 Hi, I'm Zor. Welcome to Unisor Education. Today's topic will be congruent figure in geometry. The ability to count the same properties in algebra is usually called equality. We are saying that two different numbers are equal. What does it actually mean? It means that these two objects, or generic objects in this case, have exactly the same properties. For instance, you can have two different equations, which are basically exactly the same equations after certain transformation. In geometry, we also have exactly the same kind of congruent figures. They are basically equivalent from the logical standpoint to equal objects in algebra, let's say. Sometimes I will probably use words equal just by accident. I really mean congruent, speaking about geometrical figures. So why do we actually talk about congruent geometrical objects? Well, basically it's just because to signify the properties of these two objects are exactly the same. If I will draw two different segments, this one and let's say this one, what does it mean that these two segments are congruent? Well, in case of segments, since it's such a simple geometrical object, it's just basically the length of these two segments is exactly the same. By the way, if I will have even more elementary objects, like two points, two points will always be congruent to each other, because actually there are no properties which we have to compare. As far as the segments are concerned, you can have a longer segment, or you have a shorter segment, so these are non-congrant. With points, you cannot have non-congrant points. So basically the more complex geometrical object, the more requirements we actually put into their congruence. Since we are talking about geometrical objects on the plane, we obviously can talk about circles being congruent to each other, or triangles being congruent to each other. Now, it's not simple actually to basically see that two different geometrical objects on the plane are congruent to each other. So how do we make sure that they are? Well, for this, we actually have to consider a certain process. The process during which one object is transformed into another. So there are certain transformations which we accept as a valid transformation, which if convert one object into another, basically signifies that these two objects are congruent. Now, it's not simple actually to see that these two triangles are congruent, unless we will come up with some kind of transformation. So the transformations which we are allowing to have to prove the congruence of two different objects basically are non-deforming transformations, which means that the lengths of all elements which constitute a particular object, in case of triangles, for instance, it's the lengths of all three segments which constitute the triangles. And all the angles which are angles between these two, between every two sides, remain the same. So that's what non-deforming transformations actually are. It means all lengths of all the segments and all angles between these segments. And if you are talking about circles, then radiuses of all arcs basically should remain the same in this non-deforming transformation. And just to be a little bit more precise, let's consider what kind of non-deforming transformations we can come up with. Well, the first non-deforming transformation is obviously a plane shift on the plane. Well, plane means simple. On the plane means on the plane. So the shift is basically when you shift to one particular direction, all points of one geometrical object into another location. And after that shift, if they coincide, which means all the points which one object takes belong actually to another object as well and vice versa, object number one points after this transformation are object number two's points and object number two's points are the same as object number one after the transformation. So this coinciding after the shift actually means that two different objects are congruent. Now, what other kind of transformations we can basically think about? Well, let me just give you a very simple example and you will realize what we are talking about. Here are two different segments of the same length. Now, what kind of transformation I should apply to this segment to get into this one, to coincide with this one? I can't apply any shift because no matter how shift actually is done, it will remain parallel to this and never perpendicular. So I can shift it into any location. It will not coincide with this. So we have to introduce yet another transformation which is deforming, which is non-deforming transformation, which will help us to bring these two segments together. Well, obviously it's a rotation, in this case by 9 to the group. So we can rotate the geometrical figure, whatever the geometrical figure is, around a certain point, which means every segment which connects this point to a corresponding point on that particular object will be turned on the same angle. So let's say it's 90 degrees, for instance, then this will turn into this, this will turn into this, and this will turn into this. And the whole object will be rotated. It's difficult for me to draw it, but whatever the object actually is in its rotated shape and form. But in case of a little bit simpler, a simpler geometrical figure, I can actually demonstrate it to you. Let's say you have one triangle which is this. It's easier with triangles. And then we will consider this triangle. Well, they look similar, but in order to transform one into another, you really have to turn this triangle by 90 degree counterclockwise. So this segment will turn into this one. Oh, I'm sorry. I think I made a mistake. It's not this one. So this will turn into this, and this will turn into this. Okay. So this is another triangle. The one which I drew the first was a little bit more complex transformation. But if you take this triangle and turn it around this point by 90 degree, then you will have this triangle. So this actually proves that these two triangles are congruent. So we have the shift or parallel shift. Sometimes it's called, we have a rotation. What else? Well, there is one more transformation which is very important, and it's called reflection. Well, obvious example is this. You have one triangle, and you have another triangle. Well, no matter how you turn this triangle around anything, you will not be able to turn it into this one. What you need is a reflection relative to certain axis. Now, what is a reflection? Well, you remember, in case of a rotation, we had to choose a point around which we will rotate, and then every segment will be turned by a certain angle. Well, here, instead of a point, we have to choose a line, which is called a line of reflection, and instead of connecting a point to every point which constitutes the object, we will have a perpendicular from every point on this triangle to a line of reflection. And every perpendicular is basically turned around, so this point goes into this, this point goes into this, and this point goes into this. So, basically, the whole triangle is flipped around. If you wish, you can consider it a rotation as well, but not around a fixed point on the plane, within the plane, but you really need a third dimension to rotate this geometrical figure around this axis, and you will get this particular reflection. So, we have three different types of non-deforming transformations. We have a shift, a parallel shift, we have a rotation, and we have a reflection. Now, let me just tell you a very interesting statement, which in one statement I would like to summarize basically how the whole geometry is built. So, what do we do in geometry? Usually, or in most of the cases, of course, or at least in many cases, what we usually do is we are trying to prove the congruence of two different geometrical figures based on certain properties. Now, congruent figures have all their properties exactly the same. Now, what if you don't know anything about all the properties, but you do know something about two or three particular properties of these two geometrical figures, and you know that these properties, let's say, the length of one side of the triangle is equal to the length of another side of the triangle. That's the property. So, there are certain properties which are sufficient. Do you remember logic, sufficient and necessary conditions? So, okay, so for congruence there are two different geometrical figures. Sometimes it is sufficient the equality or congruency of certain elements of these geometrical figures. Just as an example, going a little bit forward, two triangles, for instance, are congruent if two sides and an angle between them of one triangle are congruent to two sides and an angle of another triangle. So, these congruences, two sides and an angle in between. Let me just draw a little picture here. So, if this side is equal to this and this equals this and this angle equals this angle, these triangles are congruent. We will go into the details about the theorems about this, but anyway, in general, people who study geometry are saying that there are certain conditions which are sufficient for certain geometrical objects to be congruent. So, if only these properties are known about these two triangles, like this side equal to this and this to this and angle to angle, then everything, because they are congruent, then everything else is congruent or equal or whatever. So, lengths of this side will be equal to lengths of this side and this angle, a measure of this angle equals a measure of this angle, etc. So, these are all these geometrical theorems about you take certain number of properties, more or less, well, preferably a small number of properties and then you're proving that it's sufficient to having the whole geometrical figures completely congruent, which means all other properties will also satisfy congruences of other sides or other angles or radiuses or whatever else. So, many, many theorems in geometry are of this particular kind. So, let me just give you a couple of examples. I just gave you one that two triangles are congruent with two sides and an angle between them. Now, what about two circles? When two circles are congruent, well, obviously you have to have the same radius. So, if radius is the same, then you can prove that these two are congruent and you shift this to this and then every angle, every point on every angle will correspondingly be shifted to the corresponding point. So, basically because of certain equalages between certain geometrical figures you can prove that the whole circle is completely congruent. I mean, you can consider this parallelogram, etc., etc. So, all these theorems are about congruency based on certain properties. Now, what are the properties? For instance, you have two segments. When two segments are congruent, well, obviously the length will be sufficient because you can move this point to this, that will be a parallel shift. So, the whole thing will be in this. And then you will rotate one segment into another and because these lengths are the same, then the whole segment will coincide with this segment. So, we are coming up with certain transformations to basically show that there are certain sufficient conditions like lengths of these two segments sufficient to have them congruent. So, basically let me just summarize the congruency as a definition, basically, if you wish. So, two geometrical figures are congruent if there is a non-deforming transformation which can be a combination of parallel shift and rotation and reflection which turn one geometrical figure into another position which is completely coincides point by point with that other geometrical figure. So, that would include this little introduction into the concept of a congruence. We are not proving any theorems here. This is just an introduction to, I would say, a terminology or explanation of what really congruent figures are about. Thank you very much.