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Why SSA isn't a congruence postulate/criterion

There are some cases when SSA can imply triangle congruence, but not always. This is why it's not like the other triangle congruence postulates/criteria. Created by Sal Khan.

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  • blobby green style avatar for user Janette Rhee
    He didn't have a video on SSA BEFORE this one...now i'm super confused about what he's talking about. What other video was BEFORE this one? and What's a postulate?
    (9 votes)
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    • duskpin ultimate style avatar for user Joshua Choe
      A postulate is kind of like a definition or theorem, but it is something you have to accept without any "proof." For example, 2+2 is something you have to accept as 4. The 2 could be 2 apples which have billions and billions of atoms and that doesn't really equal 4 apples. Numbers are relative.
      (40 votes)
  • aqualine ultimate style avatar for user TheEpicOne
    What's the difference between an axiom and a postulate?
    (11 votes)
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    • piceratops seedling style avatar for user Rain
      The terms "postulates" and "axioms" can be used interchangeably: just different words referring to the basic assumptions - the "building blocks" taken as given (assumptions about what we take to be true), which together with primitive definitions, form the foundation upon which theorems are proven and theories are built.

      The choice to use one particular term rather than the other is largely a function of the historical development of a given branch of math. E.g., geometry has roots in ancient Greece, where "postulate" was the word used by the Pythagoreans, et.al.

      So it's largely a matter of history, and context, and the word favored by the mathematicians that introduced or made explicit their "axioms" or "postulates." "Postulate" was once favored over "Axiom", with the development of analytic philosophy, particularly logical positivism, the term "axiom" became the favored term, and its prevalence has persisted since. Perhaps I should be corrected: As Peter Smith points out, it is "more likely" that the "uptake" of the term "axiom" can be attributed to "mathematicians like Hilbert (who talks of axioms of geometry), Zermelo (who talks of axioms of set theory), etc.".

      Neither term is more formal than the other. I personally prefer "postulate" over "axiom", since a "postulate" transparently conveys (or connotes - as in connotation) that what we are calling a postulate is "postulated" as a "supposition", from which we agree to work in building theorems or a theory. In contrast, to me, the connotation of an "axiom" is that of a "law" of some sort, which MUST be followed or MUST be true, though it is no stronger than, or different from, a postulate. But again, this is simply a personal observation and preference, and the term "axiom" seems to have more "uptake" at this point in history
      (31 votes)
  • blobby green style avatar for user zappolia
    What is SSA? Or where can I find the video on it? Thanks!
    (8 votes)
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  • duskpin ultimate style avatar for user graceearp
    Is Hypotenuse Leg Therorem the same as RSH?
    (10 votes)
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  • female robot amelia style avatar for user Mohd Abdallah
    Isn't SSA actually SAS , because in my school we are studying it that way. And is there a video on how to find which property to use in the triangles .. Thank you.what is the meaning of "Wwf"
    (3 votes)
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    • leaf green style avatar for user Christopher
      No, SSA and SAS are two different things! The order of the letters matters a lot. Both of these two postulates tell you that you have two congruent sides and one congruent angle, but the difference is that in SAS, the congruent angle is the one that is formed by the two congruent sides (as you see, the "A" is between the two S), whereas with SSA, you know nothing about the angle formed by the two congruent sides: you only know that the angle formed by the second congruent side and the third side (whose length you don't know) is congruent.

      I fear I'm not being very clear (it would be easier to draw it and show you but I can't), but feel free to ask other questions if you need further explanation.
      (16 votes)
  • duskpin ultimate style avatar for user Rhys Beutler 🤽🏼‍♀️
    Why do we learn about SSA if it is not a congruence postulate?
    (2 votes)
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  • leafers ultimate style avatar for user futurestormchaser02
    At Sal says that you do not want to use SSA as a postulate. In my proofs, how can I reference it in the special case where the given angle is obtuse? Thanks!
    (3 votes)
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  • aqualine ultimate style avatar for user Olivia
    Today in class, my friend and I had a debate about whether or not SSA created congruent triangles. She proved that it did not using the example above, but I proved that it did. I said that if you know two sides and one angle across from one of the sides, you can use the law of sines to find the angle across from the other side. From there, you can find the final angle, and use the law of sines again to find the third side. From that, I concluded that SSA created congruent triangles. Can someone explain what I did wrong?
    (2 votes)
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    • mr pink green style avatar for user David Severin
      You only considered one of the two possible cases you could have when you have SSA, you showed it was okay for one case, but ignored the other case because the sin (Θ) = sin (180 - Θ) which should also work with the law of sines. What you did prove is SSS congruence.
      As a Math teacher, I enjoy these kinds of debates and the deeper thinking that goes behind them.
      (5 votes)
  • male robot hal style avatar for user Arnold
    Just an observation. If you bring the pink line in it will be longer than the green line, meaning for you to make it congruent you might have to change the angle of the green line or the length of the pink line
    (4 votes)
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  • starky tree style avatar for user Dhaval
    But, if the pink side does touch the dotted line at some other point, wouldn't it's length change
    (2 votes)
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    • female robot grace style avatar for user loumast17
      Nope, though there are only two possibilities. it will either make a triangle with three acute angles or one with two acute angles and an obtuse one like was shown in the video it might help if you have something to make triangles with physically, like sticks or something.

      If it helps, moving the pnk one would be moving it in a circle, and picturing that maybe you could see you could make it hit that green line at two different points.
      (4 votes)

Video transcript

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.