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Determining congruent triangles

Sal uses the SSS, ASA, SAS, and AAS postulates to find congruent triangles. Created by Sal Khan.

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Video transcript

What we have drawn over here is five different triangles. And what I want to do in this video is figure out which of these triangles are congruent to which other of these triangles. And to figure that out, I'm just over here going to write our triangle congruency postulate. So we know that two triangles are congruent if all of their sides are the same-- so side, side, side. We also know they are congruent if we have a side and then an angle between the sides and then another side that is congruent-- so side, angle, side. If we reverse the angles and the sides, we know that's also a congruence postulate. So if we have an angle and then another angle and then the side in between them is congruent, then we also have two congruent triangles. And then finally, if we have an angle and then another angle and then a side, then that is also-- any of these imply congruency. So let's see our congruent triangles. So let's see what we can figure out right over here for these triangles. So right in this triangle ABC over here, we're given this length 7, then 60 degrees, and then 40 degrees. Or another way to think about it, we're given an angle, an angle and a side-- 40 degrees, then 60 degrees, then 7. And in order for something to be congruent here, they would have to have an angle, angle, side given-- at least, unless maybe we have to figure it out some other way. But I'm guessing for this problem, they'll just already give us the angle. So they'll have to have an angle, an angle, and side. And it can't just be any angle, angle, and side. It has to be 40, 60, and 7, and it has to be in the same order. It can't be 60 and then 40 and then 7. If the 40-degree side has-- if one of its sides has the length 7, then that is not the same thing here. Here, the 60-degree side has length 7. So let's see if any of these other triangles have this kind of 40, 60 degrees, and then the 7 right over here. So this has the 40 degrees and the 60 degrees, but the 7 is in between them. So this looks like it might be congruent to some other triangle, maybe closer to something like angle, side, angle because they have an angle, side, angle. So it wouldn't be that one. This one looks interesting. This is also angle, side, angle. So maybe these are congruent, but we'll check back on that. We're still focused on this one right over here. And this one, we have a 60 degrees, then a 40 degrees, and a 7. This is tempting. We have an angle, an angle, and a side, but the angles are in a different order. Here it's 40, 60, 7. Here it's 60, 40, 7. So it's an angle, an angle, and side, but the side is not on the 60-degree angle. It's on the 40-degree angle over here. So this doesn't look right either. Here we have 40 degrees, 60 degrees, and then 7. So this is looking pretty good. We have this side right over here is congruent to this side right over here. Then you have your 60-degree angle right over here. It might not be obvious, because it's flipped, and they're drawn a little bit different. But you should never assume that just the drawing tells you what's going on. And then finally, you have your 40-degree angle here, which is your 40-degree angle here. So we can say-- we can write down-- and let me think of a good place to do it. I'll write it right over here. We can write down that triangle ABC is congruent to triangle-- and now we have to be very careful with how we name this. We have to make sure that we have the corresponding vertices map up together. So for example, we started this triangle at vertex A. So point A right over here, that's where we have the 60-degree angle. That's the vertex of the 60-degree angle. So the vertex of the 60-degree angle over here is point N. So I'm going to go to N. And then we went from A to B. B was the vertex that we did not have any angle for. And we could figure it out. If these two guys add up to 100, then this is going to be the 80-degree angle. So over here, the 80-degree angle is going to be M, the one that we don't have any label for. It's kind of the other side-- it's the thing that shares the 7 length side right over here. So then we want to go to N, then M-- sorry, NM-- and then finish up the triangle in O. And I want to really stress this, that we have to make sure we get the order of these right because then we're referring to-- we're not showing the corresponding vertices in each triangle. Now we see vertex A, or point A, maps to point N on this congruent triangle. Vertex B maps to point M. And so you can say, look, the length of AB is congruent to NM. So it all matches up. And we can say that these two are congruent by angle, angle, side, by AAS. So we did this one, this one right over here, is congruent to this one right over there. And now let's look at these two characters. So here we have an angle, 40 degrees, a side in between, and then another angle. So it looks like ASA is going to be involved. We look at this one right over here. We have 40 degrees, 40 degrees, 7, and then 60. You might say, wait, here are the 40 degrees on the bottom. Then here it's on the top. But remember, things can be congruent if you can flip them-- if you could flip them, rotate them, shift them, whatever. So if you flip this guy over, you will get this one over here. And that would not have happened if you had flipped this one to get this one over here. So you see these two by-- let me just make it clear-- you have this 60-degree angle is congruent to this 60-degree angle. You have this side of length 7 is congruent to this side of length 7. And then you have the 40-degree angle is congruent to this 40-degree angle. So once again, these two characters are congruent to each other. And we can write-- I'll write it right over here-- we can say triangle DEF is congruent to triangle-- and here we have to be careful again. D, point D, is the vertex for the 60-degree side. So I'm going to start at H, which is the vertex of the 60-- degree side over here-- is congruent to triangle H. And then we went from D to E. E is the vertex on the 40-degree side, the other vertex that shares the 7 length segment right over here. So we want to go from H to G, HGI, and we know that from angle, side, angle. And so that gives us that that character right over there is congruent to this character right over here. And then finally, we're left with this poor, poor chap. And it looks like it is not congruent to any of them. It is tempting to try to match it up to this one, especially because the angles here are on the bottom and you have the 7 side over here-- angles here on the bottom and the 7 side over here. But it doesn't match up, because the order of the angles aren't the same. You don't have the same corresponding angles. If you try to do this little exercise where you map everything to each other, you wouldn't be able to do it right over here. And this over here-- it might have been a trick question where maybe if you did the math-- if this was like a 40 or a 60-degree angle, then maybe you could have matched this to some of the other triangles or maybe even some of them to each other. But this last angle, in all of these cases-- 40 plus 60 is 100. This is going to be an 80-degree angle right over. They have to add up to 180. This is an 80-degree angle. If this ended up, by the math, being a 40 or 60-degree angle, then it could have been a little bit more interesting. There might have been other congruent pairs. But this is an 80-degree angle in every case. The other angle is 80 degrees. So this is just a lone-- unfortunately for him, he is not able to find a congruent companion.