 Hello and welcome to another session on triangles and in this session. We are going to take up the first very important theorem Related to triangles and this is called angles some property of the triangle The theorem says the sum of three angles of a triangle is 180 degrees Right. So this is the basic statement of the angles some property theorem of the triangle What is it once again the sum of three angles of a triangle is 180 degrees So before we take up the proof it is always good to verify this So what I've done is I have drawn a triangle ABC you can see on the screen and I've also measured the Triangles so you can see at this stage angle a which is denoted by alpha. It's 24 degrees and 24.59 degrees and Beta is 98.26 degrees and gamma is 57.15 degrees now if you add all three of them It is coming out to be 180 degrees So you might ask that it could be just a coincidence that in this case it is summing up to 180 degrees But in fact it is true for any triangles any any sort of triangle Whether whatever type we studied in the previous session for example skill in triangle isosceles Equilateral or Whether it is acute whether it is obtuse or right angle triangle so it holds for anything So let's let's try and Change the configuration a bit and see whether whatever we are saying is true Yes, obviously, however much you try and demonstrate through these demonstrations You cannot generalize that every time you will be this particular theorem will be valid So only one instance is good enough to disprove any theorem So hence we have to have a generalized through theorem proof as well Which after this validation will take up and we'll try to prove it for all cases You know so so that no one can question that hey this was possible only for one particular configuration of the triangle So we will see a generalized proof as well But before that let us first verify or try to see if there is any case Do we really get any triangle where the sum of the three three angles is not 180 degree So I'm what I'm going to do is I'm going to change the position of a so this is what I'm changing and you can see The angles are changing all the three angles are changing, but the sum remains the same Same is the case with be if I change the position of be you can see The sum is again 180 though the three individual angles are changing and Finally I can change C as well So that you can see that it is right now It is an obtuse triangle guys if you can see it's an obtuse triangle Why because one of the angles is more than more than 90 degree we can go for a cute angle triangle as well So this is a cute angle triangle all the three angles are less than 90 degrees so this is a cute angle as well Now what I'm going to do is I'm going to simulate the process that is I will let it run on its own and you keep an eye over the three angles as well as its sum and Let's try to find out if there is any case where You know you get less than 180 degrees or so far you have seen now the triangles are you know the triangle is changing its position on its own angles are changing and We are not getting any such position where The angle is or the total sum is not 180 degrees so I'll let it run for a few more moments So it will keep on Changing its position right so it is automatically moving the configuration is changing slowly and In no case. We are seeing that there is less than 180 degrees or more than 180 degrees case every triangle the sum is 180 degree all the time Right this is what I wanted to first validate so in this session We learned that or we validated the particular theorem that is angles some property for triangle Which says that the sum of the three? Angles of a triangle will be 180 degrees always and we have verified it now. It's time to prove this so in the previous part of the video We saw the validation of the given theorem The theorem was sum of three angles of a triangle is 180 degree which is also called angle some property of a triangle We saw that in the software that in different configuration of the triangle The the sum of the three angles was in indeed 180 degrees But then just few examples cannot Generalize the theorem or we cannot prove it for all the cases So hence we must have a generalized proof and that's what we are going to do So first of all learn the theorem the theorem says sum of three angles of a triangle is 180 degrees And we have to find out or we have to basically prove that it is true for any triangle So what I've done is I've drawn a triangle ABC Okay, so let me give you the proof So what is given? A triangle ABC Is given Right And what is to prove? to prove is angle A plus angle B Plus angle C is 180 degrees Okay, now you can see in the diagram I have given them a numeral Identities right so one two three four five like that so you can see which angle is what? And to prove this I am doing a construction I'm doing a construction. What is that a line? L Parallel to BC is drawn Okay, I have done this so I've drawn drawn this line Parallel to BC. How is that going to help me? Let's see now since L is parallel to BC I can say what angle five is equal to angle two Right, why is this? This is because of alternate interior angles So you learned in the previous Lines and angles chapter that if two lines are parallel and and ab is Transversal in this case So angle five will be equal to angle two alternate interior angles also Angle four is equal to angle three Right Same reason alternate interior angles now We have angle one plus angle four plus angle five Is equal to 180 degrees y sum of Angles on a line angles on a point In this case the point is a on one side of the line one side of the Line L right in this case we have seen this in the previous chapter in lines and angles chapter that If you have a line and if you have a point then on one side all the angles on that point Uh will be summing up to 180 degrees. So angle one plus four plus five is 180 degrees, isn't it? Otherwise, you can also see like say like that. So let's say this is Right, so one plus four. Let's say this is angle one plus four Angle one And angle four and this is angle five. Let's say so I am Trying in another view, right? So hence these two angles now are forming linear pair and linear pair are always 180 degrees So any of these reasoning both are same actually. So one plus four plus five is 180 degrees And we now know that one is angle four is angle three so I can write and angle five is angle two is 180 degrees Right, so let this be one and this be two so you can say from One and Right therefore angle a Plus angle b Plus angle c Is 180 degrees and hence proved Right, so the construction Of parallel line to bc was really helpful, isn't it? So that's how you prove that the sum of um three angles Of a triangle is 180 degrees. Actually, you can generalize it and I'll give you the generalized version as well so generalized version is if there is a n sided polygon and sided polygon Okay, so n is four then it becomes quadrilateral and is five becomes pentagon n is six becomes hexagon so on and so forth So if there is an n sided polygon then sum of all angles of the given polygon Now when I say all angles, I mean interior angles There could be exterior angles also we will see afterwards But always remember sum of all angles of the given polygon is given by nothing but n minus two Into 180 degrees Okay, this is the formula n minus two into 180 degrees you can check for triangle for triangles n is three three sides so three so three minus two into 180 degrees Which is simply 180 degrees now you can guess for quadrilaterals Angle some property of quadrilateral will be nothing but four minus two into 180 degrees which is 360 degrees So sum of all the four angles of a quadrilateral one two three and this four So angle one plus angle two plus angle three plus angle four is 180 degrees Okay, similarly pentagon pentagon is how much For pentagon will be it will be five minus two into 180 degrees. So you now know what is it? 540 degrees. Similarly for hexagon, it will be 720 degrees and so on and so forth So this is a generalized form and you can prove this if you know The triangle sum or angle sum property for triangle, right? So you can see you can draw draw a line diagram like that for quadrilateral. Let's say So both there are two triangles. So two triangles Interior angle is 180 180 each so it will become 360 if There was a polygon a pentagon then there will be three triangles And hence it will be three times 180 like that. So this is how you can prove also, right? So this is an information that if there are and there is an n-sided polygon Then sum of all angles of the given polygon will be n minus two into 180 degrees Okay, or many times this is written as two n minus four Into 90 degrees Correct. So whichever Suits you you can remember that right. So this is much easier to remember n minus two times 180 degrees. Okay