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