upon EF But it's given that angle A is equal to angle D Let's say this is given This is not SAS criteria because the angle is not included Angle is Because the angle is not included, right? So let me write it here Because The angle is not included Angle the angle is Not a not an included Angle so please keep this in mind only when the angle is included between the sides Then this criteria will hold. Otherwise, it will not hold. It will we will require for some other Criteria to be fulfilled before we can establish that the two triangles are similar. I hope you understood this similarity criterion, okay