 Hi, I'm Zor. Welcome to Unisor Education. I would like to talk about angles between planes. Well, obviously I would like you to watch this lecture from Unisor.com because the lecture has notes on this website. And it's very useful to read the notes. Sometimes they are even more detailed than the lecture itself. Plus, it's just a different kind of view. One thing is you're watching the lecture, another is you're reading the textbook, which notes basically, you know, they are really a textbook. Alright, so we will talk about, we continue talking about angles. And in this particular lecture, I would like to talk about angles formed by two parallel planes and another plane which intersects it. So first we will talk about terminology. Well, actually it's a lot of terminology there, so let's just be patient about it. Let's assume that two parallel planes are horizontal, like this. And I can try to draw them in some way. This is one plane and this is another plane. This is sigma and this is tau. Now we will have the plane which intersects them. And the view will be like this. So it intersects here and here. Now these are lines of intersection of this plane. That's called gamma. Now the lines are obviously divided completely. So we have many dihedral angles here. For instance, angle between this half a plane and this half a plane. Or this half a plane and this one. Or this one and this. Or this one and this. I mean there are many different angles. And I'm going to define all of them very similarly to the plane geometry situation when you have two parallel lines and the line which intersects them. So terminology is very similar. Actually it's identical. So the parallel planes are parallel planes and they correspond to parallel lines in the plane geometry. The plane which intersects both of these is called transversal. And that's exactly the same as in the plane geometry when this line is transversal. Now the angles. Remember these angles are vertical. So in this particular case it's exactly the same thing. Let's make a slightly different view basically. If you view really horizontally then these two parallel planes would be viewed as lines. And this plane if you view again from this direction would look like this one. So we will just see the edges of these planes. So it's very easy for me right now to define the angles between them. Now this line of intersection well basically it's... If you look from this you just see the point where it intersected. And here the same thing. So let me put some letters which describe half planes. This is invisible right? Okay now this line which divides plane sigma. This line is an intersection of gamma and sigma. Now we will call this sigma L for left and this is sigma right for right. Why? Because in this perspective from this view it would be on the left and on the right. This will be sigma left and this will be sigma right. Same thing with the tau. This would be tau left and this would be tau right. Tau left and tau and tau right. Now gamma will have basically two lines. This one and this line which divides in different parts. Now if I will assume this line as a division then I will have this gamma as a gamma up and this one will be gamma down. But in this particular line if this particular line divides the plane gamma I will call this piece of gamma gamma up and the bottom line will be gamma down. So based on the context, based on which edge, which line of intersection I will consider I will use the same gamma up and gamma down but mean different things. If this is the division this will be up and this will be down. If this is the division then this will be up and this is down. So let me just symbolize one of the angles. For instance the vertical angle this one which is from this plane half plane to this half plane. Now that would be sigma left. Let's call this line S and this line T. S and gamma up. So this is this angle. It's vertical with this one. So it's vertical with this angle and sigma right S and gamma down. So these angles are vertical. Left sigma S, gamma up and right S, gamma down. These two are vertical. As well as these are vertical and these are vertical and these are vertical. They are all vertical. And all the names I put in the notes I don't want right now to spend time but you understand what it is. So this is basically the definition of the vertical angles and this definition follows exactly the same logic as in the plane geometry. Now what else do we have? We have corresponding angles. Now corresponding angles in the plane geometry remember it's this one and this one. They're on one side and both let's say the upper part. So in this particular terminology it would be this angle and this angle. So it's let's say sigma left S, gamma up angle and on the tau side angle tau left T and gamma up. So that would be my corresponding angles. Then what else? Alternate interior. This angle and this angle. They are alternate on both sides of the transversal and they're both internal. Again I can put some letters into it. Now in this particular case this angle would be gamma left S, sorry, sigma left S, gamma down, left S down and alternate would be this one. It would be tau right T and gamma up. Something like this. These two angles are corresponding. Sorry, alternate interior. Now the alternate exterior are these opposite angles. In this case it's this one and this one. And what else? Consecutive interior. These are two angles on the same side of the gamma. They are supplemental. Together they are making 180 degree. So these are consecutive interior angles. Now these are all terminology. It's just definitions. So what happens if you have two parallel lines, planes, and one transversal that intersects them all. Now the obvious continuation of this discussion is well in the plane geometry we have all these wonderful theorems and properties like vertical angles are equal to each other. The alternate interior are equal, corresponding are equal and consecutive internal are supplemental to each other. It's exactly the same in the solid geometry. When you are replacing lines with planes and the line which intersects the lines would be replaced with plane which intersects the planes. All the dihedral angles which we were talking about right now have exactly the same properties. So two vertical dihedral angles are equal to each other. Two, let's say, alternate interior dihedral angles are congruent to each other, etc., etc. How can we prove it? Well let's just recall that any dihedral angle can be represented as it's corresponding linear angle. So let me go back to my previous lecture where I introduced the linear angle concept for dihedral angles. Actually, I don't need this. So if you have a particular dihedral angle, something like this, then a plane which perpendicular to the edge, the plane which goes something like this, it intersects here. This is invisible and this is visible. This is invisible, too. So it intersects this line and this. So the corresponding linear angle, which is angle between these two lines of intersection of the perpendicular to the edge, this angle represents the dihedral angle basically one to one. Equal dihedral angles correspond to equal linear angles and no matter where you position this plane, the linear angle will be exactly the same, etc. So now let's go back to our situation with two parallel planes and transversal. What I would like to do, and it's very difficult to draw, but I will try, is I would like to replace all these dihedral angles which occur in this situation with two parallel and one transversal planes. I would like to replace it with linear angles, corresponding linear angles. Here is how I would like to do it. Now I have a really challenging task of drawing. So again, two parallel planes and let's say you intersect them something like this. Actually, it would be better if I will slant it a little bit because it's not necessarily perpendicular and I draw it as a perpendicular. So that's not good. Slanted would be something like this. Okay, and this plane would be... Let me continue this. Okay, so this is my sigma, this is my tau, and this is my gamma. This is line S, this is line T. By the way, lines S and T are parallel to each other. Well, there was a theorem actually. If you have two parallel planes and one plane which intersects them both, because of intersection R parallel, we have already proven that. So S and T are parallel. Now, what does it mean? Well, let's consider what kind of edges do we have here. Well, we do have... This is an edge, right? And this is an edge. So basically... One second. Okay. So basically, when we are talking about dihedral angles, so this dihedral angle or this dihedral angle, these are edges. Now, to go to the linear angles, I have to have a plane which is perpendicular to the edge. Remember, this is the edge, and I have to have a plane perpendicular to the edge. So basically, there is a plane which I can imagine that I draw a plane which is perpendicular to this one. So it cuts basically this and cuts this plane and it cuts this and it cuts this plane. This is my plane which is perpendicular. And these are linear angles. In this case, I just draw the corresponding angles. But at the same time, I can use these as vertical angles, etc. So I'm replacing, basically, by constructing the plane which is perpendicular to this edge, which happens to be perpendicular to this one as well, because these are parallel lines. So I'm replacing every dihedral angle with the corresponding linear angle. And since all these theorems about linear angles are exactly as we know from the plane geometry, because now we can consider everything within this plane which is perpendicular to the edges. Now, within this plane, every dihedral angle is represented by the corresponding linear. So we are dealing with plane geometry here. So linear angles are basically reducing our three-dimensional problem into two-dimensional, which we have already learned that all these properties of the vertical and corresponding angles and alternate interior, etc. We know that. So by doing this, I'm actually proving the fact that whatever we know about all these angles plane geometry, corresponding vertical, alternate interior, etc., everything, all of these properties are transformed into the corresponding properties of the dihedral angles. So that's basically the properties which I wanted to mention in this lecture. And basically, that's it. That's all I wanted to do. So I'm just trying to introduce this concept of all these angles which correspond to the corresponding plane angles and have exactly the same properties. Well, that's it for today. I would recommend you to read the notes for this lecture, which contain basically the same material, and if I'm talking, let's say, about vertical angles, notes list all the vertical notes on this picture, etc. Well, basically, that's it. Thank you very much and good luck.