Why SSA isn't a congruence postulate/criterion

several videos ago I very quickly went through why side-side-angle is not why it is not a valid postulate and what I want to do in this video is explore it a little bit more and it's not called angle side side for obvious reasons because then the acronym would make would make people giggle in geometry class and I guess we don't want people giggling while they're doing mathematics so let's just think about a triangle here so let's say I have a triangle let me draw it let me say let's say have a triangle that looks something like this if I have a triangle that looks something trouble drawing straight triangles so let's say the triangle looks something like that it looks like that and let's say that we've found another triangle we've found another triangle that has a congruent side a side that is congruent to this side right over here it has and then it that is next to I guess any side on the triangles next to the other two sides next to that is a side that is congruent to this side right over here and then that side is one of the sides of an angle so it forms one of the parts of an angle right over here and that other triangle has a congruent angle right over here so this is the angle that that first side is not a part of that first side only that second side is part of this angle so this is side side angle or you could call it angle side side and giggle a little bit about it now how do we know that this doesn't by itself show that this is congruent well we'd have to show that this could actually imply two different triangles and to think about that let's say we know that the angle we know that this other triangle has that same yellow angle there which means that the blue side the blue side is going to have to look something like that it's going to have to look something like that just the way we drew it over here this side down here I'll make it a green side this green side down here we know nothing about we never said that that this is congruent to anything if we knew then we could use side side side we only know that this side is congruent this sides congruent and this angle is congruent so this green side and I'll draw it as a dotted line it could be of any length we don't know we don't know what the length is of that Green now we have this magenta side and we have another side that's congruent here so this thing could pivot over here we know nothing about this angle because it could so it can form any angle but it does have to get to this other side so one possibility is that maybe the triangles are congruent so maybe the side does go down just like that in which case we actually would have congruent triangles but the kind of aha moment - you're the reason why SSA isn't possible is that this side could also come down like this it could also come down like this there's two ways to get down to this base if you want to call it that way you can come out that way or you could kind of come in this way and so that's why that's why SSA by itself with no other information is ambiguous it does not give you enough information it does not give you enough information to say that those triangles are definitely the same now there are special cases so in this situation right over here our angle the angle in our SSA our angle was acute this is an acute angle right over here this is an acute angle and when you have an acute angle as one of the sides of your triangle the other sides of the triangle you could still have an obtuse angle remember acute means less than 90 degrees obtuse means greater than ninety degrees so you could still have an obtuse angle and so that's why this is an option so one option is that you have two other acute angles so this is also it could be acute this is also acute also acute also acute but then you had the option where this is even more acute even narrower and then this becomes an obtuse angle so that is an obtuse angle and that's only possible if you don't you can't have two obtuse angles in the same triangle you can't have two things that have larger than 90 degree measure in the same triangle and so that's why there is a possibility where if you have another triangle that looks like this if you have another triangle it looks like this and if I were to tell you and if I were to tell you very clearly that this angle right over here is obtuse if I were to tell you this angle right over here is obtuse and that is the a in the SSA so you have the angle and I would say I have another triangle where this angle is congruent to that other triangle some angle that other triangle and then one of the sides adjacent to it is congruent and then the next side over is also congruent and the next side over is also congruent and then it's not so ambiguous because we could try to draw that so let's draw that same congruent obtuse angle let's draw we know nothing about this side down here because we haven't said that that's necessarily congruent so that could be that could be of any length we do know that this triangle is going to have the same same length for this side of the angle so it looks like this it looks like this and then we know that this side let me do that in I'll do it in orange we know that this side is also going to be the same length and we haven't told you anything about this angle right over here so this side could pivot could pivot over here we can kind of rotate it over there but there's only one way now that this orange side can reach this green side now the only way is this way over here and we were more constrained or this case isn't ambiguous because we used up our obtuse angle here the a here is an obtuse one and so then it constrains what the triangle can become so I don't want to make you say oh maybe SSA in general if SSA you do not you do not want to use it as a postulate I just wanted to make it clear that there is the special case where if you know that the a and the SSA is obtuse then it becomes a little bit less ambiguous and then finally there's a circumstance so this is an acute angle where it would be ambiguous you have the obtuse angle and then you have something in between which is the right angle so where you have the a and SSA is a right angle so if you had it like this if you have a right angle and you have some base of unknown length but you fix you fix this length right over here if you know if you know that this is fixed because you're saying is congruent to some other triangle and if you know that the next length is fixed and if you think about it this next side is going to be the side opposite the right angle it's going to have to be the hypotenuse of the right angle then you know that the only way you can construct this and similar to the obtuse case is if and if you know the length of this the only way you could do it is to bring it down over here so that actually does lead to another postulate called the right right it's called the right angle side hypotenuse postulate which is really which is really just a special case of SSA where the angle is where the angle is is actually a right angle and here they wrote the angle first you could view this as angle side to side and they were able to do it because now they can write right angle and so it doesn't form that embarrassing that embarrassing acronym and this should also be a little bit common sense because if you know two sides of a right triangle and we haven't gone into depth in this in the geometry playlist but you might already be familiar with it by Pythagorean theorem you can always figure out the third side so if you have this information about any triangle you can always figure out the third side and then you can use side side side so I just want to show you this little special case but in general the important thing is that you can't just use SSA unless you have more information