-
We're on problem number four,
and they give us a theorem.
-
It says a triangle has, at
most, one obtuse angle.
-
Fair enough.
-
Eduardo is proving the theorem
above by contradiction.
-
So the way you prove by a
contradiction, you're like,
-
well what if this
weren't true.
-
Let me prove that that
can't happen.
-
Well let's see what
he did anyway.
-
He began by assuming, that in
triangle ABC, angle A and B
-
are both obtuse.
-
Which theorem will Eduardo use
to reach a contradiction?
-
OK, let me draw this, what
Eduardo is trying to do.
-
The way I'm drawing it is
actually very hard.
-
So this is actually not
drawn at all to scale.
-
So he's saying that angle A and
angle B are both obtuse.
-
So this means that this angle
is greater than 90.
-
Let's say that's angle A.
-
And this is angle B.
-
And it's also greater than 90.
-
That's what obtuse means.
-
Which theorem will Eduardo use
to reach a contradiction?
-
Well, before even reading the
choices, think about it.
-
What do we know about
triangles?
-
That all of the angles add
up to 180 degrees, right?
-
So if this is angle A, this is
angle B, and then let's call
-
this angle C.
-
We know A plus B plus C
have to be equal to
-
180 degrees, right?
-
Or another way to view it
is, C is equal to 180
-
minus A minus B.
-
Or another way you can think of
it, I'm just writing it a
-
bunch of different ways.
-
C is equal to 180 minus
A plus B, right?
-
Now, let me ask you
a question.
-
If we assume from the get-go,
as Eduardo did, if we assume
-
that both A and B are greater
than 90 degrees, what's A plus
-
B going to be at least
greater than?
-
If this is greater than 90 and
that's greater than 90, then A
-
plus B is going to be greater
than 90 plus 90.
-
So this has to be greater
than 180.
-
So if this is greater than 180,
and we're subtracting it
-
from 180, so this essentially
says if angle A is greater
-
than 90, and angle B is greater
than 90, than what we
-
can deduce is, from this
statement right here.
-
From this equation right here.
-
If this and this is greater than
90 then this whole term
-
is greater than 180.
-
So then the deduction would be
that C has to be less than
-
zero, and we can't have
negative angles.
-
So right there, that is
the contradiction.
-
And then you would say, OK,
therefore you cannot have two
-
angles that are more than
90 degrees or two
-
angles that are obtuse.
-
And that would be your proof
by contradiction.
-
Let's see if what we did
can be phrased in
-
one of these choices.
-
If two angles of a triangle are
equal, the sides opposite
-
the angles are equal.
-
No.
-
If two supplementary angles
are equal, the angles each
-
measure 90.
-
Well, we didn't use that.
-
The largest angle of a triangle
is opposite the
-
longest side.
-
No.
-
The sum of the measures of the
angles of a triangle is 180.
-
That's the first thing we
wrote down right there.
-
So it's choice D.
-
That's the theorem Eduardo used
to reach a contradiction.
-
Next question.
-
Problem five.
-
OK, this one.
-
OK, it's a big question.
-
Let me see if I can copy and
paste the whole thing.
-
I've copied it.
-
All right.
-
I think it all fits
in the window.
-
Let's see, it says use
the proof to answer
-
the question below.
-
So given that side AB is
congruent to side BC.
-
So we could say that side
is equal to that side.
-
That's given.
-
D is the midpoint of AC.
-
So that means D is equidistant
between AC.
-
So that means that AD and
DC are equal length.
-
Let me write that.
-
Prove that triangle ABD is
congruent to to triangle CBD.
-
All right, and just so you know,
congruent triangles are
-
triangles that are the same in
every way, except they might
-
have been rotated.
-
They could have been rotated
in some way.
-
If you had similar triangles,
then you could also have
-
different side measures.
-
They're just kind of the same
shape, but they could be
-
expanded or contracted
in some way.
-
If you're congruent, you have
similar triangles but they
-
also have the same
side lengths.
-
But even though they have the
same side lengths, they could
-
be flipped over.
-
Like, you can just
look at this one.
-
ABD looks like it's a
mirror image of DBC.
-
So, just eyeballing it, it
already feels like they're
-
congruent triangles.
-
Let's see how they go
about proving it.
-
So statement one, AB
is congruent to BC,
-
they give us that.
-
D is the midpoint of AC.
-
That was given, fair enough.
-
AD is congruent to CD.
-
That's because D is the
midpoint of AC.
-
We did that part right there,
definition of midpoint.
-
Fair enough.
-
BD is congruent to
BD, of course.
-
Anything is congruent
to itself.
-
So that just says the BD for
that triangle is the same
-
length as the BD for
this triangle.
-
Fair enough, reflexive
property.
-
Fancy word for a very
simple idea.
-
And then finally, they
say triangle ABD is
-
congruent to CBD.
-
OK, well from the get-go, using
these statments, we've
-
already shown that they
have the same
-
exact three side measures.
-
Both triangles have a
side of length BD.
-
Both triangles have a side
of length AD or DC.
-
And both triangles have
a side of length BA.
-
So all of their sides
are the same length.
-
That's what we know after
the first three steps.
-
So what reason can be used
to prove that the
-
triangles are congruent?
-
Well we just said, these three
steps showed that all the
-
sides are the same.
-
So this SSS that you see.
-
What reason?
-
That means side, side, side.
-
And that's just the argument
that you use in your geometry
-
class to say that all
three sides of both
-
triangles are congruent.
-
This means that you have an
angle, an angle, and a side.
-
This means that you have an
angle, and then the side
-
between the two angles.
-
And then the next angle that
all of those are congruent.
-
And this says that one of the
sides and the angle, and the
-
other side, that those
are congruent.
-
We'll probably run
into those in the
-
next couple of questions.
-
But anyway, this shows that
all three sides of both
-
triangles are equal.
-
And then so, we could say
by the side, side, side
-
reasoning, I'm not that
good with terminology.
-
By the side, side, side
reasoning, these are both
-
congruent triangles.
-
And I said, that's one of the
ways of thinking about a
-
congruent triangle, is that all
the sides are going to be
-
the same length.
-
Next question.
-
All right.
-
In the figure below, AB
is greater than BC.
-
OK, so this side is greater
than that side.
-
Although the way they drew it,
they all look the same.
-
So let's see what we can do.
-
If we assume that measure of
angle A is equal to measure of
-
angle C, it follows that
AB is equal to BC.
-
AB is equal to BC.
-
And I don't know if you've run
into this already, but you
-
learned that if you have two
angles that are congruent, or
-
if the measures are the same.
-
This is essentially saying
that angle A is
-
congruent to angle C.
-
They instead just wrote it as
that the measures of the
-
angles are equal.
-
That's what the definition of
congruence is, is that the
-
measures of the angles
are equal.
-
You could have written angle
A is congruent to angle C.
-
But anyway, if you have two
angles that are equal, then
-
the sides that are opposite
those angles are
-
also going to be equal.
-
So this side right here
is going to be
-
equal to that side.
-
And that's what they
wrote here.
-
It follows that AB
is equal to BC.
-
Fair enough.
-
Then they say this contradicts
the given statement that AB is
-
greater than BC.
-
Right, it says, it follows that
AB is equal to BC and it
-
contradicts this statement.
-
Where are they going
with this?
-
What conclusion can be drawn
from this contradiction?
-
Let's see, measure of
angle A is equal to
-
measure of angle B.
-
No, that's not the case.
-
I can think of an example.
-
These can both be 30
degree angles.
-
If these are both 30 degree
angles, add up to 60, then
-
this would have to be 120 for
them to all add up to 180.
-
And it would completely
gel with
-
everything else we've learned.
-
So, A is definitely not right.
-
That A does not have
to be equal to B.
-
Measure of A does not equal
the measure of angle B.
-
Well, they could, right?
-
All of these angles could
be 60 degrees.
-
We haven't said that B
definitely does not equal A.
-
This could be 60, that
could be 60, and so
-
could this be 60.
-
And we'd be dealing with an
equilateral triangle.
-
So I don't think that's
right either.
-
Measure of angle A is equal
to measure of angle C.
-
I see what they're
saying here.
-
Sorry, and this is my bad.
-
They're saying, AB is definitely
greater than BC.
-
Now, they said if we assume
that measure of angle A is
-
equal to measure of angle
C, it follows that
-
AB is equal to BC.
-
They didn't say that this
is definitely true.
-
They just said that if we assume
that this is true.
-
But they didn't say this
is a definite case.
-
And that's where the
contradiction came.
-
Because if we assumed it,
then AB could not be
-
greater than BC.
-
Because then AB would
equal BC.
-
So now I see what
they're asking.
-
So this is an assumption.
-
This isn't actually
proven to be true.
-
So this contradicts the given
statement that AB is
-
greater than BC.
-
Right, that's true.
-
What conclusion can be drawn
from this contradiction?
-
So we made the assumption that
the measure of angle A is
-
equal to the measure
of angle C.
-
That follows that these two
sides are equal, which
-
contradicted the given
statement.
-
Therefore, we know that the
measures of these two angles
-
cannot be equal to each other.
-
Because if they were, then we
would contradict the given
-
assumption.
-
So, we know from the
contradiction that the measure
-
of angle A cannot equal the
measure of angle C.
-
And we can't make that
assumption because it leads to
-
a contradiction.
-
So the correct answer is D.
-
All right, I'll see you
in the next video.