 Hi, I'm Zor. Welcome to Unisor Education. I will talk about certain very simple theorems, mini theorems as I call them, related to parallel lines. In the lecture which is related to the parallel lines, basically I was proving one and very, very important theorem in many different incarnations. It's a theorem about two parallel lines and transversal and basically stating that the characteristic property of the parallel lines, if there is such a situation, is that either alternate interior angles are equal or corresponding angles are equal or sum of one-sided interior or exterior is equal to 180 degrees, etc. And every theorem of this type has the corresponding converse theorem, so basically that would make actually the angles, the congruence of angles, a characteristic property of parallelism of the lines. They are always going together. It's a necessary and sufficient condition to have these angles congruent for the lines to be parallel. So I have a few problems, little problems, mini theorems as I call them, which I'm going through right now. They are in the notes. I do encourage you to try to prove them yourselves. They're very simple. And here I will just do it one by one. Okay, a perpendicular to one of two parallel lines is a perpendicular to another one. All right, so if you have two different parallel lines and you have a perpendicular to one of them, then it's perpendicular to another. Obviously, it follows from the theorem which I was talking about before. These are exterior alternate angles and since the lines are parallel, the perpendicular can be considered as a transversal. That's why these angles are congruent to each other. If this one is the right angle, then this one is the right angle. And the story, two perpendicular to the same line are parallel to each other. Very similar. If you have two perpendicular to the same line, then these two perpendicular should be considered as two lines with this line as a transversal and since these are corresponding angles, in this case, and the congruence of the corresponding angles is a characteristic property of the parallel lines, that's why these two lines are parallel among themselves. And the story, very simple theorem, as you see. You have to use this quality of parallel lines and transversal relatively freely and you should be ready to apply it to any simple problems. Okay, perpendicular and non-perpendicular is the same line. Intercept, not parallel. All right, so these are perpendicular and non-perpendicular to the same line. They must intersect. Well, consider they're not intersecting. Well, what does it mean? It means they're parallel. If they are parallel, then the corresponding angles must be congruent, which means these two angles must be congruent to each other. This one is right, that's why this one is right, which contradicts the premise of the theorem that this is not a perpendicular. Okay, next one. Two angles with correspondingly parallel lines besides are congruent or supplementary to each other. All right, so you have one angle and another angle with parallel lines. Now, this theorem states that these two angles are either congruent or supplemental to each other. Now, what I will do is the following. I will continue in both directions all four rays which make these two angles. So, these two angles, now which is obvious, that these two angles, actually these four angles must be congruent to each other. Why? Because this and this are parallel. Considering this one as a transversal, you have the corresponding angles. Now, if these two are considered to be parallel and this one as a transversal, then these two angles are corresponding and must be congruent. Similarly, these two lines and this as a transversal and these two angles are corresponding and congruent to each other. And these are vertical to all these congruent angles. And what are these guys? These are all the supplementary angles to these one. So, double-arced, all double-arced angles are supplementary to single-arced angles. So, basically we have only two angles, single-arced and double-arced angles. All other angles are basically congruent to one of these. So, among 1, 2, 3, 4, 8, 12, 16, 16 different angles which we have when we have two pairs of parallel lines, they are all either congruent to each other or supplemental to each other. There are only two basically different types of angles here. This one and this one. Everything else is congruent. So, that's why if you have just two angles, let me go back to what we have started with. You have this angle and this angle with parallel sides. Then, obviously, it's either this angle or that angle. So, in this case, you have these two congruent. In this case, obviously, you also have parallel sides. So, in this case, you have angles supplementary to each other. This one and this one. All right, that proves the point. Next, two angles corresponding with perpendicular sides are congruent or supplementary to each other. Now, very similar, but the proof is supposed to be somehow different. So, you have perpendicular and this one perpendicular. So, this perpendicular to this and this perpendicular to this. So, the statement of the theorem is that these two angles are either supplementary to each other or congruent to each other. Obviously, if you have this angle, then this will be congruent and this one will be supplementary on this particular drawing. Now, how to prove it? Okay, the best way is to use this vertex as an origin of an angle congruent to this one. So, how can we do it? Well, construct the line from this line from this point parallel to this and another line parallel to this one. So, obviously, these two angles are congruent as we have proven just before in the previous theorem or if we made a different step and consider this angle, which is also parallel to this, then we have to prove that these two are congruent. So, basically, what we have done is we have moved this angle to the common vertex with the first angle. Now, does it make our life easier? Well, just a little bit. Now, we know that this is a perpendicular and this is perpendicular, right? So, in this particular case, it is obvious that if we will make another turn of the picture, rotate the picture by 90 degrees, not the entire picture, just this particular angle. So, what would happen? Let me just make a continuation of these two lines to make it kind of easier. So, what happens if we will rotate these two lines by 90 degrees? Well, obviously, since these two lines, this one and this one, are perpendicular to these two lines and they will turn it by 90 degrees, then this will go here and this will go here and they will basically coincide. And depending on which side of the angle you take, it will be either congruence or supplementary between these two angles. So, we have proven this through the property of invariant transformation, the transformation which doesn't really change the magnitude of the angle. So, that's basically how congruence is considered in geometry. What are the congruent geometrical objects? Those which can be brought from one to another with non-deforming transformation, which are parallel shift and rotation and sometimes symmetry. So, that's how we did it in this particular case. So, angles with perpendicular sides are either congruent or supplementary to each other. The sum of three angles of any triangle is equal to two right angles with 180 degrees. Well, this is a very known fact to all students of geometry and the proof is basically quite elementary. We draw a line parallel to the base and obviously, since these are parallel then interior alternate interior angles are congruent and this one is congruent to this one. So, the sum of one, two, three interior angles of a triangle are equal to one, two, three which constitute 180 degrees since this is a straight line. Very simple. Any exterior angle of a triangle is equal by measure to a sum of two interior angles not supplemental with it. Okay, we do know, by the way, from one of the previous theorems that every exterior angle is greater than any interior not supplemental with it. Now, it's not just greater, it's equal exactly to the sum of these two. Now, how can we prove it? Well, let me think how can we prove it? What if we will draw a line parallel to this one? So, what do we have now? Now, this angle is same as this. This angle is the same as this. Basically, that's how we proved that the sum of triangles is 180 degrees. Now, let's consider this angle which is exterior angle. Now, this exterior angle obviously has this property because these two angles are vertical. And now it's obvious that this exterior angle consists of this one which is equal to this and this one which is congruent to this one. And that's why it's obvious that it's equal to sum of these. One exterior angle is equal to sum of two interior, not congruent, not supplemental to it. Okay? So, all it takes is just to come up with some additional drawing, additional line to make the whole thing quite obvious. If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent. Well, this doesn't even require a drawing because if the sum of all three angles is 180 degrees, regardless of what triangle this is, and two angles of one triangle are equal to two angles of another, then the third one is also supposed to be congruent because the sum of three is supposed to be the same in both cases which is 180 degrees. Okay, sum of two acute angles of any right triangle is equal to 90 degrees. Obviously, again, the same thing. Since the sum is 180 of all three angles, this one is 90, so these two should sum up to 90 degrees. In the right isosceles triangle, both acute angles are 45 degrees. Well, okay, if it's right isosceles triangle, which means two legs are equal in size, but that means that these two angles are congruent to each other because this is an isosceles triangle. So they are in sum 90 degrees, so each one of them is supposed to be 45 degrees, obviously, since they are equal to each other. Okay, in the equilateral triangle, all angles are 60 degrees. Okay, now we have equilateral triangle when all three sides are congruent to each other. Well, let's draw a median. Now, we know that in the isosceles triangle, a median and an angle bisector and the altitude are one and the same. So in this particular case, we have a situation when we have two different triangles, right triangles, by the way. So each one has a 180 degree if you will summarize these three angles, which means this and this in sum are 90 degrees too. So let's call this x. Now, since this angle is supposed to be equal to this one, well, actually, I don't even need this. Yeah, I'm sorry. I'm complicating the issue, which doesn't really need to be complicated. I already have, since it's equilateral triangle, then all three angles are the same, are congruent to each other. And since the sum is supposed to be 180 degrees, then obviously each one is 60 degrees. Now, why they are equal? Well, again, because it's isosceles from any side, which means this is equal to this, because these two sides are equal. This is equal to this, because these sides are equal, etc. So basically all of these angles are congruent to each other. Sum is 180 degrees, and that's why each one is 60. Okay, that's it for this set of mini theorems. I hope I wasn't too quick with all these theorems, but they are really trivial. I do encourage you to look at the Unisor.com webpage, which contains lots of educational material, including obviously these problems, these mini theorems. And what's very important, it's very useful for parents who would like to supervise the educational process of their children, since the website contains exams, and every exam is scored in somewhere or another, so the parents can actually check the score and every exam what's the maximum score, what's the score of my student, and basically make a decision to pass or fail any particular enrollment which was done for this particular student. That's it. Good luck. Thank you very much.