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Proofs with transformations

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Video transcript

- [Voiceover] This right here is a screenshot of the line and angle proofs exercise on Khan Academy, and I thought we would use this to really just get some practice with line and angle proofs. And what's neat about this, this even uses translations and transformations as ways to actually prove things. So let's look at what they're telling us. So it says line AB and line DE are parallel lines. All right. Perform a translation that proves corresponding angles are always equal, and select the option which explains the proof. All right, so let's see what they have down here. So they say perform a translation that proves corresponding angles are always equal. And then select the option that explains the proof. So they've picked two corresponding angles here. And so you see this is kind of the bottom left angle, this phi, and then you have theta right here is the bottom left angle down here, so these are corresponding angles, line FB is a transversal, and they already told us that line AB, what did they call it, did they call it DE? And line DE, are indeed, parallel lines. So we wanna prove that these two things, that the measure of these two angles are equal. So, there's many ways that you can do this geometrically. And we do that in many Khan Academy videos, but this one, they offer us the option of translating, of doing a translation, so let's see what that is. So I press the translate button, and when I move this around notice it essentially translates these four points, which has the effect of translating this entire intersection here. So if we take, this point where my mouse is right now, that is the point D, and I'm translating it around. If I move that over to B, what it shows is, because under a translation the angle measures shouldn't change. So when I did that, this angle, so it's down here, theta is the measure of angle CDF. And so when you move it over here, this right over here should be the same. This angle's measure is the same as CDF, I'm just translating it. And when you move it over here, you see look, that's the same exact measure as phi. So this is one way to think about it. I just translated the point D to B, and then it really just translated angle CDF over angle ABD. So to show that these have the same measure. Or at least to feel good about the idea of them having the same measure. So let's see which choices describe that. So let me, I'm having trouble operating my mouse. All right, the translation mapping point F to point D, so point F to point D, point F to point D, we didn't map point F to point D. So this is already looking suspect, produces a new line which is a bisector of segment DB. A new line which is a bisector of segment B, okay, this doesn't seem anything like what I just did, so I'm just gonna move on to the next one. Since the image of a line under translation is parallel to the original line, that's true, the translation that maps point D to point B, that's what I did right over here, maps angle CDF to ABD. And that's what I did, I mapped angle CDF to angle ADB, that'e exactly what I did right over there. So this is to ABD, translations preserve angle measures, so theta is equal to phi. Yup, that one looks pretty good. The translation that mapped point D to E. I didn't do that, I didn't take point D and move it over to E like that, that didn't really help me. Let's just keep reading it just to make sure. Produces a parallelogram, that actually is true, if I translate point D to point E it does, I have this parallelogram constructed, but it really doesn't help us establishing that phi is equal to theta. So that one I also don't feel good about. So and it's good because we felt good about the middle choice. Let's do one more of these. So they are telling us that line AOB, and they could have just said line AB, but I guess they wanted to put the O in there to show that point O is on that line, that AOB are colinear. And COD is our straight lines, all right, fair enough. Which of these statements prove vertical angles are always equal? So vertical angles would be the angles on the opposite sides of an intersection. So in order to prove that vertical, so for example angle AOC, and angle DOB, are vertical angles. And if we wanted to prove that they are equal, we would say well their measures are gonna be equal, so theta should be equal to phi. So let's see which of these statements actually does that. So this one says segment OA is congruent to OD. OA is congruent to OD. We don't know that, they never even told us that. So I don't even have to read the rest of it, this is already saying, I don't know how far D is away from O, I don't know if it's the same distance A is from O. So we can just rule this first choice out. I can stop reading, this started with a statement that we don't know, based on the information they gave us. So let's look at the second choice. If ray OA and ray OC are each rotated 180 degrees about point O, they must map to OB and OD respectively. If two rays are rotated by the same amount, the angle between them will not change, so phi must be equal to theta. So this is interesting, so let's just slow down and think about what they're saying. If ray OA and OC are each rotated 180 degrees, so if you take ray OA, this right over here, if you rotate it 180 degrees, it's gonna go all the way around and point in the other direction, it's going to become, it's going to map to ray OB. So I definitely believe that. OA is going to map to ray OB, and ray OC, if you rotate it 180 degrees, is going to map to ray OD. And so this first statement is true. If ray OA and ray OC are each rotated 180 degrees about point O, they must map to ray OB and OD respectively. And when people say respectively, they're saying in the same order. That ray OA maps to ray OB, and that ray OC maps to ray OD. And we saw that, ray OA maps, if you rotate it all the way around 180 degrees, it'll map to OB, and then OC if you rotate 180 degrees, will map to OD. So I'm feeling good about that first sentence. If two rays are rotated by the same amount, the angle between them will not change. Yeah, I could, yeah. Especially if they are rotated around, yeah, I'll go with that. If two rays are rotated by the same amount, the angle between them will not change. So if we rotate both of these rays by 180 degrees, then we've essentially mapped to OB and OD. Or another way to think about this angle, angle AOC, is going to map to angle BOD. And so the measure of those angles are going to be the same. So phi must be equal to theta. So I actually like this second statement a lot. So let's see this last statement. Rotations preserve lengths and angles. AB is congruent to CD. Actually, we don't know whether segment AB is congruent to CD, they never told us that. We don't know how far apart these things are. So we know that phi is equal to theta. So this statement right over here is just suspect. And so actually I don't like that one. So I'm gonna go with the first one which is, it takes a little bit of visualization going on, but if you took angle AOC and you rotated it 180 degrees which means take the corresponding rays, or the rays that make it up, and rotate them 180 degrees, you get to angle BOD. And the angle between those rays, or the measure of the angles we were just talking about, shouldn't change. So I feel really good about this second choice.