 Hi, I'm Zor. Welcome to Unisor education. I would like to continue talking about solid geometry and in particular lines and planes and their correspondence and different objects you can make from them. This lecture is part of the advanced math course for teenagers presented on Unisor.com which you actually are encouraged to watch from this website because the site contains not only the reference to the lecture itself but also notes which can be used as a textbook. Alright, so we are talking about angles between the planes today. So I have this little plan which I'm going to go through one item after another. Mostly this is terminology and very little properties. I would like just to introduce you to a concept of angles between the planes. Now, obviously I assume that angles between the lines is part of the plane geometry which you know. Alright, so angles between the planes. We will start with, well, let me just make a little reference back. Remember when we were talking about angles between the lines, it's actually better if you take a point and have two half lines and then the angle between them. Why? Because if you have complete lines there are too many angles here, right? So we will do the same thing with planes. So I will start with introduction to a concept of a half plane. Well, let's say the plane is the surface of this board. Now, if you draw a straight line it divides the plane in two halves. Well, you can call it left and right or depending on what the orientation etc. Well, in this case I will use left and right. So the left half plane is this part and I usually include the line itself also in the half plane. Now, let's say my plane is called sigma. So my left half plane would be sigma 1 and the right one will be sigma 2. And this is the line D which divides these. So what I can say is that sigma 1 union with sigma 2 is an entire plane sigma. And sigma 1 intersection with sigma 2 is my line D. So this is where they intersect because this line is included in both half planes. And if you unite them together you will get an entire plane. So this is the half plane. This is the concept of a half plane. Now, fine, this is done. But remember angle between the lines is you need two lines, right? Or two half lines. In this case we need two half planes to introduce an angle. So let's assume we have the following situation. We have one half plane and another half plane. So this one would be sigma and this would be tau. And what's important is that these two half planes have one common line which basically was the one which we used to form the half plane. Remember when the whole plane we just draw a line which separates it in two halves. So this line is called actually the edge of the half plane. So these two half planes sigma and tau have common edge. We call it line D. Well, the combination of these two half planes is basically an angle between two planes which is called dihedral angle. Which means from obviously to some ancient language translated as the angle between two planes. So this is just the concept. Now, what can we say about these two things? That the combination of two half planes with a common edge is a dihedral angle. And you can actually use the symbolic like something like you remember when you were talking about angles. Let's say you had angles, you had point B, A, C. The symbolics would be A, B, C. This is the angle. So I'm trying to be more or less in the same way. And I can say that symbolically this particular angle can be written as sigma d tau. Well, I mean as well as anything else. We can obviously put some letters and use the letters. But in this case it's more I think symbolically correct because it's half plane edge and another half plane. So that's basically the definition of the dihedral angle. So this is the edge and these are two faces. So this is the face and this is the face. So two faces and the edge basically form the dihedral angle. Okay, now this is the definition. Now let's talk about the way how we can compare and measure dihedral angles. Here is what I suggest. Let's assume that you have another plane which is perpendicular to the edge. I think that's the proper way of drawing it. Something like this. No, actually this is right. This is solid. So these are basically... Okay, this is a plane gamma which is perpendicular to the edge at this point. And obviously... Actually this is also solid. And is that the right? Yeah, I think this is right. Well, so the plane gamma is perpendicular to the edge D at point, let's call it A. And obviously it intersects both sigma and tau. Let's call this line S and this line T. Now the angle which is formed by lines S and T of the intersection of the plane gamma perpendicular to the edge and corresponding faces of the dihedral angle also forms some kind of an angle. But in this case it's a plane angle because these are two lines. They are intersecting at point A so obviously they belong to the plane gamma. And they form certain angle. And this angle is called a linear angle of the dihedral angle given before. Now, what's interesting is that since plane gamma is perpendicular to edge D edge D would be perpendicular to any line on the plane gamma which passes through the point of intersection A which means D is perpendicular to S and D is perpendicular to T. So this is the perpendicular and this is the perpendicular. So D is perpendicular to both of them. Now, what if we will have another plane gamma which is intersecting and perpendicular to the edge? Let's say above it or below it. Well, obviously two planes perpendicular to the same line D must be parallel. We have already learned that. So this new plane gamma let's say delta. It will be just parallel. Now, will the linear angle formed by plane delta and faces of this dihedral angle be different? Well, the answer is no. Why? Very simple reason. If you have another plane let's put it a little bit above this and it will intersect the same edge D a little bit higher. Let's say at point B. Now, this plane will intersect face tau along this line and face sigma along this line and these two will be parallel to these two because this line and this both belong to the plane tau and both are perpendicular to the D which means they are parallel. Now, these two lines both belong to the plane sigma and both perpendicular to the D which means they are parallel and two angles with corresponding with parallel sides are congruent. So it will be exactly the same angle which means that given the dihedral angle you completely determine the value of the corresponding linear angle because its value doesn't really depend on where exactly you put this plane gamma perpendicular to D. Now, again, if you have already given to you dihedral angle then the linear corresponding linear angle is completely defined. It's one and only. Now, how about in reverse? For instance, you have a linear angle. Does it define the dihedral? Well, let's put it this way. If you have a linear angle you can always draw a plane through these two lines S and T which is gamma. You can always put a perpendicular one and only gamma and one and only perpendicular to both S and T or perpendicular to the entire plane gamma actually at point of intersection of these two lines. So D is uniquely defined and then you have a line D and line T and these two lines intersecting obviously perpendicular to each other they define the plane tau and S and D define the plane gamma. So again, if you have a linear angle defined by S and T you can completely rebuild back the dihedral angle. So they both uniquely define each other. Dihedral angle has one and only linear angle and linear angle has one and only dihedral angle and both can be effectively constructed from one or another. Now, what does it mean actually? Well, it means that equality or congruence if you wish between dihedral angles can be completely reduced to the equality or congruence between the corresponding linear angles and the other way around congruence or equality between linear angles corresponds to the dihedral angles, linear congruence or equality which means we can actually measure the dihedral angle in terms of corresponding linear angles. For instance, we can say that a particular dihedral angle is acute. What does it mean? Well, it means that the corresponding linear angle is acute or that the two half planes are perpendicular to each other. Well, it means actually that the corresponding linear angles are perpendicular to each other or the angle is of two. So basically I would like to consider that linear angles are a complete equivalent basically of the dihedral angles. To learn something about the dihedral angles it's sufficient actually to do it for corresponding linear angles and then the story will be completed. Okay, what else? So I talked about measure. Okay, one more thing. Let's talk about measurement between two lines, two cat lines actually, two rays. Let's say you have two rays S and T. How many angles do they form? Well, four. Well, this is one angle. Now, this is another angle. And then each angle can be bidirectional. There is a positive and there is a negative direction of the angle measurement. Remember, counterclockwise would be positive direction and clockwise would be negative direction. It's kind of complex. And for dihedral angles, I mean obviously we can use exactly the same thing. Considering the directions and both angles from T to S or from S to T. But traditionally we will just simplify this issue. We will always consider angle between two half planes as being less than 100 and 80 degrees. So we will always, so we will not consider this situation. Only cuter perpendicular, no, sorry, only less than 180 degrees angles. And then we will probably disregard the direction of the positive or negative counterclockwise, clockwise, etc. In three dimensions, it doesn't really make much sense. So we will always consider these angles as positive, some positive number. And measured in degrees, it will be always less than or equal to 180. And the corresponding measure is the measure between these two rays, which are linear angle of this dihedral angle. Well, that's basically it. That's my introduction to what is dihedral angle, what's linear angle of the dihedral angle and their correspondence to each other. Now you know which two planes or half planes are forming the acute angle or obtuse angle or perpendicular to each other. It's all measured by the corresponding linear angle. Okay, that's it for today. The properties and some theorems will be in the next lecture. I do encourage you to read the notes to this lecture. It introduces you to some symbolics because I like to use mathematical symbols. And also what's very important, if you are a registered student on Unisor.com, then you can actually have enrollment in certain courses, certain topics, etc. And then what's very important is you can take exams. Exams are free, so basically you can take it as much as you want until you get the perfect score. It's very important because the solving problems and passing some tests and exams is the purpose of the entire course. Now that's it for today and thanks. Good luck to you.