Congruence postulates
-
Congruent Triangles and SSS
-
SSS to Show a Radius is Perpendicular to a Chord that it Bisects
-
Other Triangle Congruence Postulates
-
Two column proof showing segments are perpendicular
-
Finding Congruent Triangles
-
Congruency postulates
-
More on why SSA is not a postulate
-
Perpendicular Radius Bisects Chord
-
Congruent Triangle Proof Example
-
Congruent Triangle Example 2
-
Congruent triangles 1
-
Congruent triangles 2
More on why SSA is not a postulate SSA special cases including RSH
⇐ Use this menu to view and help create subtitles for this video in many different languages.
You'll probably want to hide YouTube's captions if using these subtitles.
- 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
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
|
Have something that's not a question about this content? |
This discussion area is not meant for answering homework questions.
Discuss the site
For general discussions about Khan Academy, visit our Reddit discussion page.
Flag inappropriate posts
Here are posts to avoid making. If you do encounter them, flag them for attention from our Guardians.
abuse
- disrespectful or offensive
- an advertisement
not helpful
- low quality
- not about the video topic
- soliciting votes or seeking badges
- a homework question
- a duplicate answer
- repeatedly making the same post
wrong category
- a tip or feedback in Questions
- a question in Tips & Feedback
- an answer that should be its own question
about the site
Share a tip
Suggest a fix
Have something that's not a tip or feedback about this content?
This discussion area is not meant for answering homework questions.