 So the notions of isometry are associated with the notions of symmetry, and in particular the symmetry we're important here involves reflections. If I have an object and some line of reflection, what I can do is I can reflect the object across that line of reflection, and so I've now reflected it, and if I view both of the objects together as a single thing, then this line of reflection also forms a line of symmetry, or we call that line of reflection a line of symmetry. Conversely, if I have an object, I can find a line of symmetry that'll allow me to reflect one part of the object onto the rest of the object. So for example, here is a totally abstract and random drawing of something, and let's see if we can find any line of symmetry. So we want to find a line that can be used as a line of reflection to reflect part of the object onto the rest of the object. So let's draw a random line and see what works. So I'll draw a random line, so there's my random line, and I'm going to see if that line can be used to reflect part of the object onto the rest of the object. So I'll take part of the object, and I'll reflect it across that line. And, well, that doesn't seem to cover up, that doesn't seem to be the rest of the object. So that line doesn't work, and since it doesn't work, then it's not a line of symmetry. And so we should try a different line. OK, so maybe I'll draw a line like this. And again, I'll take a line, and I'll see if I can use that line as a line of reflection to take one part of the object, reflect it across the line, and see if I can get the other part of the object. And well, there it is. So this part of the object reflected across the line gives me the other part of the object. And so this line is a line of symmetry. Now, since we found one line of symmetry, we should check to see if there are any other lines of symmetry, because it's possible that a figure might have more than one line of symmetry. So let's draw another random line and see if we can use this line to reflect part of the object onto the rest. So I'll take part of the object, and I'll reflect it across that line. Well, that doesn't work. I'll take another line. I'll take part of the object and see if it'll reflect onto the other. And again, that doesn't work. And after a couple more tries, we might be able to convince ourselves that this is actually the only line of symmetry. Well, because reflections are isometries, then every measured property of a figure is going to be duplicated, which means that the line of symmetry can be used to discover useful information about a figure. So for example, let's take a rectangle. This rectangle has a couple of lines of symmetry. For example, here, this horizontal line, I can take part of the rectangle, reflected across the line, and get the other part of the rectangle. There's another line of symmetry, a vertical line of symmetry, and I can take part of the rectangle, reflected across this line, and get the other part of the rectangle. So I have two lines of symmetry there. Now let's take a look at that horizontal line of symmetry. Because it's a horizontal line of symmetry, because it's a line of symmetry, I can reflect this onto the bottom half, which means that this side also gets reflected down to the bottom. And all measured properties are going to be the same. So whatever the length of this line is, it's going to be the same as the length of this line. And this leads us to a theorem in geometry that says if you have a rectangle, the opposite sides of the rectangle have the same length. And we can use this to discover other useful properties of the figure.