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Evan S.

Classify each statement as true or false. $A, B,$ and $C$ are collinear.

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Amrita B.

State Theorem $1-2$ using the phrase one and only one.

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okay. For our last similarity video, we're gonna look at three different proofs of how you can prove triangles similar to each other and how you can actually go beyond that. And use your knowledge after you prove to triangles are similar to get some difficult additional information about the triangles. So here, here's what we have were given that J and P those angles are equal. And so I like to mark my pictures up. So I know that J and P are congruent. But if you remember from the congruence videos like the markup also, some things that the picture gives us indirectly, for example, these two angles or vertical so they're equal. Oh, well, you know what? We have angle, angle, similarity just like that. So I could start off and say, You know what? Here's my first statement Angle J equals angle P. And my reason is given angle to is I could say I'm gonna go a little bit more advanced than before. I'm gonna say angle. J L K and Angles P. L m are vertical. My reason for that okay, is definition vertical alliance, Hercules apartment definition of vertical angles. Then I could say angle J L K equals angle P l m. Because what do we know about all vertical angles? All vertical angles are equal. Some people like to just go right here and to skip to that because they've they've already kind of talked about it in class or as a group that they're just gonna except when they see two crossing lines or line segments, that they have vertical angles and that we know vertical angles are equal because they've already proven that. And they kind of have, like, a a given or acceptance. I went ahead and do the extra step just to throw it in there because I've now proven the two things I need for angle, angle similarity. I literally have one last thing to say and that is, you know, statement for triangle J K. L is similar to triangle P l M. And my reason for that is angle, angle, similarity. And that's an example of a similarity proof, just like in Gruen staring. We're gonna mark everything up, but we know or were given. We're gonna mark additionally the things that the figure gives us. And then we're gonna look for the pieces of info that we need. So for congruent storm that's always three pieces you decide on a side on the side or a side at an angle on the side and so forth. For triangle similarity. You either need a side side, side, a side angle side or, in this case, just untangled. So let's look at another example. Let's say I have two right triangles and I have Triangle X, Y Z and ABC. Here's what I'm given. I'm given the triangle X Y Z and triangle A B. It is a lot nicer. Triangle A B C are right triangles. I'm also given that X y over a b is equal to y Z over BC. So this is our given. This is what we're given were asked to prove that triangle y x z and I just realized I have Tuesday's there. Oh, my gosh, that should be a y. So we're gonna prove that why x Z is congruent to triangle. Be a see I said could grow up And I also meant similar. Okay, so what do we know here? We know that these air both right angles and then we know that X Y is to a B, just as y z is to B C And folks, if it should be very clear with a side angle side similarity situation. So here's what we're gonna say. Okay, step one triangles X Y Z and triangle ABC are right triangles. This is given, and typically the middle angle is referred to as you're right angle. So you could say something like again, depending on what class you're in and how particularly your being. You might need a couple of steps to say this, but I'm gonna go ahead and say angle, why equals angle B? And my reason is gonna be all right. Angles are equal. That's my reason. Okay, so I have the Anglo established. Then I could tell you that step three X Y is to a B as y z is to B C. And that would be give it Okay, so just like we marked up the picture. Now I can go ahead and say triangles y x z and triangle B A C or similar. And my reason is going to be sorry about this at a new style is, um, trying to get used to it? This is a side angle, side similarity. And now we're done. That would be the way you prove those two triangles are similar. One more example. Okay, so let's suppose similar to the first one. You had something like this and we had, let's say triangle T and A P. We'll call this Q and S and are and let's say, were given. Sometimes I just put g with the Colin. They're given that Petey is parallel to s R. Now we want to prove that peak you is toe sq, as tech is toe arc you. So if you remember in our congruence proofs, we had to get to a C p, C T c or a definition of congruent triangles situation in order to talk about sides Air Ingles being equal similarly here, no pun intended. We have the first prove the triangle similar. And then we can improve sites proportional. So let's talk about what we have. What we given were given that Petey is paralleled s are those are symbols for parallel and as soon as we do that we have, like these two angles congruent we have these two angles can grow because we have alternate into your angles. We also by the picture, have these angles can grow because they're vertical, so there's a whole heck of a lot going on here. And clearly we have angle angle and it's up to you what you want to use. You could use one of the alternate interior angle Paris and the vertical angle. Paris. You could use both pairs of alternate interior angles. Don't need to say angle, angle, angle that's overkill and unnecessary and ideological. You just need to say angle angle, pick repairs you want. Once we have angle, angle, similarity, we prove the triangles or similar. Then we can state that the size of the triangles are proportional to each other. So let's go ahead and do that. So my first step I'm gonna say PT is parallel toe s are And my reason for that is that it's given once I do that here. So I'm gonna say angle T is equal to angle are I'm also gonna say angle P is angled their equal, the angle s and the reason for that is both or the same. It's that parallel lines imply alternate interior angles are equal. We saw that in the previous video. Now what I can say is I'll need to see my picture here to make sure I'm doing this right is I can say and we'll put this a different color. The T Q. P is congruent to our Q S. So statement three Triangle T P. Q is similar. I think it's I could grow. I meant similar again. T Let's say, in this case, I said T p Q. So I'm gonna have to do our sq. And the reason for that is going to be angle, angle, similarity. And as soon as I have a similarity statement, check this out to see what I'm trying to do here is I'm trying to prove Look at this peak, you is to ask you look at this peak, you is to ask you look at that similarity statement as T Q is to argue. Take our cue. The similarity statement has everything in order. Appropriately, that's how we know have done something right. So then I can say Peak is two sq as tech is toe r. Q. And my reason is going to be definition as recently before. When in doubt, say definition, but definition of similar triangles. So this is an example a third example of a similarity proof. But we've extended it. Not only do we approve the triangles similar, we also went beyond that. And then we're able to talk about a relationship between the two triangles based upon definition of similar triangles. Hope this helped. I'll see you later. Next topic.

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