Common Core Geometry Unit 2 Lesson 7 Basic Rigid Motion Proofs

We talked about why that was, right? Rigid body motions preserve distance or length, and they preserve angle measurement. So they keep the shape of an object exactly the same, and therefore the two objects should be able to be placed on top of each other. We have three types of rigid motions that we're concerned with. Reflections, rotations, and translations. That's it. And for much of what we do, reflections and rotations will get the job done, but we'll also want some translations in there as well. Let's move on and talk a little bit more about rigid body motions. Because you need to know the properties of rigid body motions, both the properties for all types of rigid body motions and ones that are specific to rotations or to reflections or to translations. So let's talk about those. Now, before we take away the screen, how many of them do you remember? All right, let's take a look at them. And let's kind of go through these one by one. 

Number one, all rigid motions map lines to lines, segments to segments and raise to raise. So a line isn't going to suddenly turn into a ray, a segment isn't going to suddenly turn into a circle or something like that. Number two, all rigid motions preserve distance and angle measurements. That is by far the most important of all the rigid body motions. Now let's get into some that are very specific. Number three, rotations of a line a 180° about a point not on the line produces a parallel line. That's a very, very important property. If I ever rotate a line about a point knot on the line by 180° only, then it'll produce a parallel line. On the other hand, number four says rotation of a line a 180° about a point that does lie on the line, simply produces the same line. That one's not quite as important, but it could come up. Number 5, translation of a line, not along the line, produces a parallel line. That seems pretty similar to number three, but again, what we mean is if we take a line segment and we simply translate that line segment, then we're going to get a parallel line. 

Number 6, when a point is reflected across a line, the segment connecting the point and its image is perpendicularly bisected by the line. Now that last one, which is very, very important, is the very definition of what a reflection is. All right? So using these properties and things that we know in a particular problem, let's do a little bit of reasoning. All right, first problem, let's take a look. Exercise number two in the diagram below, it is given that AB and CD lines a, B, and CD, intersect at point E angle pairs that are across from each other in this diagram are known as vertical angles. Vertical angles are exceptionally important in geometry because they have a nice property. And that property is that they're congruent, that they have the same measure. Now we could certainly experiment with that with our protractors and things like that. But we're going to use rigid body motions right now to prove that a pair of vertical angles are congruent, are equal in measure to each other. Let's take a look at letter a use tracing paper to verify that angle BEC, this angle. Is congruent to angle AED. This angle. What rigid motion did you use? 

All right, so take a moment, pause the video for as long as you need to, then come on back to me and talk to me about what rigid body motion you used to map this angle, right? Angle BEC. Sorry, onto this angle angle AED. Take a bit. All right. Let's go through it. Now, there's a number of different things that you could do, including a reflection. But what I would use is I haven't used a rotation. By 180° about point E let's take a look at that. Let me get rid of those blue. Oh, that's kind of cool. Get rid of that. And let me click on this angle. If I rotate it by a 180°, and then it ends up landing on that angle. All right? Again, important to understand why that 180° rotation works. But that's really the second part of the problem. Here, we're going to work on sort of how we rotate or how we map an angle onto another angle. Now keep in mind, that all an angle is, is the connection of two rays at their starting point, right? Specifically in this problem, let me rotate this thing back. Overshoot, right? We've got ray EC, and we've got ray EB, right? So when I rotate an angle, what I'm really doing is just rotating two rays, all right? Granted at the same time. 

So let's take a look at the reasoning in the second part of the problem. Letter B although there are a few different rigid motions that would map angle BEC onto angle AED. We will use a rotation 180° rotation of BEC about point E for our proof. Finish each statement with a reason. All right? So letter I when ray EB ray E B is rotated 180°. 180° about point E it falls on ray AE and when EC is rotated a 180° about point E it falls on ED because why? Why would that be? Because sometimes if I rotated this thing, a 180°, it wouldn't land on that. All right. But in this particular scenario, it does. So why is that? Think about this for a moment. Yeah? It's very, very simple. Okay? When I rotate this 180°. Right? Then EB lands on AE simply because a E and B. Are collinear. And whoops. Wow, that's kind of cool. Let's do that. And C, E. And T. Are collinear. Now again, this could be a tricky point for some people. So let's pause and just think about this for a moment. When I have three points that are co linear, co linear. So just kind of random. One, two, three. Let's say those three points lied in a straight line. Then what would happen, let me make those points a little bit bigger. Is that this then is a 180° angle. And so if I rotate one ray about that point, it's got a land on the other right. That's just what an angle is. Right? So again, think about this for a moment. 

The reason this worked is that when I rotated, the red angle, when I rotated each ray of this red angle by a 180°, they had to then lie on top of or coincide with the rays of this angle. Now, I can say thus fancy word just meaning, hey, take a look at what I just did. This now leads to me being able to say this because so thus angle BEC is congruent to angle AED because why? Well, the reasoning that ends almost all of these proofs is the following because a rotation. Preserves angle measurements. Again, very important. You see, there are tons and tons of transformations out there. Think about taking an image on your camera or somewhere else and modifying it. Putting a filter on it, doing something funny with it. All of those are transformations. So I could take virtually any angle I want. Let's say I had this nice obtuse angle like that. And I could then take an acute angle. Two angles that are clearly not congruent. And I could come up with some geometric transformation that mapped this angle onto that angle. 

All right, but it wouldn't be a rigid body motion because it wouldn't preserve the angle size. It couldn't, these two are clearly different angle sizes. All right, the key here is that by using a rigid body motion, one that doesn't change the angle measurement. I'm able to map this ray onto this one. And this ray onto that one and because I can map the two rays of one angle onto the two rays of the other angle, the two angles have to be congruent because it was a rigid body motion. Very, very important. All right. Write anything down that you might need to on this slide. I know it's a little bit messy. But kind of an important one. All right, let's keep moving on. Now, besides understanding the reasoning that goes into this problem, into the previous problem, we also want to understand the results of it. And a huge result that we just came up with, one that you'll see time and time again in this course is that vertical angle pairs are congruent. They have the same measure. And again, what this means is any time I take two lines and I cross them. Then angles that are crossed from one another have the same size. 

These two angles have the same size. And those two angles have the same size. And we can immediately use this in a very simple algebraic problem like the one I have up on the board. Let's go over the problem and let me erase this at the same time. Exercise number two. In the diagram below, line RT and line WZ intersect at point S if the angles have the measures is labeled in the terms of X, then which of the following is the measure of angle Z ST. Zest. All right, why don't you go ahead and try to figure that problem out? And then we'll talk about it. Okay. Now you want to be very careful in this problem. Because even if you have no idea what you're doing, you might start it off correctly, but you may not finish it. So what do we know? We know that these two angle pairs are vertical angles. And therefore they're measures 8 X -12 must be equal to each other, 6 X plus 24. This leaves us with a relatively easy equation to solve. I'll just subtract 6 X from both sides and add 12 to both sides. All right, when I do that, of course, the 6 X's and the 12s cancel out. And I'm left with two X is equal to 36. I can divide both sides by two, and I'll get X is equal to 18. Now, beware, oftentimes when students do an algebraic procedure and they get an answer and they say, hey, look, 18. Choice one in the blank. Except that's not what the problem asks for. 

The problem didn't say, tell me what the value of X is. Specifically, the problem asked for the measure of angle, Z ST. This angle. Now, that's not one of the ones marked, but what we can certainly do is we could certainly take this particular angle, 8 X -12. And we could substitute the 18 in. Of course we might have to use our calculators to do that. I'm going to do it. Longhand. 8 times 18, four 64, 8, one 44. So we've got one 44 -12. And that equals 132°. Again, be careful. Don't go with the one 32. It's not the angle you want. In fact, I want angle Z ST, and that's going to be 180. Minus my one 32. Which is going to be 48°. Choice two. The algebra in this problem isn't hard. The geometry in this problem isn't particularly hard, right? We have to just know that these two angles are congruent and therefore we set them equal to each other. That allows us to solve for X, then we have to keep our eye on the ball, always know what we're solving for, right? Use that 18 to figure out the value of W ST, one 32, subtract from one 80, and we have our 48. Vertical angles are congruent. They have the same angle measurement, which is again very simple to see by using a 180° rotation of one of those angles on top of the other. 

Okay, let's keep working with this congruence reasoning. The last thing that we're going to be doing today is we're going to be looking at some congruence reasoning with parallel lines. Again, we've talked about parallel lines a little bit in this course. And you also talked about them in 8th grade math. There's a lot of terminology with parallel lines. So let's take a look at this. Whenever you have two lines that are parallel, for instance, like this strangely purple line and blue line, these two lines being parallel. Cut by what's known as a transversal. What happens is 8 different angles get created. All right? You can see them. All 8, I've got them numbered. Now, there's lots of different terminology, but two pieces that are going to be important today are corresponding angles. And alternate interior angles. Corresponding angles mean basically two angles that are in the same place on the T that's created. I don't know if you want to call it the T or the X or whatever. But when I look at these two, right? Angle 5 corresponds to angle one. Angle 6 corresponds to angle two. 7 to three and 8 to four. They're in the same place on the X I'm going to go with X instead of T. 

The other piece of terminology that's important are what are called alternate interior angles. Some students will think of these as the angles of the Z in other words, if I have this going on, then these are alternate interior angles. And these are alternate interior angles. The angles of the Z and what I'd like to now do is take a look at Y corresponding angles that are congruent and why alternate interior angles are congruent based on rigid body motions. Let's get into that. All right, kind of a worry problem. Let me take a look. In the following diagram, lines a, B, and CD are parallel and cut by transversal line M, which intersects AB and CD respectively at E and F point H is the midpoint of segment EF. In the diagram, angle CFE and BF BEF are called alternate interior angles. So we just talked about that. CFE. And BEF. Alternate interior angles. Verify using tracing paper that these two angles are equal. What rigid motion could you use to prove this? Mark the angles as equal. All right, I'm going to move that my little red angle out of there. So I can then delete these two things out of there. Okay. So no proving anything right now.

I just want to use a rigid body motion. And tracing paper to show that this angle is congruent with this one. And there I've just marked the two angles. So take a moment and think about this. All right, if this problem looks familiar, it is. We did something very, very similar. One could argue almost identical to this. Back when we did our lesson on rotations, okay? And the plain fact is that you can actually prove we're not going to prove. We're just kind of experimentally that rotation are rotate angle, let's see. I'm going to go, we'll go BEF by 180° about point H. We even looked at why that's true, but let's take a look at it. This is kind of fun. If I take this angle and I rotate it a 180° about point H, it maps up to that angle. All right? And it's really helpful to be able to visualize this to be able to take that Z and rotate it that 180° to see that this segment falls on that segment, and that line falls on that line. And again, what I'm really doing is I'm taking two rays of one angle, and I'm mapping them onto two rays of another angle. But we're not going to go through the formalistic proof of that because quite frankly, we really did do that a couple of lessons ago in the rotation lesson, which if you haven't watched it, maybe go back and take a look. But let's go on to the next part of the problem. 

Letter B also, FEB and are called corresponding angles. Verify using tracing paper that these two angles are equal. Mark these angles is equal. All right, so I'm going to rotate this thing back down. Because now we're talking about FEB, which is this angle here. Now, if you're wondering why am I not calling it HEB? Well, quite frankly, it doesn't matter. Do you know what I mean? Whether or not I call it FEB GEB right or HEB, it's the same angle, because we're talking about rays that extend forever. It doesn't really matter as long as I've got a point on this ray. And F is a point on that ray. But I want to now see whether or not that angle is congruent to that one. All right? So take a moment with tracing paper to ensure that. Now in all likelihood, we're tracing paper. You end up tracing this angle out and then just taking your trace and paper, picking it up, slapping it down and making sure that they lie on top of each other. For me, what would I do? I would just kind of slide this angle up. And I see, yeah, the two angles are congruent. The red angle now lies entirely on top of that one, so they must be congruent. And it's interesting because even just the motion of what I'm doing to put the two angles on top of each other kind of gives me a hint about what comes next, which is what rigid body motion are we going to use to prove that corresponding angles on parallel lines are congruent. And quite frankly, what we're going to use is a translation. 

And I've reproduced the diagram down here below. All right? Letter C to prove the corresponding angles are equal. We will use a translation to show that angle FEB can be mapped, sorry about that. To show angle FEB can be mapped onto angle. Finish statements with reasons below. Okay? Now, what I've done is I've actually taken the two rays of angle FEB, and I've kind of broken them up so that I can sort of manipulate this one all by itself and this one all by itself. So let's take a look at letter I if EB, this guy. If EB is translated in the direction of EF, remember, a translation needs a direction. So EF just means move it in the direction of that ray. Such that point E gets mapped onto point F, EB must lie on FD because Y well, let's actually do that translation. I'm going to take this and I'm going to move it up. I'm going to translate it so point E lands on point F all right, but I can see that it now lies on FD, but why does it have to lie in FD? Why is it? Why isn't it maybe that like, oh, it doesn't look like this. Then it wouldn't lie on FD. So why not? Well, let me just kind of get rid of that. There's a good reason for it. Because when EB is translated. Its image. Is parallel. To EB. And passes through F I'll even say point F. Now, strangely enough, this actually goes back to Euclid's parallel postulate. Right? When I take ray E B and I translate it up so that point E gets mapped on to point F what I know is I know that I have a ray that starts at point F and runs parallel to EB. But there is only one line that goes through point F that's parallel to EB. 

That's what Euclid's parallel postulate says. So since I know that the image of EB E prime B prime, whatever, passes through F and is parallel to AB, or to EB, it must lie on FD, because we know these two line segments are parallel. But that's only one of the two rays. Let's take a look at number two, or double I or whatever you call it. If EF, this guy right here, if EF has translated in the direction of EF. Such that point E gets mapped to point F like this. Then EF must lie on FG because E F and G are collinear. And that's a little bit weird, but we can translate a line or a ray in this case. In the same direction as the ray itself. And what will happen is it will simply stay on the ray. If I have some random line, and then I take a portion of that line, and I just translate it in this direction, well, it will simply stay on the line. So why does EFG lie on FG? Well, because EF and G are collinear. And we're basically done, right? I translated EB up to FD. I translated EF up to FG, right? All with the same translation, right? I'm translating in the direction of EF so that point E gets mapped up to point a half. It's got to be the same translation for both of them and it is. 

Therefore, I can say that the two angles are congruent because why? Because a translation. Preserves. Preserves angle measurements. And it's kind of neat. We take this angle and we simply shift it up in the direction of the transversal. And map it onto the other angle. And because of the properties of both rigid body motions, which preserve angle measurement, and a property of translations, which maps a parallel line, sorry, align to another line that's parallel to it. We can now show that corresponding angle pairs created by parallel lines. And transversals are congruent. All right. A little bit of tricky reasoning, but it helps us know this fact. We'll revisit it eventually. Let's wrap up. All right, so today what we did was we looked at more sort of rigorous reasoning when it came to rigid body motions and proving specifically in this lesson that certain angle pairs are congruent. 

We saw that vertical angle pairs were congruent. Alternate interior angles created by parallel lines were congruent, and corresponding angle pairs created by parallel lines are congruent. Wordy. And we did all of this by using properties of rigid body motions. In the next lesson, we'll be looking at why triangles can be proven congruent, given certain conditions by using various rigid body motions. For now, I'd like to thank you for joining me for another common core geometry lesson by E math instruction. My name is Kirk weiler, and until next time, keep thinking and keep solving problems.