Amal Kumar in this video I am going to discuss exterior angles of a triangle in details there are basically two key things to remember one is that some of exterior angles of a triangle is 360 degrees some of exterior angles of a triangle is 360 degrees and the second one is that the exterior angle is always equal to the sum of removed interior angles let's try to understand these two rules and then we'll move forward to take up some examples so what I'm trying to say here is that if I make a triangle let's say something like this so in this triangle we'll call that as an exterior angle and then let me extend this side like this and let us say that the angles which the triangle form are x y and z so these are the three exterior angles what we're trying to say here is that some of the measure of angle X plus y plus C is always equal to 360 degrees now it is actually very easy to prove it let me show you how if I translate this particular line move it right here right so I'm drawing a parallel line here so this line is parallel to Y Z correct in that case what do you observe you will notice that this angle Y is that angle correct we already know what angle X is let me highlight that also angle axis this angle now if I translate another line which is which is this one then you can actually see that if you compare that position then angle Z is basically equal to this this is Z are you getting my point correct so what do you notice here is that the angle why is that one so we have angle X Y in C now that forms a full circle right so we know a full circle makes how much 360 degrees so you can see for yourself that just by shifting a particular line of a triangle add a vertex other than where it crosses third vertex you can see that all angles add up to one full circle and therefore the sum of exterior angles is 360 degrees now the sum of exterior angles of not only triangle any polygon is 360 degrees so that is very important to understand any polygon so you could actually do the same exercise and then you can prove it now let's look into the second which we may call as a theorem right so these are also called theorems since they are always true exterior angle is always equal to sum of remote interior angles why is it so well let me sketch another triangle to prove it let's say we have a triangle here whose angles are interior angles are ANP we want to see what this exterior angle X is right so this is the this is the exterior angle let me call this angry and these two here are remote interior angles so if you look from this position then when we say remote interior angles we mean these two away from it so let us try to prove this also right so every theorem has a proof let's call this angle as less my in that case Y is equal to 180 degrees minus sum of a and P correct since sum of all three angles in a triangle is 180 we knew that we know that sum of all three interior angles in a triangle is 180 degrees great so we are using this particular knowledge we write in Y should be equal to 180 minus a plus B only then the sum will be hundred and eighty right okay now second thing which you know is a straight line right a linear pair so this is a straight line a straight line always make maybe I don't what angle 180 degrees correct so this angle is also 180 perfect so X plus y should also be 180 now what should X be equal to so X should be equal to 180 degrees takeaway y right takeaway Y which is 180 degrees minus a plus B now when you open this bracket 180 degrees minus 180 degrees becomes 0 and that minus makes this as positive so what you get here is x equals to a plus B correct is that clear so I'd like you to do these missing steps and verify that the exterior angle is indeed equal to sum of these two interior lines so I hope with that both these points are absolutely clear now let's take a few examples and practice some questions based on exterior angles and triangles now I will draw a few triangles and you need to calculate the exterior angle okay so this triangle let's say this is one of the triangles then we have another triangle here okay let's let's say like this and let's give them angles to these triangles now here I am saying that this angle here and that angle is 40 degrees and 50 degrees you need to find the exterior angle X if the second case I am taking an equilateral triangle all sides are equal and you to find this exterior angle X so basically in the examples you need to calculate the exterior angles from the given formation so let me draw two more triangles here I'm taking a sauce this triangle this time that is to say that these two sides are equal and you to find the exterior angle X let me give this angle here as let's say 20 degrees you need to find what X is well in all these drawings they are not as per this scale right they are not as per this scale now let me draw a right triangle this time let's say that is the right triangle and in this particular case we are given an exterior now we need to find the exterior angle X and let's say we are given this triangle angle as let's say 40 degrees this is the right triangle you need to find what is the exterior angle so let this be our questionable one find X the exterior angle now from the second theorem which you learn overages exterior angle is sum of remote and interior angles you could actually solve all these questions correct so let's do it so in the first case X should be equal to 40 plus 50 try the drawing doesn't show that but what we get here is 90 degrees you know it looks as if it is much much more than 90 degrees mainly because these angles are not 40 and 50 right the drawings are not to the scale right let me write down they create confusion you go as per rules now all equal angles means that it is equilateral triangle all these angles are 60 degrees right so we could write X as 1 80 minus 60 degrees which is 120 degrees perfect you will also find x equals to 60 plus 60 get the same result 120 degrees so that gives you a clear idea of just adding these two works right now this one is easy let me write x equals to 90 plus 14 so we get 130 degrees here it is a bit tricky we're given angle 20 how do we find the other angles since it is an isosceles triangle we know let's say these angles are a and this has to be a then right both are a because I sauce this triangle so what we know here is that 2a plus 20 degrees is equal to 180 degrees and if I take away 20 from here I get 2a equals to 180 degrees minus 20 which is 160 degrees so so that gives us a as 100 sorry 160 degrees divided by 2 which is 80 degrees so is 80 degrees for us now you can write down what X is so we get x equals to 80 degrees plus 20 that is 100 degrees do cela so if I gave you and I sauce this triangle you should know that two angles are going to be exactly same you need to first find those two angles and then get the result so I hope these steps are clear now let's take question number two where we'll analyze exterior angles with reference to other exterior here is the second question you need to find exterior angle X so I'll make few sketches for you these are not as per the scale but they should help you to get the right answer right so let's say we have a triangle whose some exterior angles are known to us right so let's say this angle here is around 150 degrees and the angle here let's take this angle is let us say around 80 degrees in that case you need to find the angle X okay so that is your first question the second question for you is kind of like this very similar let's say we have a triangle which is of this type in which we are given two interior angles we need to find the value of x and we are also given that these two sides are equal so we actually give you one interior angle let this angle be 15 degrees okay let me make it okay that will cause difficulty let's make this 30 degrees for easier calculations okay so we have this angle as 30 degrees you need to find the exterior angle we did this question earlier rise is very similar to the previous question now here is the third and the last question let us say we again have a triangle this time I'm making a right triangle you need to find the exterior angle X when you know that this angle here is 30 degrees you can actually pause the video answer this question and then look into my situations the first one the sum of all these exterior angle should be 360 degrees right so we are going to use that which is let me write down sum of all exterior angles is 360 so that is what we are going to use to answer the very first question so that really means that X should be equal to 360 degrees take away 150 degrees and then take away it it is correct so that's what it is so let's calculate this we have 360 minus 180 I'm sorry minus 1 50-80 that gives us 130 degrees so the answer is 130 degrees so this value of x is 130 degrees good now question number 2 is kind of straightforward I forgot to mention here this is right triangle so in this particular case exterior angle is equal to sum of interior angles right so so we are using the second rule which is exterior angle is equal to some of a remote interior okay so that means X is 90 degrees plus 30 degrees so it should be 120 degrees perfect now the last question here is for you to do we are given I sauce this triangle that really means that these two angles should be equal right so let us say these two angles are a in that case what should a be right we know what angle is 30 so we know 180 degrees is equals to twice a plus 30 correct three angles sum will be 180 so that means 180 degrees minus 30 is twice a 150 degrees is twice a so a is half of 150 degrees which is 75 degrees so a 75 degrees so both A's are 75 each now X should be equal to sum of these two right so add them up so what we get here is that X is equals to 30 degrees plus 75 degrees which gives us 105 degrees as the answer is that clear so that is how you can actually solve exterior angle questions I hope that makes sense let's take few more questions now now here is a test question for you I would like you to pause the video copy the question answer and then look into my suggestions the question is find the sum of angle PQR PQRS this angle and p RQ which is that angle if Q M equals to M R equals 2 p.m. that is Q M is equal to M so there's a midpoint is equal to p.m. so these three sides are equal now you're to need to find sum of angle Q and hour you can actually pause the video answer this question and then check with my solution you are given four choices 70 degrees 90 degrees 180 degrees and none of the above now to begin with what we can do is we can just assume some angle a here right so if this is e and we have an isosceles triangle then we know that angle R is equal to I mean this a we have assumed this right now this angle should be equal to angle R p.m. right our p.m. since they are ice offs is triangle so this is also a good so these two are same as us now what is angle pmq now p.m. q is an exterior angle let's call it X is an exterior angle so p.m. Q which is X is equals to twice a a plus a you can add them up right because some of remote interior angles perfect now these two sides are equal that really means that angle Q is also equal to this angle correct so let's call those angles as B let's call these angles as B in that case what is B equal to in that case what is B equal to then we can find what B plus a is is that clear right so so in this triangle so what we see here is that in the Triangle Q p.m. what we have is that to be plus X is equal to 180 degrees good to be plus X is 180 degrees excess to a right so we could write this as 2 B plus 2a equals to 180 degrees what I did was I substituted x equals 2 2 way and you see that correct so 2 B plus 2 a is 180 degrees if I divide by 2 what do I get so in this particular equation look at it if I divide by 2 then becomes B plus a equals 2 I am dividing everything by 2 90 degrees this makes sense / - so what I'm doing now is / - right so when you divide that 2 cancels this 2 cancels 180 divided by 2 is 90 degrees so we get our result that a plus B is 90 degrees so the sum of these two angles is indeed 90 degrees does make sense - correct so that is how you can prove it but if you are given this question in multiple choice what you can also do is you would actually take an example let us say if I put here some value let's say if I put here 40 degrees in that case this will be 40 degrees that will be 40 plus 40 80 degrees now this should be add 200 right so each will be how much each will be 50 degrees and if you add 50 and 40 what do you get when you add them you get 90 degrees so that is a faster way quickly so that is your tip so quick way to do it is put some values and check for yourself if you get some result with matches then it's true I hope that makes sense I hope you enjoyed this learning process about exterior angles concluding with an excellent example feel free to write your views and comments and if you like and subscribe to my videos that would be great thanks for watching and all the best