 Hello everyone. So we will start in a short while. I think there are lots of people already logged in so today's class is for Triangles revision, so we'll try and revise Triangles and if you have any query you can always post it in just for a confirmation whether you guys are able to hear my voice So you can just reply back in The chat section so so that I am aware that you guys are there and you can hear me Is there any problem in viewing the slides and all Are you able to see the screen? So I have shared my screen with you guys. So if you're not able to then let me know Okay, so so I think lots of people have already logged in. So let's start the session so today we are going to try we are going to summarize the entire triangles chapter and If at all you have any specific Requirement, then you please enter that in the chat section so that if not in the live class we can definitely take that up in Let's say, you know, it we can take it offline. So now Okay, so in triangle chapter if you see there are three broad categories of Let's say there are three topics which we have learned so This is triangles So in triangles chapter if you see Okay, sorry for the glitch guys. So here we start. So basically Triangle chapters if you see is categorized or you can actually see there are three broad category of topics So the first part we dealt with something called basic proportionality theorem Basic proportionality Proportionality theorem Then we have This is also called Thales theorem. So if you if you know, we have discussed this in the class. So Thales theorem so this is the second part of it and Then we had similarity criteria. So if you see we had similarity of triangles similarity of triangles and the criteria related to that and then third part is Pythagoras theorem and its application Pythagoras theorem the age-old Pythagoras theorem and Problems related to that. So this is in a nutshell problems related to these Concepts, okay. So in your board paper whenever you get to see Any triangle or you know geometry related questions So if you know that in your boards board syllabus geometry has two subparts primarily where one is questions related to triangles will be there and second one is circles and There will be questions there could be questions which will be requiring you to know both the concepts and The same question could you could have applications of both the Both the sub topics soon So today we are going to discuss what all did we study in triangles and how to approach problems whenever You're going to face these and let's say in your board paper or mock papers In the days to come so whenever you you know see a geometry problem Then you know that there these are the concepts which are going to be Implemented there now questions related to ratio. So hence these are the things these are the catch words You have to you have to be sure of so ratios then there is you know, if you see parallel lines Parallel lines in in problems. Let's say in a given figure. There is you know There's a figure and there's a triangle There are two triangles like that and then one parallel line is this another parallel line is this like that if you see such kind of a You know figures. Let's say this is a b c and d and E f g let's say this is a figure given and something related to this figure There is a question which involves ratios parallel lines or maybe let's say square square terms Square terms, this is how you should be knowing so square terms Let's say they'll ask you let's say ac square is equal to bc into cd and things like that Whatever so whenever you see such kind of relations to be proved You know that this has got to do something with Triangles and either of these three either of these three concepts either Thales theorem similarity of triangles or Pythagoras theorem would be Applicable so keep that in your mind now. Let's go and revise all the basic theorems and results which you Must be having in your mind before attempting all these questions really do triangles What we have done is we have compiled all the major theorems which are there and the results which are there and which are useful for triangles chapter and Later on you can you know get these slides for your own You know reference and we have also planned separate sessions on problem solving which will be Which are you know which are being recorded right now and they will be uploaded on your own our YouTube channel So you can always reference the problems offline so let's begin and This is So basically we are going to deal with triangle summary, so this is the first slide so basically there are 30 slides We'll try and you know Complete as many slides as possible and if possible we are going to complete the entire revision today itself So first slide is two figures having the same shape, but not necessarily the same size are called similar figures We'll run through these concepts. We know that two circles Two circles are always Give me a moment Yeah, so two circles are always similar why because if we have two circles we have You know, they are similar we know that two circles are always similar so All congruent all congruent figures are similar, but the converse is not true. That means if there are two congruent Congruent triangles or any two congruent Geometric figure they will be similar, but the vice versa is not true. Why what is congruence congruence means if all the angles and all the sides corresponding angles and corresponding sides must be equal must be equal. This is my congruence So this side is equal to this side This side is equal to this side and this side is equal to this side and all the corresponding angles are equal then we say that The two figures are Two figures are congruent, but For similarity, you just need scaled one one geometrical figure Scaled down or scaled up visa we are with respect to the other one, right? So that's that's similarity So hence all congruent figures are similar, but the converse is not true So these two triangles might be similar, but they are not congruent. Why because the corresponding sides Corresponding angles are same are same, but corresponding sides are not equal corresponding sides are not equal So hence if you see this is a b c a b c and this is d e f This is d e f then a b is not equal to not equal to d e Okay, so though though they are similar, but they are not Converse, so this is point number one next is Yeah, two polygons have the same number of sides are similar if so this is the criteria for similarity if you see So this is a this is the criteria for similarity two polygons having the same number of sides are similar If they are corresponding angles are equal and their corresponding sides are proportional in the city that is in the same ratio So we'll start with our triangle our triangles only so let's say this is a bc again same thing a bc and D e f then these two triangles are similar These two triangles are similar only one condition is first of all all the corresponding angles should be equal So angle a must be equal to angle d angle b must be equal to angle e and Angle c must be equal to angle f This is criteria number one and criteria number two is a b upon d e is equal to bc upon e f is Equal to c a upon f d these is criteria number one criteria number two So both the criteria are met then to given this is I have shown shown it for triangles, but it is You know same it holds similarly for any other polygon as well. So corresponding angles must be same and corresponding Corresponding angles must sides must be proportional. This is the criteria for similarity next So let's go to the next slide. It says if a line is drawn parallel to one side of a triangle To intersect the other two sides in distinct points, then the other two sides are divided in the same ratio. So this is nothing but Thales theorem if you if you remember this is nothing, but if a line is drawn parallel to a Triangle, so let's have a triangle So let's say I have a triangle here triangle and This is a bc. This is Thales theorem. So or basic proportionality cell theorem. So this can also be written as this is my basic proportionality Proportionality Basic proportionality theorem right or it's also called Thales theorem whenever while solving the problem You know, I have seen that many of you write just BPT. That's okay But then You whenever you are using the theorem for the first time you have you should it's a good practice to always mention the acronym Which you are going to use later on So, you know, this is a common practice now What does it say if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points? That means let's say if I draw a line Like that then the other two sides are divided in the same ratio. Let's say this is D and E So we know by BPT. What does it? What does it say ad upon dv is equal to a e upon ec Upon ec and I'm not going to the proofs of it, but I'll just tell you the you know The basic way of proving it. So basically have to join the e and drop perpendicular from e to let's say This is perpendicular dropped from point e on side AB, which is let's say D and then if you remember we used the concept of Oh Area of triangles to you know, you have to you have to join this as well. So And then you have to drop up a particular like that So if you if you see what was the you know, I'll just give a brief Hint of what exactly the proof was about. So basically we found out that we took first of all triangle ratio of triangle area of triangle a Ed over triangle b De now these two triangles. This is area of we are trying to find out the ratio of the area now in these two triangles base is You know the height is same ed same and hence this was equal to nothing, but ad upon ad upon db why because this will be similar The triangle area was nothing but half into base into height height is same so it is reduced to ratio of the Ratio of only the base is similarly. You have to take the another other triangle. So that is area of triangle a De area of triangle ade divide by area of triangle Dce dce here also if you see dce here also the ratio will be nothing but a e upon a upon EC why Because the heights are same So it will be in the ratio of their bases now if you see triangle ade a Or sorry triangle dce dce dce is this triangle this area this area is This area is equal to the area of this area Why because the base is same d is the base for same and they are between the same parallel line So hence from that logic area of triangle bd e is equal to area of triangle dce Hence these two ratio become become equal. So hence that was that is how it was It was proved. So this is a brief proof for Okay next so let's go to Next slide. Oh, it seems some screen is not clear. Okay, let me just check the settings once just a minute Okay, screen is not clear. So Just a minute guys. Give me a second. I'll just Try and fix this. Okay, give me a second please I'll just fix it. Give me a moment Just bear with me for a second actually to reduce latency. We had done Some tweaking but never mind