I have another one of these fill-in-the-blank triangle
proof exercises.
But this time, instead of doing it
on my own little digital notebook,
I'm going to do it right on the exercise
because there actually are tools so you can mark up
the diagrams.
So first they tell us that segment BC is parallel to DF,
figure not drawn to scale, fair enough.
And it's always important, if we're going to do a proof,
to know what we're going to prove.
We're going to prove the triangle EBD-- so that's
this triangle down here-- is congruent to triangle EBC.
So this one right over here, that this triangle
is congruent to that triangle.
That's what we're going to try to prove.
So let's see what givens they gave us.
They say that-- and you see when you scroll over it,
they show you which two angles they're
talking about-- that angle EBD, or BDE,
is congruent to angle ACB.
So let me mark those.
So I just click once to show that those two are congruent.
Then they say that these two segments, DE and AC
are congruent.
So let me mark those.
You just click on them.
Click on them once it does one slash mark.
If you click on it more than once
it will do more than one slash mark.
Then we also know that these two are congruent, BED and BAC.
So I'm going to put two arcs here
to show that they are not necessarily
congruent to these other two.
So I did those.
Now we have this one, BED-- which we've already marked,
that's BED right over there-- is congruent to CEF.
So that's just telling us that CEF, this angle,
is the same as this one.
So I can put the double hash marks there.
And then they tell us that BE is congruent to EF.
And then finally, they tell us that angle DBE
is congruent to CFE.
I'll put three.
DBE is congruent to CFE.
So now let's try to do our proof.
And they've already kind of done a scaffold of our proof.
And we need to essentially fill in the blanks.
So this first thing, they're comparing two triangles
and they say these two triangles are
congruent because of angle side angle.
So where can we even make that statement?
Well we have this triangle right over here.
We can say it's congruent to this triangle
up here by angle side angle.
Or, we could say that this triangle
is congruent to this triangle, because you have this angle
side angle, this angle side angle.
So it's not clear, based on just this blank,
which one they want us to describe.
Let's go further down in the proof
and maybe it'll become more obvious.
Well then they have this blank, right over here.
Blank equals blank because corresponding parts
of congruent triangles are congruent.
Once again, it's probably based on which two triangles
we picked to be congruent.
This is going to be one of them.
But we don't know whether it's going
to be that one, or that one, depending
on how this proof is going down.
Then they say that triangle ABC is congruent to EBD.
So then they say that this triangle
is congruent to this triangle, right over here.
Well, if down here, they're making that statement,
then up here they must want us to say that this triangle is
congruent to this triangle, right over here.
So we can say that EBD, the double arc, double side,
triple side is going to be congruent to double arc,
double slashes, triple side, so EFC, because of ASA.
There you go.
And it filled it in which is a good indication
that we did it right.
Let's see, what else can we say?
So now, remember, we're trying to prove that this triangle
right over here is congruent to this triangle.
So based on this, we just said that this triangle on the left
is congruent to this triangle on the right.
Given that they're congruent, what is useful information so
that we can get information about this middle triangle?
Well this middle triangle shares a side-- shares side EC,
with this triangle over here.
So we should say that side EC is congruent to something
over here, which will help us set up
that this middle triangle is congruent.
So based on the fact that EBD and EFC are congruent,
EC's corresponding side would be what?
Well, it's the side that's opposite the triple angle.
So it's going to correspond to side DE.
So we could say that DE is congruent to EC
because they're corresponding sides.
They're both the sides opposite this triple angle.
Segment DE is congruent to segment EC.
There you go, and that worked out.
And then they say triangle ABC and EBD-- so that's this one
and this one-- are congruent because--
and we already went over this.
There's multiple ways that you could
say that could-- well actually, no, there's only one.
They have this angle side angle.
That's why they are congruent.
So we have angle side angle congruence.
And then we can say that BC is congruent to BD.
Well those are corresponding sides of congruent triangles.
Those are both the sides opposite the double arc.
Opposite the double arc, and then up here,
opposite the double arc.
So that's because they are corresponding parts
of congruent triangles.
And then finally, so let me mark those off.
So that is congruent to that, corresponding parts
of congruent triangles.
And then finally, we can prove the statement.
And actually you could have probably guessed
what this last one was because this was the end of the proof,
that EBD is congruent to EBC.
And that's, once again, because you have side side side.
We've shown that these three sides
are congruent to these three sides.
It's congruent to EBC.
And you want to make sure you get the order right.
So E is the angle opposite the three sides on this triangle.
And it's also the side opposite the three sides on that one.
And then, corresponding to B, B is opposite the single side.
And over here B is opposite the single side.
So it's going to be EBC.
It's congruent to EBC because of side side side.
And let's check our answer.
Drum roll, and it's right.
Very good.