I've taken a screen shot from our fill in the blank proof
exercises.
And I'm going to work it out, essentially, on digital paper.
And then we can fill in the blanks on the actual exercise.
So they tell us that BC is parallel to DF.
The figure is not drawn to scale-- fair enough.
Then they tell us that angle CEF is congruent to angle BAC.
CEF, so that's this right over here.
CEF is congruent to BAC.
So it's congruent to that over there.
Then they tell us that CFE is congruent to ABC.
Angle CFE is congruent to angle ABC.
Then they tell us that segment CF is congruent to BC.
Then they tell us that BDE is congruent to ACB.
So that's that right over there.
Then they tell us DE is congruent to AC.
And then finally, they tell us that-- let
me do that in a different color--
so BED is congruent to BAC.
Let me make sure I get it.
BED-- so it's this angle right over here--
is congruent to BAC.
BAC is this blue one over there, so we already
have a marker for that.
So I'll write the blue one just like that.
So that's that piece of information as well.
Now we're ready for the proof.
Now whenever you do any proof, it's
good to know what you're trying to prove.
So let's just cut to the chase and see
where this proof actually goes.
So in the last statement of the proof,
they say triangle BCA is congruent to triangle BCE
because-- and we have to figure out
why we can say that based on everything
that this proof has proven beforehand.
But so we're essentially just proving
that triangle BCA, which is this triangle right up here-- this
is BCA-- that it is congruent to triangle BCE.
BCE is this triangle-- this downward-pointing triangle.
So this is what we're trying to prove.
This is congruent to that right over here.
This triangle is congruent to that triangle.
So that's important to keep in mind.
And all we know about these two triangles
is it they share a side in common, which
is equal to itself by the reflexive property.
But we don't have any other information.
We're going to have to figure out whether other sides are
congruent or maybe other angles or something else
in order to actually make this statement.
So let's see what they've done in the proof.
So first they make a statement.
And we don't need to correct it, but it's
good to follow the narrative of their proof.
BCA is congruent to BDE because of angle, side, angle.
So let's see, we have an angle, side, angle; angle,
side, angle.
This triple arc corresponds to D here, corresponds to C there.
And you see that even over here.
The way they wrote it, D is the middle here,
and C is the middle there.
So they do correspond to each other.
That makes sense.
This side, DE, should correspond to CA.
And you see that over here.
DE corresponds to CA based on the order it's written.
E comes right after D. A comes right after C.
And angle E should correspond to angle A, and it does.
Angle E corresponds to angle A, and it does.
So this is completely legitimate-- angle, side,
angle; angle, side, angle It's a little bit hard
based on how they're oriented.
It would've been easier if they were oriented in the same way.
But this is legitimate.
By angle, side, angle, these two triangles are congruent.
Then they say, something is equal to something
because corresponding parts of congruent triangles
are congruent.
And this is a little bit harder because they're probably
making a statement-- we just proved that this triangle
right over here is congruent to this triangle up here.
And so if we're going to say that two corresponding things
are congruent, it's probably two corresponding things
in these two triangles.
And what is corresponding in these two triangles that
will help us get to our end goal of saying
that this top triangle is congruent to this bottom one?
Well the only information that this side triangle right over
here, that BDE shares with the central triangle is this side.
So maybe we can say that this side corresponds to some side
up here.
And it does because they're congruent.
And then that'll give us information
for this middle triangle.
So let's just think about that.
And it may or may not be right, but it's a good working guess.
And there's different ways to tackle these.
So this side over here, if we look at how this is-- BE
should correspond to BA.
BE-- and remember, it's between the blue
and the not triple green side.
So these two should correspond based
on the information right over here.
Let me do this in a different color.
This thing should be congruent to that thing right over there.
And that gives us interesting information
because now we've built up two congruent sides
of this middle one and this top one.
So let's write that as our working
thesis of what they want here.
So we can say BE-- well, we could say the length is equal--
but we could even say BE is congruent to BA
because corresponding parts of congruent triangles
are congruent.
And this is not an easy thing to do.
But the real hint is they just told us
that these two things are congruent.
And we want some information about this middle triangle.
And they share a common side here.
So it seems reasonable enough.
Hopefully that gets us to the right end point.
Now they tell us that triangle BCA is congruent to FCE.
And we started at F. We started at B. That's
the double angle side.
Then you went across this one side,
and you went across this side with the magenta marker.
So that seems right because of angle, angle, side.
Angle, angle, side-- angle, the blue angle,
double magenta angle, and then this single, magenta slash
side.
Angle, angle, side-- angle, angle, side.
That seems legitimate, and it should
unless there was some type of error in this code.
And then they say AC is congruent to CE
because-- well we just figured out that this top triangle is
congruent to this one down over here.
So it's probably going to be because of corresponding sides
of congruent triangles.
And we can verify it.
If this is congruent to that, then let's see,
AC is between the blue angle and the non-magenta angle.
And EC is between the blue angle and the non-magenta angle.
I haven't marked it here.
And so we have AC is indeed congruent to CE.
And now this is interesting, and this was useful
because this shared a side with this middle triangle
that we care about.
And so we now show that this top triangle, all the sides
have the same lengths as all the sides down here.
So this was actually useful information
that we figured out, so I think that was the right way
to go about it.
Oh, I need to finish this.
AC is congruent to CE because corresponding parts
of corresponding angles.
Let me write it in black.
And when we actually fill it in, there
will be a pull-down-- so corresponding parts.
And then finally, we're in the home stretch.
We're ready to say that BCA, this top triangle,
is congruent to triangle BCE.
All their sides are the same.
So this is side, side, side congruency.
So now let's verify our answer.
So let's take out the actual exercise.
And so let's go down here.
The first thing I want to type in is that BE is equal to BA.
So let me type that in, BE-- and if this is right,
it'll actually just process it-- is equal to BA.
And it is.
There you go.
Segment BE is congruent AB.
So that's how we know we've got that right.
I wrote it as BA, but that's the same thing.
And that's why it processed it.
And then AC is congruent to CE because of corresponding parts.
There you go.
And then finally, these two triangles
are congruent because of side, side, side congruence.
And now let's verify our answer.
I feel good about it, and I am ready to check my answer.
Yes, it is correct.