Other Triangle Congruence Postulates SSS, SAS, ASA and AAS postulates for congruent triangles. Showing AAA is only good for similarity and SSA is good for neither
Other Triangle Congruence Postulates
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- We now know that if we have two triangles and all of their corresponding sides
- are the same, so by side-side-side, the corresponding sides, all three of the corresponding sides have the same length,
- we know that those triangles are congruent. What I want to do in this video
- is explore if there are other properties that we can find between the triangles
- that can help us feel pretty good that those two triangles would be congruent.
- So side-side-side works, what about angle-angle-angle?
- So let me do that over here, what about angle-angle-angle?
- So what I'm saying is, let's say I have a triangle like this
- and I have a triangle like that, and if we know that this angle is congruent to that angle,
- if this angle is congruent to that angle, which means that their measures are equal.
- Or/and that angle is congruent to that angle, can we say that these are two congruent triangles?
- At first case, it looks like maybe it is, at least the way I drew it here.
- But when you think about it, you can have the exact same corresponding angles
- having the same measure or being congruent.
- But you can actually scale one of these triangles up and down and still have that property.
- For example, if I had this triangle right over here, it looks similar,
- and I'm using that in just the everyday language sense,
- it has the same shape as these triangles right over here, and it has the same angles
- that angle is congruent to that angle, this angle down here is congruent
- to this angle over here, and this angle over here is congruent to this angle over here.
- So all of the angles in all three of these triangles are the same. The corresponding angles
- have the same measure. But clearly, clearly this triangle right over here is not the same.
- It is not congruent to the other two. The sides have a very different length.
- This side is much shorter than this side right over here, this side is
- much shorter than this side over there, and this side is much shorter over here.
- So with just angle-angle-angle, you can not say that a triangle has a same size and shape.
- It does have the same shape, but not the same size.
- So this does not imply congruency. So angle-angle-angle does not
- imply congruency. What is does imply, and we haven't talked about this yet,
- is that these are similar triangles. So angle-angle-angle implies similar.
- Let me write it over here, it implies similar triangles.
- And similar, you probably are used to the word in just everyday language,
- but similar has a very specific meaning in geometry.
- And similar things have the same shape but not necessarily the same size.
- So anything that is congruent, because it has the same size and shape, is also similar.
- But not everything that is similar is also congruent. So for example, this triangle is similar,
- all of these triangles are similar to each other, but they aren't all congruent.
- These two are congruent if their sides are the same. I didn't make that assumption, but if we
- know that their sides are the same, then we can say that they are congruent.
- But neither of these are congruent to this one right over here because this is clearly much larger.
- It has the same shape, but a different size.
- So we can't have an AAA (angle-angle-angle) postulate, or an AAA axiom
- to get to congruency. What about side-angle-side? Let's try this out.
- Side-angle-side (SAS), so let's start out with one triangle right over here.
- Let's start off with a triangle that looks like this, I have my blue side, I have my pink side,
- and I have my magenta side. And let's say that I have another triangle that has
- this [triangle's] blue side, it has the same side the same length as that blue side.
- So let me draw it like that, it has the same length as that blue side,
- so that length and that length are going to be the same. It has a congruent angle
- right after that. So this angle and the next angle for this triangle are going
- to have the same measure, or they are going to be congruent. Then the next side is going to have
- the same length as this one over here. So that's going to be the same length as this over here.
- So it's going to be the same length.
- And we don't know, and because we only know that... two of the corresponding sides have the same length
- then the angle between them, and this is important, the angle between the two corresponding sides
- are also of the same measure, we can do anything we want with this last side on this one.
- We can essentially, it's going to have to start right over here.
- We can start it from this point, and we can pivot it to form any triangle we want.
- But we can see, the only way we can form a triangle is if we bring the side all the way over here
- and close this right over there.
- And so we can see just logically that if we have, for two triangles
- they have one side, that has the length the same, the next side has the length the same
- and the angle in between them, so this angle, let me do that in the same color.
- This angle in between them, this is the angle, this A is this angle and that angle.
- It's the angle in between them. This first side is in blue, and this second side right over here is in pink.
- I know it's already written in pink.
- So we can see that if two sides are the same length, two corresponding sides are the same length
- and the corresponding angle between them, they have to be congruent.
- There's no other place to put this third side.
- So SAS, and sometimes it's once again called a postulate, an axiom, or if it's proven sometimes it's called a theorem.
- This does imply that the two triangles are congruent.
- So we will give ourselves this tool in our toolkit. We had the SSS postulate, now we have the SAS postulate.
- Two sides are equal and the angle in between them, for two triangles
- corresponding sides and angles, then we can say that it is definitely, these are congruent triangles.
- Now what about, now I'm just going to try to go through all of the different combinations here.
- What if I have something, what if I have angle-side-angle?
- So let me try that. So what happens if I have angle-side-angle?
- So let's go back to this one right over here.
- Actually let me just redraw a new one for each of these cases.
- So angle-side-angle, so I'll draw a triangle here.
- So I have this triangle, so this would be maybe the side, that would be the side.
- Let me draw the whole triangle, actually, first.
- Let me draw one side over here, and then let me draw one side over there
- and that this angle right over here, I'll call it, I'll do it in orange
- and this angle over here, I will do it in yellow.
- So if I have another triangle that has one side having equal measure
- so I'll use it as this blue side right over here.
- So it has one side that has equal measure. And the two angles on either side of that side, or either end of that side
- are the same, will this triangle necessarily be congruent?
- We're just going to try to reason it out. These aren't formal proofs.
- We're really just trying to set up, what are reasonable postulates,
- or what are reasonable assumptions we can have as we, in our toolkit as we try to prove other things.
- So, that angle, let's call it that angle right over there, is going to be,
- it's going to have the same measure in this triangle.
- And this angle right over here in yellow is going to have the same measure on this triangle right over here.
- So regardless, so I'm not in any way constraining the sides over here.
- So this side, this side right over here could have any length.
- It could have any length, but it has to form this angle with it.
- So it could have any length and it can just go as far as it wants to go
- and in no way have we constrained what the length of that is.
- Actually let me mark this off too, so this is the same as this.
- So that side can be anything, we haven't constrained it at all.
- And once again this side could be anything, we haven't constrained it at all
- but we know it has to go at this angle. So it has to go at that angle.
- Well once again, there's only one triangle that can be formed this way.
- We can say all day that this length can be as long as we want or as short as we want,
- and this one can be as long as we want or as short as we want.
- But the only way that they can actually touch each other and form a triangle
- and have these two angles is if, is if they are the exact same length as these,
- as these two sides right over here.
- So these two, this side will actually have to be the same as that side
- and this will have to be the same as that side.
- Once again, this isn't a proof, I'd call it more of a reasoning through it
- or investigation, really just to establish what are reasonable baselines or axioms or assumptions or postulates that we can have.
- So for my purposes, I think ASA does show us that two triangles,
- two triangles are congruent.
- Now let's try another one, let's try angle-angle-side. Let's try angle-angle-side.
- And you know in some geometry classes,
- and maybe if you have to go through an exam quickly you might memorize,
- okay side-side-side implies congruency, that's kind of logical.
- And side-angle-side implies congruency, and so on and so forth.
- I'm not a fan of memorizing it. It might be good for time pressure.
- It is good to sometimes even just go through this logic.
- If you're like, wait, does angle-angle-angle work?
- Well no, I can find this case that breaks down angle-angle-angle.
- Will these work? Just try to verify for yourself, do they make logical sense why they would imply congruency?
- Now let's try angle-angle-side. Let's try angle-angle-side.
- So once again, let's say, let's have a triangle over here.
- It has some side, so this one's going to be a little bit more interesting.
- So it has some side, that's the side right over there.
- And then it has two angles, so let me draw the other sides of this triangle.
- I'll draw one in magenta and then one in green.
- And it has, there's two angles, and then the side.
- So let's see, you have this angle, you have that angle right over there.
- Actually I didn't have to put the double, since that's the first angle that I'm,
- so that angle, which we'll refer to as that first A.
- Then we have this angle, which is that second A.
- So if I know, if I know that there's another,
- that there's another triangle that has one side having the same length.
- Let me draw it like that, it has one side having the same length.
- It has one angle on that side that has the same measure.
- So it has a measure like that.
- And so this, this side right over here could be of any length.
- This side over here could be of any length, we aren't constraining what the length of that side is.
- But whatever the angle is on the other side of that side is going to be the same as this green angle right over here.
- So, for example, it could be like that, and then we could have the green side go like that.
- It could be like that and have the green side go like that.
- And if we have, so the only thing we're assuming
- the only thing we're assuming is that this is the same length as this
- and that this angle has the same measure as that angle
- and that this measure is the same measure as that angle.
- And this magenta line can be of any length
- and this green line can be of any length, we in no way have constrained that.
- But how - can we form any triangle that is not congruent to this?
- Because the bottom line is, this green line is going to have to touch this one right over there
- and the only way it's going to touch that one right over there is if it starts,
- is if it starts right over here. Starts right over here.
- Because we're constraining this angle right over here, we're constraining that angle.
- And so it looks like angle-angle-side does indeed imply congruency.
- So that does imply congruency.
- So let's just do one more, just to kind of try out all of the different situations.
- What if we have, and I'm running out of a little bit of real estate right over here at the bottom.
- What if we tried out side-side-angle?
- So once again, draw a triangle. It has one side there, it has another side there
- and then, I'm not gonna do those hash marks just yet.
- So one side, then another side, and then another side.
- And it, what happens if we know that there's another triangle
- that has two of the sides the same and then the angle after it?
- So for example, it would have that side, just like that
- and then it has another side, but we're not constraining the angle.
- We aren't constraining, we aren't constraining this angle right over here.
- But we're constraining the length of that side.
- So let me color-code it. So that blue side is that first side.
- Then we have this magenta side right over there.
- So this is going to be the same length as this right over here.
- But let me make it at a different angle to see if I can disprove it.
- So let's say it looks like that. Or actually, let me make it even
- let me make it even more interesting. Let me try to make it,
- Let me try to make it like that.
- So it's a very different angle, but now it has to have the same angle out here.
- It has to have that same angle out here, so it has to be roughly that angle.
- So actually it looks like we can draw a triangle that is not congruent that has the same,
- two sides being the same length and then an angle that's different.
- For example, this is pretty much that. I made this angle smaller than this angle.
- These two sides are the same. This angle is the same now, but what the byproduct of that
- is that this green side is going to be shorter on this triangle right over here.
- So these aren't, you don't necessarily have congruent triangles with side-side-angle.
- So this is, this is not necessarily congruent.
- Or similar, it gives us neither congruency nor similarity.
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