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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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  • male robot hal style avatar for user Lawrence
    How would triangles be congruent if you need to flip them around?
    (5 votes)
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    • piceratops ultimate style avatar for user abassan
      Congruent means the same size and shape. It doesn't matter if they are mirror images of each other or turned around. If you could cut them out and put them on top of each other to show that they are the same size and shape, they are considered congruent. Rotations and flips don't matter.
      (54 votes)
  • mr pants teal style avatar for user TenToTheBillionth
    in ABC the 60 degree angle looks like a 90 degree angle, very confusing.... :=D
    (10 votes)
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  • leaf red style avatar for user Zinxeno Moto
    how are ABC and MNO equal? both of their 60 degrees are in different places
    (9 votes)
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    • leafers seedling style avatar for user Fieso Duck
      Basically triangles are congruent when they have the same shape and size. So if you have two triangles and you can transform (for example by reflection) one of them into the other (while preserving the scale!), the two triangles are congruent.
      If you flip/reflect MNO over NO it is the "same" as ABC, so these two triangles are congruent.
      (11 votes)
  • blobby green style avatar for user Michael Rhyan
    Can you expand on what you mean by "flip it". I cut a piece of paper diagonally, marked the same angles as above, and it doesn't matter if I flip it, rotate it, or move it, I cant get the piece of paper to take on the same position as DEF.
    (6 votes)
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  • marcimus pink style avatar for user charikarishika9
    does it matter if a triangle is congruent by any of SSS,AAS,ASA,SAS?
    (10 votes)
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    • male robot hal style avatar for user Julian Mydlil
      Your question should be about two triangles. This is because by those shortcuts (SSS, AAS, ASA, SAS) two triangles may be congruent to each other if and only if they hold those properties true.
      SSS: When all three sides are equal to each other on both triangles, the triangle is congruent
      AAS: If two angles and a non-included (you can think of it as outside) side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.
      ASA: If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.
      SAS: If any two angles and the included side are the same in both triangles, then the triangles are congruent.
      (2 votes)
  • mr pants orange style avatar for user Daniel Saltsman
    Is there a way that you can turn on subtitles?
    (4 votes)
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  • spunky sam red style avatar for user jloder
    why doesn't this dang thing ever mark it as done
    (4 votes)
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    • piceratops ultimate style avatar for user Aaron Fox
      IDK. It happens to me though. When it does, I restart the video and wait for it to play about 5 seconds of the video. Then I pause it, drag the red dot to the beginning of the video, push play, and let the video finish. For some unknown reason, that usually marks it as done. I hope it works as well for you as it does for me.
      (7 votes)
  • blobby green style avatar for user bahjat.khuzam
    Why are AAA triangles not a thing but SSS are? If you can't determine the size with AAA, then how can you determine the angles in SSS?
    (2 votes)
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    • hopper cool style avatar for user Iron Programming
      Two triangles that share the same AAA postulate would be similar. But if all we know is the angles then we could just dilate (scale) the triangle which wouldn't change the angles between sides at all.

      If we know that 2 triangles share the SSS postulate, then they are congruent. This means that they can be mapped onto each other using rigid transformations (translating, rotating, reflecting, not dilating).

      There is only 1 such possible triangle with side lengths of A, B, and C. Note that that such triangle can be oriented differently, using rigid transformations, but it will 'always be the same triangle' in a manner of speaking.

      Angles tell us the relationships between the opposite/adjacent side(s), which is what sine, cosine, and tangent are used for.

      Hope this helps!
      - Convenient Colleague
      (4 votes)
  • starky sapling style avatar for user Mercedes Payne
    what does congruent mean?
    (1 vote)
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  • blobby green style avatar for user saawaniambure
    would the last triangle be congruent to any other other triangles if you rotated it?
    (2 votes)
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    • duskpin ultimate style avatar for user Ash_001
      It would not. I see why you think this - because the triangle to the right has 40 and a 60 degree angle and a side of length 7 as well. But here's the thing - for triangles to be congruent EVERYTHING about them has to be the exact same (congruent means they are both equal and identical in every way). This is not true with the last triangle and the one to the right because the order in which the angles and the side correspond are not the same. One of them has the 40 degree angle near the side with length 7 and the other has the 60 degree angle next to the side with length 7. If you need further proof that they are not congruent, then try rotating it and you will see that they are indeed not congruent.
      (2 votes)

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.