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# Proofs with transformations

CCSS.Math:

## Video transcript

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 a 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 a B 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 performer that proves corresponding angles are always equal and then select the option that 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 a bottom-left angle down here so these are corresponding angles line F B is a transversal and they already told us that line a B and what do they call it do they call it de and line de are indeed parallel lines so we want to prove that these two things that these two things that the measure of these two angles are equal so there's many ways that you can do this geometric 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 this these four points which has the effect of translating this entire intersection here so if we take so this point that where my 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 this angle so what's down here theta is the measure of angle C D F and so when you move it over here this right over here should be the same this angles measure is the same as C D F I've just trans 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 but I just I just took I just translated the point D to B and then it really just translated angle C D F over angle a B D 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 having trouble operating my mouth all right the translation mapping the translation mapping point F to point D so point F to point D point F to point D we did map point F to point D so this is already looking suspect produces a new line which is a bisector of segment D be a new line which is a segment a bisector of segment BD okay this this doesn't seem anything like what I just did so I'm just going to 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 map's point D to point B that's what I did right over your maps angle C D F to a BD and that's what I did I mapped angle C D F to angle abd that's exactly what I did what I did right over there so this is two abd translations preserve angle measures so theta is equal to Phi yeah that one looks pretty good the translation that Maps Point D to e I didn't do that I didn't I didn't take I didn't take point D and move it over to e like that that didn't really help me that would let's just keep reading it just to make sure produces a parallelogram that actually is true if I if I translate point D to point a it does I have this parrot this this parallelogram get 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 lie that line a OB a ob and they could have just they could have just said line a B but I guess they wanted to put the O in there to show that point o is on that line that a OB are collinear and C OD 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 that they are equal we would say well their measures are going to 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 o a is congruent to OD o a is congruent to OD we don't know that we don't they never even told us that so we know this is I don't even have to read the rest of it this is already saying I don't know how far de is away from oh I don't know if it's the same distance as a is from O so we can just rule this first choice out I can just 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 Rayo a and Rhea 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 let's just let's just slow down and think about what they're saying if Ray Oh a and OCR each rotated 180 degrees so if you take ray o a this ray right over here if you rotate it 180 degrees it's going to go all the way around and point in the other direction it's going to become ray it's going to it's going to map to ray OB so I definitely believe that Oh a is going to map to ray OB and OC r AOC if you rotate 180 degrees is going to map to Ray o D and so this first statement is true if Ray o a and O and Rhea 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 Oh a that rayo a maps to Ray OB and that Ray OC maps to Ray Oh Dee and we saw that Ray Oh a maps if you've rotated all the way around 180 degrees it'll map to OB and 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 I yeah especially if they are if they are if they are rotated around yeah I'll go with that if two rays are rotated by the same amount the angle between them not change so if we rotate both of these Ray's 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 DB OD and so that the the measure of those angles are going to be the same so Phi must be equal to theta so actually like this second statement a lot so let's see this last statement rotations rotations preserve lengths and angles a B is congruent to CD actually we don't know whether segment we don't know whether segment a B 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 a so this this this statement right over here is just suspect and so actually I don't I don't like that one so I'm going to go with the first one which is it's a little you know takes a little bit of visualization to 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 it 180 degrees you get to angle B OD and the angle between those rays or the measure of the angles we're just talking about shouldn't change so I feel really good about this second choice