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Intro to Congruent Triangles

A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted ? ABC. In Euclidean geometry any three points, when non-collinear, determine a unique triangle and a unique plane (the triangle's plane). This article is about triangles in Euclidean geometry, and in other geometries.

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all right, my fellow geometries, we have now come to one of the most classic topics in geometry, and that is the concept of congruent triangles. This is gonna be a Siris of videos that's gonna cover a wider a range of examples because it's such a big and traditional topic. Before we could talk about congruent or congruence. See, I want to talk about a few symbols equals equivalent. Approximate. And the simple we're gonna talk about you might have heard mentioned before congruent If you're equal, that means four equals for one value. When numeric value equals another numeric value, one length equals another length. The approximate symbol. Pardon me. The equivalent symbol, which is like three basically bars that you would see means things that aren't equal but equivalent. For example, 1 p.m. or one o'clock in 13 o'clock in 25 o'clock bills both represent all of those represent basically the one o'clock hour. They're not the same, but they're equivalent the approximate symbol Well, but if you're close to the right, answer your approximately. You're sort of equal to a value That's the approximate symbol is that congruent. Symbol is saying that you are physically the same. You have the same physical length. The objects are physically the same thing. If we're gonna prove triangles to be congruent, for example this triangle here and this triangle here and I said that, you know, side A B is congruent decided CD. What I mean is there the same length? Yes, they have the same physical link, which means their length is equal. But the segment itself is equal to the other physical segment and that makes it can grow up. We could talk about sides being congruent. We could talk about angles being congruent and for the most part, we can just think of congruence is equal. That's probably a really easy way to think of congruence e equal. But instead of like just a number we're talking about also a physical thing, a segment, an angle, something that can't be measured. But it is an object that can be measured. And so if we can show that one triangle has all of the same sides come grew into all the same sides of another and the angles as well Well, those triangles are going to be congruent toe one another Now it's not always going to be the case that we need to show every piece of one triangle congruent every piece of the other triangle is actually going to be some minimum amount of of information required for us to be able to make our claims. And so a we're gonna have to figure out what is the minimum amount of information that we need be. We're gonna talk about these things called congruent statements and see why do we even care about this? Well, geometries so born in interwoven with logic, reasoning and argument in terms of the classical steak not getting in a fight with your friend over who they think is right. But the classical sense of arguing, which is talking back and forth and trying to prove a point logically and so proof is a huge part of geometry, and there's lots different proofs. You can have flow charts, and you can have, um, column proofs, and you can have paragraph proofs and however you wanna be, however you want to write, how do you want to set it up? As long as you using the rules of logic and setting it up so that somebody can read your proof and say Yes, I agree. I can see your steps and I could see your reasoning. That's backing up your steps. Then you have a geometric proof. And so we're gonna try Thio easier away into what is concurrency? How do you write a congruent statement? And then eventually, um, talk about how what's the minimum requirements and how would you formally approved two triangles group. So first, let me start off with two triangles that I'm gonna claim are congruent, meaning they're the exact same size and shape. It's important to say size and shape, because later on, we're gonna talk about similar triangles, which we're gonna be the same shape, but not necessarily the same size. So we talk about two triangles that air congruent same size, same shape. So let's pretend these triangles are congruent. I have triangle A, B, C and triangle X Y c. I'm gonna go ahead and put my marks A B and X Y are congruent. I have this tick marks that indicate that B, C and Y z and then a C and XY. The different number of tape marks indicate which sides match up. So I could say that triangle A B C is congruent to and here's the really important part. You have to put the letters of the second triangle in the exact same order of the first. So if I went from a to B to C, we better say Triangle X. Why, Z? Why is that? Well, hey, we want to be consistent. But be if we didn't have the picture, the reader would be able to know that side A b is congruent decide x y at that angle, see would be congruent angles e because the letters and so forth the order or in the same respective orders and so I wouldn't need a picture per se. Now that's not the only way you could write it as long as you're consistent, it doesn't matter. You could say triangle C a b instead, which is the same triangle. Just a different name is congruent to triangle in this case Z x y. So that would be another congruent statement. Now, from this, from the congruent statement you can make, you could make statements like All right, Well, based upon what you've written here, I could say angle see is congruent to angles e I could say side B A is congruent notice that won't be a that I'm going to write Side Y X Order is very important. It comes down to the logic. It comes down to the order and making sense of things. So this would be an example of what it means to be congruent, not just like two numbers being equal, but we're talking about two objects of the same size and same shape. And what is a congruent statement? How do you write a congruence statement? The order of importance? And then what is it congruent statement indicate to us. So this is the intro of what it means to be congruent. And by the way, you don't have to actually have just congruent triangles. You can have congruent polygons. So in general, congruence e just means whatever shape you're dealing with, there's another shape that's the exact same size and shape. What that means is every respective matching side is congruent to each other, and every matching angle is congruent to each other. So it's like having two identical objects all the same. It's like having two cars off the line, same make and model. Everything's the same. They have the same color, the same trim, the same parts the same, you know, all the same extras it'll be Those two cars would be considered congruent. Similar would mean if you had a car and then a model car that you made and all the pieces of the same. The model car has all the pieces that are the same as the real car, but just all scaled down by the same factor. That would be a similar situation. Concurrency would mean everything is a 1 to 1 ratio, all the sides of equal to decide, respectively. All of the angles are able to all the other angles, respectively. So that's what congruence he is. I'll be back with more details. And what's the minimum information for us approved. Specifically, triangles are concluding to each other.