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## Worked examples

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# CA Geometry: Proof by contradiction

## Video transcript

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.