 So, one of the key concepts in geometry is the notion of a rigid transformation. And we can actually use this to define what geometry actually is. And the idea is, geometry in general is the study of properties that are invariant, that don't change under a predefined set of allowed transformations. So, the idea is that we have something, whatever it is, and this is something that will not change as long as we allow, use a set of allowed transformations of our geometrical objects. Now, the simplest type of geometry, plane geometry, these transformations fall into two primary categories. The first and more important one are the rigid transformations. And a rigid transformation is something that leaves both lengths and angles unchanged. These are also known as isometries. The other type of transformations are known as conformal transformations. And these are transformations that leave angles unchanged, but in the course of a conformal transformation, it's possible that a length might change. Now, if you think about what our definition of geometry is, properties invariant, then our rigid transformations will tell us that geometry, plane geometry, is at the very least going to study lengths and angles, because those are unchanged under our transformations, at least under the rigid transformations. Now, when we include conformal transformations, lengths might change, but then angles won't be. So, angles, very definitely part of plane geometry, because our transformations that were allowed, the rigid transformations, the conformal transformations, those leave angles unchanged. So, angles, very much part of plane geometry. Length's pretty much a solid part of plane geometry. We have to mess around with the conformal transformations a little bit later on, but certainly lengths will also be part of plane geometry. Now, what about those rigid transformations? So, there's a couple of different categories, and the important ones are the following. First off, we have a translation. So, the idea is that in a translation, what we're going to do is, we're going to take every point in a figure, and we're going to move it a specified distance in a specified direction. So, for example, I'm going to take my figure, and I'm going to move it one unit vertically. And so, again, here's my specified distance, one unit, whatever that unit happens to be, one foot, one inch, one mile, whatever, and I'm going to move it a specified distance, and I'm going to move it in a specified direction, in this case vertically. So, let's go ahead and do that. So, I have my figure, I'm going to move every point one unit vertically. I'm going to take the entire figure, I'm going to move it one unit vertically, and I may end up with a transformed figure that looks like that. Another important type of rigid transformation, an isometry, is a rotation. So, in a rotation, every point in a figure is going to be rotated as a specific amount around a specified point. So, for example, rotate the figure shown a quarter turn clockwise around the point shown. So, here's my point, there's my specified point, and my rotation of a specific amount that's a quarter turn clockwise around the point shown. Now, this is probably easiest to do if you imagine turning the paper. Now, since you're watching this on the screen, you can actually turn the screen a quarter turn clockwise, but I can't do that quite as effectively. So, let's see, I'll do that rotation, and maybe I'll end up looking something like this. Okay, so now we know what it looks like, so let's see if we can draw that. So, the idea is I'm going to turn this thing a quarter turn clockwise around this point. That's going to rotate it over to here, and so when I actually draw that, I'm going to end up with a slow rotation too, something about like that. So, that's basic rotation. All right, so next we have a reflection. So, when I have a reflection, what I'm going to do is I'm going to take every figure in a point, and I'm going to reflect it across a specified line. So, for example, I'm going to take this figure, I'm going to reflect it across this line shown. And if you think about this as a mirror, then what I'm going to get is I'm going to get the mirror image of this object. So, when I do that reflection, I end up with an object that looks something like that, and so there's my reflection. Now, there's one last type of transformation that is usually included, which is something called the glide reflection. And so, what we're going to do with the glide reflection, I have two components to it. First of all, I'm going to reflect it around a specific line, and then I'm going to translate it parallel to the line, some specific distance. So, for example, I'm going to reflect the figure across the line, and I'm going to shift it one unit to the right. So, there's my specified line, there's my specified distance. So, I reflect the figure across the line, and I'm going to move that one unit to the right, and there's what I get as the result of my glide reflection.