 Welcome again to another session on geometry. We are in triangles and its angles chapter and in this session, we are going to Deal with another very important theorem, which is called exterior angle theorem. So let's start So what I'm going to do is I am going to draw the figure as I you know explain the given theorem So what is angle exterior angle theorem? So, you know what an exterior angle to a triangle is So let's first draw a triangle. So here I'm drawing a triangle ABC This is ABC. So this is ABC and let me extend BC So what I'm going to do is I'm going to draw a line from B to C like that so I have extended it and Let me mark a point over Extended BC. Let's call this point B. Okay, and what I'm going to do is I'm going to measure this angle so angle B C and a okay, so you can see this comes out to be 147 point nine nine, right? now, let's see what We achieve here. So We I'm going to measure another angle. So which one is B a and C okay, so beta is seventy nine point eight degrees. You can see that okay, and one more angle that is C B and a Now I have measured angle a and angle B and angle ACB ACD, which is the external angle So, let me write something here. So what I'm going to write is So I'm going to measure I'm going to check what is What will happen if I add gamma and beta? So let me write that so in this you can also learn how to actually use this tool to Validate any geometric, you know relationship. So C a B triangle C a B plus angle, what is that angle? a b c is it a b c which is nothing but in my Definition I have taken that as beta and gamma. So let me add beta beta Here is beta and I will add a gamma to it So I've added beta plus gamma. Let's see what happens. Okay, so here is you can check that, you know, I have added C a B and ABC and the sum of these beta plus gamma is actually equal to ACD That's what external angle exterior angle theorem suggests that exterior angle of a triangle is equal to sum of The two angles which two angles interior opposite angle so opposite to this angle here Are two angles what a and b is it it? So if you add a and b you will get the external angle a CD, I hope this is clear. So let's try to you know move around this point and See what happens every time it is matching or not So let me keep it here and we see that indeed it matches So C a B plus a B C is actually equal to 118.83 degrees. So it works for any Any configuration any configuration, right? See if it is 90 and 68.2 So 158.2. So I hope you understood this and the subsequent portion of the Video, let's try to prove this theorem Hello guys. So we saw the Validation of the given theorem in the previous part of this session. So now let's try and prove this session So what is given given is exterior angle theorem This is a statement given if a side of a triangle is produced the exterior angle. So formed is Equal to sum of their two interior opposite angles. So let's first understand. What does it mean? So exterior angle what is exterior angle? So let's say this is exterior angle. Let us call it X Okay, and X is equal to sum of the two interior opposite angles So these are interior opposite. This is let's say Y and This is it Okay, so you can identify very easily. So this angle is the supplement of the exterior angle So this will not be the interior opposite angle. So interior opposite angle will be the other two angles apart from this Let's say this W. Okay, so for X Y and Z are the interior opposite angles Let's say if you had this external angle this one this one Okay, so for this what will be the interior opposite angle? So for this interior opposite will be Y and W Y and W and let's say if you had This exterior angle. So for this one the interior opposite will be Z as and W right for this It was Y and W. So I hope you understood what is meant by exterior angle and interior opposite angles now What is to be proven to prove? to prove You have to prove that X is equal to Y plus Z X equals to Y plus Z. Let's see how to prove. What do we know basically? So we know there's a triangle so and there are angles involved So one thing is very clear that angle some property would be used here So can I not say that X plus Y not X plus Y? Sorry We have to say that Y plus Z Plus W is how much this is 180 degrees, isn't it angle some property? So let's write one and this is because of angle angle some Property of a triangle property of a Triangle right angle some property of a triangle is 180 degrees now also Also, can we not write that W plus X or we can write W W plus X W plus X is 180 degrees Right. This is equation number two Right. So from here, what can you say? You can say W is equal to 180 degrees Minus X. Let's say equation number three and now what you do from One and three what I'm going to do is I'm going to substitute W for this given relationship So from one one is Y plus Z and instead of this W. I will write 180 degrees 180 degrees minus X and The RHS of one was 180 degrees, right? So what did I do? Basically? I replace this W By this W 180 minus X. Okay. I hope this is clear So if you simplify, what will you get? Y plus Z minus X Plus 180 degrees is equal to 180 degrees. So clearly you can cancel these two 180 degrees the same So what do I get? I get Y plus Z is equal to X This is what is the external angle theorem Okay, similarly, you can establish that angle Let's say angle this one. Let me call this as X dash So you can say X dash will be equal to Y plus W Y plus W and let's say this one is X double dash. This one is X double dash. This angle is X double dash So similarly, you can say X double dash is equal to Z plus W. So all of these are true all of these are true Another important aspect to this theorem is if you see since let me write it here since since X is equal to Y plus Z then clearly X is greater than Y and X is greater than Z because X Y and Z all are positive quantities So X is greater than Y and X is greater than Z, right? That means what do I infer we infer that Exterior angle. So let me write it exterior angle exterior angle off of a triangle off a triangle is greater than greater than then either off either off Interior interior Opposite interior opposite Angles right so this is another very important Fact about this exterior So what did we learn in this we learned that Exterior angle is equal to sum of interior opposite angles and Exterior angle is always greater than either of the interior opposite angles and there are six exterior angles So this theorem holds for each one of them. There's no particular Exterior angle for which it holds it holds for all of them