Triangle congruence postulates/criteria

We now know that if we have two triangles and all of their corresponding sides are the same, so by side, side, side-- so if the corresponding sides, all three of the corresponding sides, have the same length, we know that those triangles are congruent. What I want to do in this video is explore if there are other properties that we can find between the triangles that can help us feel pretty good that those two triangles would be congruent. So side, side, side works. What about angle, angle, angle? So let me do that over here. What about angle angle angle? So what I'm saying is, is if-- let's say I have a triangle like this, like I have a triangle like that, and I have a triangle like this. And if we know that this angle is congruent to that angle, if this angle is congruent to that angle, which means that their measures are equal, or-- and-- I should say and-- and that angle is congruent to that angle, can we say that these are two congruent triangles? And at first case, it looks like maybe it is, at least the way I drew it here. But when you think about it, you can have the exact same corresponding angles, having the same measure or being congruent, but you could actually scale one of these triangles up and down and still have that property. For example, if I had this triangle right over here, it looks similar-- and I'm using that in just the everyday language sense-- it has the same shape as these triangles right over here. And it has the same angles. That angle is congruent to that angle, this angle down here is congruent to this angle over here, and this angle over here is congruent to this angle over here. So all of the angles in all three of these triangles are the same. The corresponding angles have the same measure. But clearly, clearly this triangle right over here is not the same. It is not congruent to the other two. The sides have a very different length. This side is much shorter than this side right over here. This side is much shorter than that side over there. And this side is much shorter over here. So with just angle, angle, angle, you cannot say that a triangle has the same size and shape. It does have the same shape but not the same size. So this does not imply congruency. So angle, angle, angle does not imply congruency. What it does imply, and we haven't talked about this yet, is that these are similar triangles. So angle, angle, angle implies similar. So let me write it over here. It implies similar triangles. And similar-- you probably are use to the word in just everyday language-- but similar has a very specific meaning in geometry. And similar things have the same shape but not necessarily the same size. So anything that is congruent, because it has the same size and shape, is also similar. But not everything that is similar is also congruent. So for example, this triangle is similar-- all of these triangles are similar to each other, but they aren't all congruent. These two are congruent if their sides are the same-- I didn't make that assumption. But if we know that their sides are the same, then we can say that they're congruent. But neither of these are congruent to this one right over here, because this is clearly much larger. It has the same shape but a different size. So we can't have an AAA postulate or an AAA axiom to get to congruency. What about side, angle, side? So let's try this out, side, angle, side. So let's start off with one triangle right over here. So let's start off with a triangle that looks like this. I have my blue side, I have my pink side, and I have my magenta side. And let's say that I have another triangle that has this blue side. It has the same side, same length as that blue side. So let me draw it like that. It has the same length as that blue side. So that length and that length are going to be the same. It has a congruent angle right after that. So this angle and the next angle for this triangle are going to have the same measure, or they're going to be congruent. And then the next side is going to have the same length as this one over here. So that's going to be the same length as this over here. So it's going to be the same length. And because we only know that two of the corresponding sides have the same length, and the angle between them-- and this is important-- the angle between the two corresponding sides also have the same measure, we can do anything we want with this last side on this one. We can essentially-- it's going to have to start right over here. You could start from this point. And we can pivot it to form any triangle we want. But we can see, the only way we can form a triangle is if we bring this side all the way over here and close this right over there. And so we can see just logically for two triangles, they have one side that has the length the same, the next side has a length the same, and the angle in between them-- so this angle-- let me do that in the same color-- this angle in between them, this is the angle. This A is this angle and that angle. It's the angle in between them. This first side is in blue. And this second side right, over here, is in pink. Well, it's already written in pink. So we can see that if two sides are the same, have the same length-- two corresponding sides have the same length, and the corresponding angle between them, they have to be congruent. There's no other one place to put this third side. So SAS-- and sometimes, it's once again called a postulate, an axiom, or if it's kind of proven, sometimes is called a theorem-- this does imply that the two triangles are congruent. So we will give ourselves this tool in our tool kit. We had the SSS postulate. Now we have the SAS postulate. Two sides are equal and the angle in between them, for two triangles, corresponding sides and angles, then we can say that it is definitely-- these are congruent triangles. Now what about-- and I'm just going to try to go through all the different combinations here-- what if I have angle, side, angle? So let me try that. So what happens if I have angle, side, angle? So let's go back to this one right over here. So actually, let me just redraw a new one for each of these cases. So angle, side, angle, so I'll draw a triangle here. So I have this triangle. So this would be maybe the side. That would be the side. So let me draw the whole triangle, actually, first. So I have this triangle. Let me draw one side over here. And then let me draw one side over there. And this angle right over here, I'll call it-- I'll do it in orange. And this angle over here, I will do it in yellow. So if I have another triangle that has one side having equal measure-- so I'll use it as this blue side right over here. So it has one side that has equal measure. And the two angles on either side of that side, or at either end of that side, are the same, will this triangle necessarily be congruent? And we're just going to try to reason it out. These aren't formal proofs. We're really just trying to set up what are reasonable postulates, or what are reasonable assumptions we can have in our tool kit as we try to prove other things. So that angle, let's call it that angle, right over there, they're going to have the same measure in this triangle. And this angle right over here in yellow is going to have the same measure on this triangle right over here. So regardless, I'm not in any way constraining the sides over here. So this side right over here could have any length. It could have any length, but it has to form this angle with it. So it could have any length. And it can just go as far as it wants to go. In no way have we constrained what the length of that is. And actually, let me mark this off, too. So this is the same as this. So that side can be anything. We haven't constrained it all. And once again, this side could be anything. We haven't constrained it at all. But we know it has to go at this angle. So it has to go at that angle. Well, once again, there's only one triangle that can be formed this way. We can say all day that this length could be as long as we want or as short as we want. And this one could be as long as we want and as short as we want. But the only way that they can actually touch each other and form a triangle and have these two angles, is if they are the exact same length as these two sides right over here. So this side will actually have to be the same as that side. And this would have to be the same as that side. Once again, this isn't a proof. I'd call it more of a reasoning through it or an investigation, really just to establish what reasonable baselines, or axioms, or assumptions, or postulates that we could have. So for my purposes, I think ASA does show us that two triangles are congruent. Now let's try another one. Let's try angle, angle, side. Let's try angle, angle, side. And in some geometry classes, maybe if you have to go through an exam quickly, you might memorize, OK, side, side, side implies congruency. And that's kind of logical. Side, angle, side implies congruency, and so on, and so forth. I'm not a fan of memorizing it. It might be good for time pressure. It is good to, sometimes, even just go through this logic. If you're like, wait, does angle, angle, angle work? Well, no, I can find this case that breaks down angle, angle, angle. If these work, just try to verify for yourself that they make logical sense why they would imply congruency. Now, let's try angle, angle, side. Let's try angle, angle, side. So once again, let's have a triangle over here. It has some side. So this one is going to be a little bit more interesting. So it has some side. That's the side right over there. And then, it has two angles. So let me draw the other sides of this triangle. I'll draw one in magenta and then one in green. And there's two angles and then the side. So let's say you have this angle-- you have that angle right over there. Actually, I didn't have to put a double, because that's the first angle that I'm-- So I have that angle, which we'll refer to as that first A. Then we have this angle, which is that second A. So if I know that there's another triangle that has one side having the same length-- so let me draw it like that-- it has one side having the same length. It has one angle on that side that has the same measure. So it has a measure like that. And so this side right over here could be of any length. We aren't constraining what the length of that side is. But whatever the angle is on the other side of that side is going to be the same as this green angle right over here. So for example, it could be like that. And then you could have a green side go like that. It could be like that and have the green side go like that. And if we have-- so the only thing we're assuming is that this is the same length as this, and that this angle is the same measure as that angle, and that this measure is the same measure as that angle. And this magenta line can be of any length, and this green line can be of any length. We in no way have constrained that. But can we form any triangle that is not congruent to this? Because the bottom line is, this green line is going to touch this one right over there. And the only way it's going to touch that one right over there is if it starts right over here, because we're constraining this angle right over here. We're constraining that angle. And so it looks like angle, angle, side does indeed imply congruency. So that does imply congruency. So let's just do one more just to kind of try out all of the different situations. What if we have-- and I'm running out of a little bit of real estate right over here at the bottom-- what if we tried out side, side, angle? So once again, draw a triangle. So it has one side there. It has another side there. And then-- I don't have to do those hash marks just yet. So one side, then another side, and then another side. And what happens if we know that there's another triangle that has two of the sides the same and then the angle after it? So for example, we would have that side just like that, and then it has another side. But we're not constraining the angle. We aren't constraining this angle right over here, but we're constraining the length of that side. So let me color code it. So that blue side is that first side. Then we have this magenta side right over there. So this is going to be the same length as this right over here. But let me make it at a different angle to see if I can disprove it. So let's say it looks like that. Or actually let me make it even more interesting. Let me try to make it like that. So it's a very different angle. But now, it has to have the same angle out here. It has to have that same angle out here. So it has to be roughly that angle. So it actually looks like we can draw a triangle that is not congruent that has two sides being the same length and then an angle is different. For example, this is pretty much that. I made this angle smaller than this angle. These two sides are the same. This angle is the same now, but what the byproduct of that is, is that this green side is going to be shorter on this triangle right over here. So you don't necessarily have congruent triangles with side, side, angle. So this is not necessarily congruent, not necessarily, or similar. It gives us neither congruency nor similarity.