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Congruent triangles & the SSS postulate/criterion

Video transcript
Let's talk a little bit about congruence, congruence And one to think about congruence, it's really kind of equivalence for shapes So, when in algebra when something is equal to another thing it means that their quantities are the same But when we're all of the sudden talking about shapes and we say that those shapes are the same, the shapes are the same size and shape then we say that they're congruent And just to see a simple example here: I have this triangle, right over there and let's say I have this triangle right over here And if you are able to shift, you are able to shift this triangle and flip this triangle, you can make it look exactly like this triangle As long as you're not changing the lengths of any of the sides or the angles here But you can flip it, you can shift it, you can rotate it So you can shift, let me write this, you can shift it, you can flip it and you can rotate If you can do those three procedures to make these the exact same triangle, then they are congruent And if you say that a triangle is congruent, let me label this So, let's call this triangle ABC Now let's call this D, let me call it XYZ XY and Z So, if we were to say, if we make the claim that both of these triangles are congruent So, if we say triangle ABC is congruent And the way you specify it, it almost look like an equal sign But it's equal sign with a curly thing on top Let me write it a little bit either So, we would write it like this If we know that triangle ABC is congruent to triangle XYZ That means their corresponding sides have the same length And their corresponding angles have the same measure So, if we make this assumption or someone tells us that this is true then we know, for example, that AB is going to equal to XY The length of segment AB is gonna be equal to the segment of XY And we could do this like this, and I'm assuming this are the corresponding sides And you can see that actually we've defined these triangles A corresponds to X, B corresponds to Y and C corresponds to Z right over there So, side AB is gonna have the same length as XY Then you can sometimes if you don't have the colors you can denote it just like that These two length are- or this two lines segments have the same length And you can actually say this, you don't always see this written this way You could also make the statement that line segment AB is congruent to line segment XY But congruence of line segments really just means that their lengths are equivalent So, these two things mean the same thing If one line segment is congruent to another line segment that just means the measure of one line segment is equal to the measure of the other line segment And so we can go thru all the corresponding sides If these two characters are congruent, we also know that BC, we also know that the length BC is gonna be the length of YZ Assuming those are the corresponding sides And we can put these double hash marks right over here to show that these lengths are the same And when we go the third side, we also know that these are going to be has same length or the line segments are going to be congruent So, we also know that the length of AC is going to be equal to the length of XZ Not only do we know that all of the sides, the corresponding sides are gonna have the same length If someone tells that a triangle is congruent We also know that all the corresponding angles are going to have the same measure So, for example: we also know that this angle's measure is going to be the same as the corresponding angle's measure, and the corresponding angle is right over It's between these orange side and blue side Or orange side and purple side, I should say And between the orange side and this purple side And so it also tells us that the measure of angle is BAC is equal to the measure of angle of YXZ Let me write that angle symbol, a little less like that, measure of angle of YXZ YXZ We can also write that as angle BAC is congruent angle YXZ And once again, like line segment, if one line segment is congruent to another line segment It just means that their lengths are equal And if one angle is congruent to another angle it just means that their measures are equal So, we know that those two corresponding angles have the same measure, they're congruent We also know that these two corresponding angles I'll use a double arch to specify that this has the same measure as that So, we also know the measure of angle ABC is equal to the measure of angle XYZ And then finally we know that this angle, if we know that these two characters are congruent, then this angle is gonna have the same measure as this angle as a corresponding angle So, we know that the measure of angle ACB is gonna be equal to the measure of angle XZY Now what we're gonna concern ourselves a lot with is how do we prove congruence? 'Cause it's cool, 'cause if you can prove congruence of 2 triangles then all of the sudden you can make all of these assumptions And what we're gonna find out, and this is going to be, we're gonna assume it for the sake of introductory geometry course This is an axiom or a postulate or just something you assume So, an axiom, very fancy word Postulate, also a very fancy word It really just means things we are gonna assume are true An axiom is sometimes, there's a little bit of distinction sometimes where someone would say "an axiom is something that is self-evident" or it seems like a universal truth that is definitely true and we just take it for granted You can't prove an axiom A postulate kinda has that same role but sometimes let's just assume this is true and see if we assume that it's true what can we derive from it, what we can prove if we assume its true But for the sake of introductory geometry class and really most in mathematics today, these two words are use interchangeably An axiom or a postulate, just very fancy words that things we take as a given Things that we'll just assume, we won't prove them, we will start with this assumptions and then we're just gonna build up from there And one of the core ones that we'll see in geometry is the axiom or the postulate That if all of the sides are congruent, if the length of all the sides of the triangle are congruent, then we are dealing with congruent triangles So, sometimes called side, side, side postulate or axiom We're not gonna prove it here, we're just gonna take it as a given So this literally stands for side, side, side And what it tells is, if we have two triangles and So I say that's another triangle right over there And we know that corresponding sides are equal So, we know that this side right over here is equal into, like, that side right over there Then we know and we're just gonna take this as an assumption and we can build off of this We know that they are congruent, the triangle, that these two triangles are congruent to each other I didn't put any labels there so it's kinda hard for me to refer to them But these two are congruent triangles And what's powerful there is we know that the corresponding sides are equal Then we know they're congruent and we can make all the other assumptions Which means that the corresponding angles are also equal So, that we know, is gonna be congruent to that or have the same measure That's gonna have the same measure as that and then that is gonna have the same measure as that right over there And to see why that is a reasonable axiom or a reasonable assumption or a reasonable postulate to start off with Let's take one, let's start with one triangle So, let's say I have this triangle right over here So, it has this side and then it has this side and then it has this side right over here And what I'm gonna do is see if I have another triangle that has the exact same line, side lengths is there anyway for me to construct a triangle with the same side lengths that is different, that can't be translated to this triangle thru flipping, shifting or rotating So, we assume this other triangle is gonna have the same size, the same length as that one over there So, I'll try to draw it like that Roughly the same length We know that it's going to have a size that's that length So, it's gonna have a side that is that length Let me put it on this side just to make it look a little bit more interesting So, we know that it's gonna have a side like that So, I'm gonna draw roughly the same length but I'm gonna try to do it in a different angle Now we know that's it's gonna have that looks like that So, let me, I'll put it right over here It's about that length right over there And so clearly this isn't a triangle, in order to make it a triangle, I'll have to connect this point to that point right over there And really there's only two ways to do it I can rotate it around that little hinge right over there If I connect them over here then I'm going to get a triangle that looks likes this Which is really a just a flip, am I visualizing it right? Yeah, just a flip version You can rotate it a little back this way, and you'd have a magenta on this side and a yellow one on this side And you can flip it, you could flip it vertically and it'll look exactly like this Our other option to make these two points connect is to rotate them out this way And the yellow side is gonna be here And then the magenta side is gonna be here and that's not magenta The magenta side is gonna be just like that And if we do that, then we actually just have to rotate it We just have to rotate it around to get that exact triangle So, this isn't a proof, and actually we're gonna start assuming that his is an axiom But hopefully you'll see that it's a pretty reasonable starting point that all of the sides, all of the corresponding sides of two different triangles are equal Then we are going to- we know that they are congruent We are just gonna assume that it's an axiom for that we're gonna build off, that they are congruent And we also know that he corresponding angles are going to be equivalent