Congruence postulates
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Congruent Triangles and SSS
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SSS to Show a Radius is Perpendicular to a Chord that it Bisects
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Other Triangle Congruence Postulates
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Two column proof showing segments are perpendicular
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Finding Congruent Triangles
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Congruency postulates
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More on why SSA is not a postulate
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Perpendicular Radius Bisects Chord
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Congruent Triangle Proof Example
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Congruent Triangle Example 2
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Congruent triangles 1
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Congruent triangles 2
Congruent Triangle Example 2 Showing that segments have the same length
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- Let's say given this diagram right over here
- We know that the length of segment AB is equal to the length of AC
- so AB which is this whole side right over here
- The length of this entire side as a given
- is equal to the length of this entire side right over here
- So that's the entire side right over there
- And then we also know the angle ABF, ABF is equal to angle ACE
- or you could see their measures are equal or this just implies
- that they are congruent so they have the same measures
- It's equal to angle ACE so this angle right over here is congruent
- to that angle right over there
- Or you could say that they have the same measure
- Now the first thing that I want to attempt to prove in this video
- is whether BF, is whether BF has the same length as CE
- Does BF have the same length as CE
- So let's try to do that
- So we already know a few things I could do
- it two column professionally
- let me just do it just so that
- in case you have to do two column proofs
- in your class you can see how to do it more formally
- So let's try out our statements
- Over here I'm going to write I'm going to write my reason for the statement
- So let me just rewrite this kind of formal two column proof
- so we know AB is equal to AC so this is statement 1 and this is given
- We know statement 2 that angle ABF is equal to angle ACE
- Once again that was given
- Now the other interesting thing on each of these ,we have an angle
- and we have a side, each of these triangles
- And then what you can see is both of the triangles
- and when I say both of the triangles
- I'm talking about triangle ABF and triangle ACE
- and they both share this vertex at A and point A is a vertex for both of these
- So, we could say angle A we say BA, let's call it BAF, angle BAF
- We could say is equal to angle BAF
- or we could say is equal to angle CAE
- That makes it little bit clear that we're dealing
- with two different triangles right here
- But it really is the exact same angle
- It's equal to itself right there that's our third statement
- and we could say that it's obvious
- Some people would call this the reflexive property
- It's obvious that an angle is equal to itself
- And so we could say it's obvious
- or we could call it maybe the reflexive property
- that an angle is clearly reflexive obviously equal to itself
- even if we labeled it different way
- this angle is going to be the same measure
- And now we have something interesting going on,
- we have an angle, a side, and an angle
- So, we end up having is that triangle so by angle-side-angle
- we have the triangle BAF so our statement number 4
- I'm running out of space right here I'll go down right here
- Statement right here is triangle BAF triangle BAF
- Let me kind of highlight it a little more blue right here
- BAF so that's this entire triangle right over here
- And half of the trick of some
- of this problems is seeing the right triangle
- So we started with this white angle we went through the side
- that we knew then we went to this orange angle right over here
- BBA I'm sorry we started at this angle
- then we work to this orange angle across the side E that we know
- is congruent to that side over there
- And then we went to the side the aim of the vertex is not labeled
- So at triangle BAF
- we now know is going to be congruent Congruent to triangle
- we start at the white angle
- go to the orange angle then go to the unlabeled angle
- It's going be congruent to angle to triangle CAF
- So, this is kind of a messily drawn version
- but you can get the idea
- These two triangles are going to be congruent
- C A sorry CAE I should say
- is congruent to triangle CAE
- White angle, orange angle, and then the unlabeled angle
- on that triangle right over there
- And this comes straight out of angle-side-angle
- This comes straight out of ASA
- and these are the two angles and so this is the side in between
- so it comes out of the statements 1, 2, and 3
- And so they're congruent, we know that corresponding sides are going
- to be congruent so we know our statement 5
- We should do this a little bit neater
- Our statement 5, we now know that BF is equal to CE
- BF is equal to CE
- And this comes straight out of statement 4,
- or we could say, corresponding sides,
- sides congruent, corresponding sides are congruent
- Now let's take it up another notch
- Let's see if we can prove whether ED is equal to EF
- So let's just keep going down this and see if we can prove
- whether ED is equal to EF
- I put a question mark there
- 'cause we haven't necessarily proven it yet
- So I'm gonna prove that this little short line segment EF is equal to DF
- Sorry not EF is equal to DF, ED is equal to DF
- So let's see if we can prove this right over here
- So the interesting thing that
- we might at first it might not be so obvious
- You know, how do we figure out some type of congruency over that
- but we do already have some information here
- We know that BAF is congruent to CAE
- So we also know that this side right over here
- Let me do it with the color I haven't used yet
- Let me see I have been using a lot of the colors in my pallet
- so it's getting a little too
- So we know from these two congruent triangles that side AE
- Side AE which is part of CAE
- We know that AE is going to be equal to AF
- That these two sides are congruent and the reason why is
- 'cause they're corresponding sides of congruent triangles
- AF is the side opposite the white angle on BAF triangle BAF
- And AE is the side opposite the white angle on triangle CAE
- which we know are congruent
- So we know that AE is equal to AF
- And once again this comes from statement 4
- and we could even say corresponding sides congruent
- Same reason as we gave right up here
- Now, what's interesting here is you know
- this isn't even a triangle that we're seeing up here
- but this information that these two characters
- are congruent help us with this part over here
- Because we know that BA or as to say we know that AB is equal to AC
- that was given and so we know that EB
- Let me write it over here
- and make it a little bit messy right over here
- Statement 7 will give us will give us some space
- we know that BE is going to be equal to CF
- Let me write that down, we know that BE is equal to CF
- And why do we know that let me put the reason right over here
- Let me try to clean up my work a little bit
- This column has been slowly drifting to the left
- But how do we know that BE is equal to CF?
- Well we know that the length of BE is equal to the length of BA minus AE
- or I could just say AB I could that's how
- I call it up here so it's equal to
- AB minus AE is the same thing
- based on these last few things that we saw
- As saying AC minus AF cause AB is equal to AC so that's equal to AC
- and AE we already showed as the same thing as AF
- AC minus AF and AC minus AF is the same thing as CF right over here
- is equal to CF right over there
- And we know that because and we know this from statement 1,
- we know it from statement 5 and we know it from statement 6
- Actually we didn't need we didn't need statement 5 there
- Let me see we just need 1 and 6
- So, let's say we need this is from 1 and 6 is what we had to do there
- So, we just know that look this side is equal to that side
- This little part is equal to that part
- So if you subtract the big part minus the little part
- This right over here is going to be equal to this right over here
- So, that's all we're showing
- So, this yellow side is equal to this yellow side right over here
- Now the other thing that we know
- And this is straight out of vertical angles is that this angle EDB
- is going to be congruent to angle FDC
- So let me write that down again
- So 8, we know that angle EDB is going to be equal to angle FDC
- That comes straight out of vertical angles are congruent
- or their measures are equal
- And now all of a sudden we have something interesting again
- We have the Orange angle-white angle-side
- So we know that these two smaller triangles are congruent
- So, now we know I don't want to lose my diagram
- We know that triangle BED, we know so statement number 9
- we know that triangle BED is congruent
- So, BED is this one we know that BED is congruent to triangle
- Now we want to use the same sides white angle yellow side
- then orange angle
- White angle white angle let me be careful here, white angle
- So B is white angle, E is the unlabeled angle,
- and then D is the labeled angle the orange labeled angle
- So, you want to start C unlabeled angle orange angle
- so triangle CFD
- And this comes straight from once again,
- orange angle white angle side
- So angle-angle-side orange sorry,
- orange angle white angle side
- So this come straight angle-angle-side congruency
- And since we've now shown that this triangle is equal
- to that triangle we know that their corresponding sides are equal
- And then this is our homestretch
- We now know since these two triangles are congruent
- We now know that ED is equal to DF
- because they're corresponding sides
- And I could write that right over here ED is equal to DF
- And once again the reason here is
- the same thing up here corresponding
- so we know our statement 9 which means they're congruent
- And corresponding sides congruent and we are done
- So that was a pretty involved problem
- But you see once again you go step by step
- just try to figure out each triangle and you eventually get it
- But really the hard part isn't so much the realizing which postulate
- to use or how to apply them necessarily
- But seeing the triangle seeing that
- there's some information there
- Seeing that you could figure out BE by subtracting it from AE minus BE
- Seeing that there are two triangles kind of overlapping in the star
- or arms or whatever you might want to call it
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?
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