 I'm Zor. Welcome to the user education. I will continue talking about triangles in particular about exterior angles of the triangle. Well, first of all, what is exterior angle? Well, consider you have a triangle and these angles are for obvious reason called interior angles. Well, exterior angle is something different. Consider you are extending one of the sides beyond the vertex in some direction. Then the continuation of this side with one other side from the same vertex actually makes an angle which is called exterior. Now, if I will extend this particular side, this is also exterior angle. By the way, equal to the first one because they are vertical. Now, similarly, I can extend towards this direction beyond this vertex and have this angle as an exterior one or extend this particular side and consider this angle as exterior. Finally, we have two other continuations and two other exterior angles, this and this. So, as you see, we have six different exterior angles, and they're making pairs actually of equal to each other, congruent angles because they are vertical. Now, let's talk about one particular property of the exterior angle and let me just leave one of those. Six little below six. So, let's say we have a triangle called ABC and we have extended line AC side AC beyond vertex C to equal to G. And we consider exterior angle BCG. Now, the theorem about the exterior angle. So, the theorem is that any exterior angle is greater than any of the interiors, not supplemental to it. So, the supplemental for this exterior is this. So, we are not considering this supplemental interior angle. We are considering two other angles and the theorem states that any one of those is smaller than this. All right, now let's talk about proving for this. Since we are trying to be as rigorous as possible in this course, I will present a proof which is actually first time presented by Euclid in one of his books, 300 Years BC. Now, first let me just explain how the whole logic actually goes. And then I will comment on it. So, here is what Euclid has offered as a proof of this statement. Well, let's take the middle point of BC and connect with A and go further. This is E to point F for a distance equal to AE and connect these two. Now, Euclid suggested to consider these two triangles. This one and this one. So, the statement is they are equal because BE equals to EC since E is the middle point of BC. So, these two sides of these two triangles are congruent segments. Now, since AE and EF also are segments of the same size, because we have made this, that's the way how we constructed the point F. We measure this distance AE and the continuation of AE just put the F in the same distance. So, this is also congruent segments, sides. And obviously, this angle is, these angles are vertical and that's why they are equal. So, both triangles are congruent to each other by side angle, side axle. Fine, this is done, no problem with that. Next statement is because these two triangles are equal, congruent, I'm always mixing equal, equal to congruent. Anyway, because they are congruent, we can say that the corresponding angles are also congruent. And corresponding to this angle B is this angle, angle ACF. So, instead of proving that the exterior angle ECG is greater than angle ABC, we will prove that it's bigger than ECF, this angle, which is equal to this. How do we do that? Well, here is where logic offered by Euclid might have certain questions. He said the following. Since the point F lies inside the angle BCD, ray CF makes an angle with CB part of the entire angle BCD or BCD. So, this angle is part of this angle. And that's why, since it's part of the exterior angle, it's smaller and that's why the exterior angle is bigger than B, which was supposed to be proven. Well, this logic is not like exactly 100% reverse because we have to really talk about what does it mean for a point to be inside the angle. We also have to elaborate on the question of a part being smaller than the whole. This particular angle, even if it's a part, why is it smaller than the bigger exterior angle? But I believe that these are very difficult nuances in logic, which I probably would not address right now. Euclid did not address them. And it's probably sufficient to just take this explanation as it is without getting very strict on rigorousness of these concepts of point being inside the angle. We kind of understand what it is. And I would like you, with me, to give Euclid for this more intuitive rather than 100% rigorous proof. But it's an interesting kind of standpoint and it probably would be interesting to put the whole proof on a more rigorous foundation. But that is not actually our purpose. Since our purpose is to exercise the brain, this is enough for an exercise. So we have proven, almost proven, that exterior angle ECG is greater than ABC interior angle, not supplemental, is it? Now similarly, we can prove that this angle, BAC, is also smaller. To this purpose, we actually have to draw a slightly different picture and instead of considering this exterior angle, we can consider this exterior angle and then compare it with A and build slightly different triangles. Instead of going this way, we will go from B through the center of AC to this point. That would be point F. So anyway, the proof will be exactly the same in this particular case. And we will prove that this angle is equal to this one, which is part of the exterior angle. Exactly the same type of watch. So that's why exterior angle, either this or this, and they are equal to each other, exterior angle is always bigger than any interior not supplemental to it. Basically, it's a very, very short and very simple theorem. It just requires one additional construction drawing. However, what's interesting is that this particular theorem has a lot of very important implications. And for instance, one of the simplest implications is that in the right triangle, both angles, which are not the right angle, both non-right angles are acute. Why? Because exterior angle is also 90 degree and it's bigger than these guys. That's why they are smaller than 90 degree. Same thing if it's not a right triangle but a choose triangle. Again, this is, since it's a 2's, this angle is greater than 90 degree. So this one is smaller than 90 degree, but still they are even smaller. That's why they're also acute. So these are just two little consequences. Well, that's it for this little theorem. I would like to mention again Unisor.com as the site where parents might find it some very useful techniques for supervising education of their children. They can enroll them in certain lessons or programs or lectures and have them take exams and check the scores and even participate in like passing or failing this or that particular educational material. So I do recommend to go to the site. It has lots of very interesting material. Well, that's it for today. Thank you very much.