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CA Geometry: Proof by Contradiction

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

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Video Language:
English
Team:
Khan Academy
Duration:
11:57

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