More on why SSA is not a postulate SSA special cases including RSH
More on why SSA is not a postulate
- 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 its 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 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 say I 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.
- It looks like that. And let's say that we found another triangle,
- we found another triangle that has a congruent side.
- A side that is congruent to this side over here.
- That is-- 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.
- 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 the second side is part of this angle.
- So this is side-side-angle, or you could call it the angle-side-side
- and then giggle a little about it.
- Now, how do we know this doesn't buy itself show this is congruent?
- Well, we would 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 this other triangle has the same yellow angle there.
- Which means that the blue side,
- the blue side has 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 this is congruent to anything.
- If we knew, then we could use side-side-side.
- We only know this side is congruent, this side is congruent,
- and this angle is congruent.
- So this green side (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, and 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 "A hah!" moment here, or 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 are two ways to get down to this base that
- you want to call that way. It could come down that way
- or it could kind of come in this way. And so that is 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, the angle was acute. This is an acute angle right over here.
- This is an acute angle right over here.
- And when you have an acute angle as one of the side of
- your triangle. The other sides could still have an obtuse angle.
- Remember, acute means less than 90º and obtuse means
- greater than 90º.
- So you could still have an obtuse angle, and
- so that is why this is an option.
- So one option is 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 have the option
- where this is even more acute, even narrower, and this becomes
- an obtuse angle.
- So that is an obtuse angle, and that is only possible
- --you can't have two two obtuse angles in the same triangle.
- You can't have two things that have larger than 90º measure
- in the same triangle.
- And so that is why, there is a possibility where if you have
- another triangle that looks like this.
- If you have another triangle that looks like this, then
- And if I were to tell you, and if I were to tell you
- very clearly that this angle is obtuse,
- and 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'm going to say I have
- another triangle, where this angle is congruent to that
- other triangle, some angle in 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 can 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 is necessarily congruent.
- So that could be of any length.
- We do know that this triangle is going to have the same
- same length for this side.
- above the angle, so it looks like this--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 over here.
- So this side could pivot over here.
- We could kind of rotate it over here.
- But there is only one way now that this orange side
- can reach that 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--
- do not want to use it as a postulate.
- I just wanted to make it clear that there is this special case
- where if you know the A in the SSA is obtuse, then it becomes
- a little bit less ambiguous.
- And so finally there is a circumstance where
- this is an acute angle where it would be ambiguous.
- You have the obtuse angle and something in between,
- which is the right angle, 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 are saying it 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 is 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 then 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, called the Right Angle Side Hypotenuse postulate.
- Which is really just a special case of SSA where the angle is.
- where the angle is actually a right angle
- and here they wrote first. You can view this as
- Angle Side Side and they're able to do it because
- now they can write Right Angle, so it doesn't form that embarassing
- acronym. And this can also be a little bit common sense
- because if you know two sides of a right triangle,
- and we haven't gone into depth with this
- in the geometry playlist,
- but you might already be familiar with it.
- By pythagorean theorem, you can just 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 special case, but in general,
- the important thing is that you can't just use SSA
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