 In this video where you go into state and prove the exterior angle theorem as a theorem of congruence geometries But let's first make sure we understand the vocabulary behind what is a what's an remote interior angle What's exterior angles of triangle and such so imagine we have three points a b and c Which these are three distinct points which are not co-linear Therefore they make a triangle so we have angle a angle b and angle c so we have this triangle in mind And so this definition will make sense I should say in an order to geometry the exterior angle theorem will be a theorem of congruence geometry Because it'll have to say it says something about the relative size of these angles We need a notion of congruence to say something about that But the definition of an exterior angle and interior angles make sense in any order geometry Okay, and so let me actually switch this around just to make it easier to see on the screen I'm gonna call this point b and this point c and the reason for that is I want to extend the line a C so that there's a new point d that's over here again This is I just moved the point so it fits better on the screen here So the angle the angle d a b which is this angle right here This is what we refer to as an exterior angle to the triangle So I want to say there's a lot of exterior angles to a triangle The idea is you extend a line and you look at this one right here These two exterior angles are vertical angles and therefore are congruent to each other So if we refer if we refer to the exterior angle associated to the vertex a it doesn't matter Which will we talk about because as as vertical angles are congruent? It's unique up to congruence same thing over here. We can talk about the exterior angle associated to vertex b We can talk about the exterior angle associated to vertex c and again. This is unique up to Congruence which one we're referring to so we have this exterior angle The angle d a b it's it's exterior relative to the angle a that's part of the triangle And so then every triangle of course has three interior angles you see am I here now? Since this exterior angle is connected to one of the interior angles of the triangle the other two exterior Interior angles excuse me or what we refer to as the remote interior angles So the angle a b c is a remote interior angle and the angle b c a is also a remote interior angle So the exterior angle theorem of a triangle will make a connection between this exterior angle d a b with the remote interior angles a b c and b c a So let's make that statement explicit right here in a congruence geometry So we do need to have the actions of incidence between this and congruence here In an x in a congruence geometry an exterior angle of a triangle is always greater than each of its remote interior angles So remember we we can make sense on what it means to say that an angle Is greater than another angle right so just as a reminder if we had two angles And if we can translate one of those angles inside of the other one kind of like this Then of course we say that the angle that got translated inside of the other Is smaller than that so again this idea of ordering of angles was something we can talk about in in a in an order to geometry but The proof of this statement is going to use the alternate here angle theorem And that's why the a congruence geometry is the necessary place We need to be in for the exterior angle there even though this statement makes sense for any order geometry We can only guarantee it's true in a congruence geometry because of the alternate here angle theorem Okay, so let's redraw the picture that we have on the screen a moment ago So we're gonna have the the line and a triangle and some exterior angles like so So we'll call this point right here a we're gonna call this point over here D This point right here C and this point right here B So again the exact same picture we saw a moment ago, and so the point a is between the points C and D and the angle Dab is an exterior angle to the triangle All right, so we're gonna first show that the angle ABC which we're just going to call it angle B for short because after all when you look at this diagram There's no ambiguity in what angle B is going to be so the first thing we want to show is that angle ABC Aka angle B is actually smaller than the angle Dab now to do that To be smaller than it. There's actually three possibilities right given any two angles The first one could be smaller than the second They could both be congruent to each other or the first could be larger than the second So to prove that angle B is less than angle Dab the exterior angle We're going to then consider the other possibilities We're going to consider well, what if the angle B is congruent to the exterior angle and we'll also consider the possibility where the angle B is Larger than the angle Dab so we'll consider the two other possibilities and derive contradictions therefore By process of elimination you'll have to conclude that angle B is less than the exterior angle That's gonna be our plot that we go with going on with this proof right here so note that Angle B and the exterior angle Dab are alternate interior angles to the line AC so I want you to think about it for a little bit. We're going to try to extend the lines to make it a little bit more apparent So we extend the line AB in that direction we can extend the line AB in that direction Okay, we're gonna extend the line BC in that direction we're gonna send the line like so So we've extended each of these lines and so then come back to the picture. We're looking at we're assuming for the moment, right? That angle B is congruent to the angle the angle Dab like so and so with these lines in play here. We have this line BC We have this line AC the line AB transverses the two when we talked about transversals We didn't require they have to be parallel And so we have these congruent alternate here angles well by the alternate here angle theorem Because we have these congruent alternate here angles. That means the line AC must be parallel to the line That must be parallel to the line it says AB there it should be it should probably say BC that's a typo I'll fix that in just a second But clearly those two lines cannot be parallel to each other because they both intersect each other at the point C For which that then shows us that the first case is impossible the angle B cannot be congruent to the angle The angle B can't be congruent to the exterior angle Dab So like I said, I'm gonna fix this right here. That should be the line AC intersects CB It was corrected. I've needs to be corrected there as well C is on both of those points Both of those lines so okay that two angles can't be congruent to each other Well, the other possibility was what if B is Larger than the exterior angle because again, we're trying to show that B is less than the exterior angle in that situation Well, if B was larger than the exterior angle then that means there exists some point Let's call it F that lives inside the interior of the angle ABC Such that the angle ABF is congruent to the it's that's then congruent to the exterior angle D AB So F lives inside the interior of the angle ABC. We'll call this F And if we then extend this Line like so whoops, let me do it straight We get this ray and in fact actually want to go the other way around So we have this line BF right here And so we're assuming under this situation that the angle ABF is congruent to the angle D AB like so, but I want you to then consider the same basic picture we had before so the line AB is acting like a transversal to the line AC and this time to the line BF Like so for which then by the alternate to your angle again It's got to be that the line BF is parallel to the line AC As is written right here. Now, of course when you look at the diagram, it seems like that doesn't quite seem right Is there don't they intersect right here? Well, how do we know that point of intersections there? Now the ray BF is Interior to the angle ABC because F was an interior point So the ray emanating from the vertex of the angle to any interior point will also be interior So by the crossbar theorem since BF is an interior angle to ABC by the crossbar theorem The ray BF must intersect the line segment AC and this is going to happen at some point of intersection Call that point G like so therefore G is the intersection between the lines BF and the line AC Because after all of the ray the ray can be extended into the line BF and the segment AC can be extended into the line AC So G is the intersection of those two lines But they're supposed to be parallel by the alternate to your angle theorem which gives us a contradiction So it can't be you can't have that B is greater than DAB the exterior angle we also can't have that B is congruent to the exterior angle DAB So that leaves only one possibility The only possibility left because we have a total order on the angles in an order to geometry It has to be the case that angle B is less than the angle DAB For which angle B was one of the wrong interior angles. So that's that's what we want to prove of course We have to also consider the situation about angle C, right? But if we put some point E over here like we mentioned before the angle EAC is a vertical angle to the angle DAB So those two exterior angles are congruent to each other and by mimicking the argument We just did a moment ago. We can then argue that the remote interior angle C is less than the exterior angle EAC Thus finishing the proof of the exterior angle theorem That also brings us to the end of lecture 17 which in this lecture We prove the ultra interior angle theorem and then clearly in this video We prove the exterior angle theorem if you learned anything about these about interior and exterior angles in a congruent geometry Like these videos subscribe to the channel to see more More math videos like this in the future and if you have any questions post your comments below and I'll be glad to answer