Triangle congruence from transformations
Several videos ago, I very quickly went through why side-side-angle 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 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's have a triangle that looks something like this. If I have a triangle that looks something-- I have trouble drawing straight triangles. So let's say the triangle looks something like that. And let's say that we've found another triangle that has a congruent side, a side that is congruent to this side right over here. I guess any side on a triangle is 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. 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 is 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 this is congruent to anything. If we knew, then we could use side-side-side. We only know that this side is congruent and this side is 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 what the length is of that green side. Now we have this magenta side. We have another side that is congruent here. So this thing could pivot over here. We know nothing about this angle so it could 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 this side does go down just like that, in which case, we actually would have congruent triangles. But the kind of aha moment here, or the reason why SSA isn't possible, is that this side, could also come down like this. There's two ways to get down to this base, if you want to call it that way. It can come out that way or it could kind of come in this way. And so that's why SSA by itself with no other information is ambiguous. 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. 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 90 degrees. So you could still have an obtuse angle. So that's why this is an option. So one option is that you have two other acute angles. So this could be acute. This is also acute, also acute, also acute. But then you have the option where this is even more acute, even narrower, and then this becomes 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, and if I were to tell you very clearly that this angle right over here is obtuse-- and that is the A in the SSA. So you have the angle. And I were to say I have another triangle where this angle is congruent to that other triangle, some angle of that other triangle, and then one of the sides adjacent to it is congruent, and then the next side over is also congruent, then it's not so ambiguous. Because we could try to draw that. So let's draw that same congruent obtuse angle. We know nothing about this side down here because we haven't said that that's necessarily congruent. So that could be of any length. We do know that this triangle is going to have the same length for this side of the angle. So it looks like this. And then we know that this side-- let me do that in orange. We know that this side is also going to be the same length. We haven't told you anything about this angle right over here. So this side 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, in general, SSA, 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 in the SSA is obtuse, then it becomes a little bit less ambiguous. And then finally, there's a circumstance that 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 in 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 this length right over here-- if you know that this is fixed because you're saying it's 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, 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 angle side hypotenuse postulate, which is really just a special case of SSA where the angle is actually a right angle. And here, they wrote the angle first. You could view this as angle-side-side. And they were able to do it because now they can write "right angle," and so it doesn't form that embarrassing acronym. And this would also be a little bit common sense. Because if you know two sides of a right triangle-- and we haven't gone into depth on 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 wanted 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.