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## Properties & definitions of transformations

Current time:0:00Total duration:8:45

# Proofs with transformations

## 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.