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Angles Formed by Parallel Lines and Transversals

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    Let's say we have
    two lines over here.
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    Let's call this line
    right over here line AB.
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    So A and B both
    sit on this line.
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    And let's say we have
    this other line over here.
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    We'll call this line CD.
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    So it goes through point C
    and it goes through point D.
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    And it just keeps
    on going forever.
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    And let's say that these lines
    both sit on the same plane.
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    And in this case, the
    plane is our screen,
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    or this little piece
    of paper that we're
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    looking at right over here.
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    And they never intersect.
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    So they're on the same
    plane, but they never
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    intersect each other.
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    If those two things are
    true, and when they're not
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    the same line, they
    never intersect
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    and they can be
    on the same plane,
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    then we say that these
    lines are parallel.
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    They're moving in the
    same general direction,
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    in fact, the exact
    same general direction.
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    If we were looking at it from
    an algebraic point of view,
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    we would say that they
    have the same slope,
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    but they have
    different y-intercepts.
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    They involve different points.
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    If we drew our
    coordinate axes here,
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    they would intersect that
    at a different point,
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    but they would have
    the same exact slope.
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    And what I want to
    do is think about how
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    angles relate to parallel lines.
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    So right over here, we have
    these two parallel lines.
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    We can say that line AB
    is parallel to line CD.
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    Sometimes you'll
    see it specified
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    on geometric drawings like this.
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    They'll put a little
    arrow here to show
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    that these two
    lines are parallel.
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    And if you've already
    used the single arrow,
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    they might put a double arrow to
    show that this line is parallel
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    to that line right over there.
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    Now with that out of the
    way, what I want to do
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    is draw a line that intersects
    both of these parallel lines.
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    So here's a line that
    intersects both of them.
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    Let me draw a little
    bit neater than that.
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    So let me draw that
    line right over there.
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    Well, actually, I'll do
    some points over here.
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    Well, I'll just
    call that line l.
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    And this line that intersects
    both of these parallel lines,
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    we call that a transversal.
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    This is a transversal line.
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    It is transversing both
    of these parallel lines.
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    This is a transversal.
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    And what I want to think about
    is the angles that are formed,
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    and how they relate
    to each other.
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    The angles that are
    formed at the intersection
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    between this transversal line
    and the two parallel lines.
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    So we could, first
    of all, start off
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    with this angle right over here.
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    And we could call
    that angle-- well,
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    if we made some labels
    here, that would
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    be D, this point, and
    then something else.
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    But I'll just call it this
    angle right over here.
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    We know that that's going to be
    equal to its vertical angles.
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    So this angle is
    vertical with that one.
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    So it's going to be equal to
    that angle right over there.
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    We also know that this
    angle, right over here,
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    is going to be equal to its
    vertical angle, or the angle
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    that is opposite
    the intersection.
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    So it's going to
    be equal to that.
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    And sometimes you'll
    see it specified
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    like this, where you'll see a
    double angle mark like that.
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    Or sometimes you'll
    see someone write
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    this to show that
    these two are equal
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    and these two are
    equal right over here.
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    Now the other
    thing we know is we
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    could do the exact
    same exercise up here,
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    that these two are going
    to be equal to each other
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    and these two are going
    to be equal to each other.
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    They're all vertical angles.
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    What's interesting here is
    thinking about the relationship
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    between that angle right
    over there, and this angle
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    right up over here.
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    And if you just look at
    it, it is actually obvious
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    what that relationship
    is-- that they
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    are going to be the same
    exact angle, that if you
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    put a protractor
    here and measured it,
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    you would get the exact
    same measure up here.
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    And if I drew
    parallel lines-- maybe
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    I'll draw it straight
    left and right,
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    it might be a little
    bit more obvious.
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    So if I assume that these
    two lines are parallel,
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    and I have a transversal
    here, what I'm saying
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    is that this angle
    is going to be
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    the exact same measure
    as that angle there.
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    And to visualize that, just
    imagine tilting this line.
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    And as you take
    different-- so it
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    looks like it's the
    case over there.
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    If you take the line like this
    and you look at it over here,
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    it's clear that this
    is equal to this.
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    And there's actually
    no proof for this.
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    This is one of those
    things that a mathematician
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    would say is intuitively
    obvious, that if you look
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    at it, as you tilt
    this line, you
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    would say that these
    angles are the same.
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    Or think about putting
    a protractor here
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    to actually measure
    these angles.
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    If you put a
    protractor here, you'd
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    have one side of the
    angle at the zero degree,
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    and the other side would
    specify that point.
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    And if you put the
    protractor over here,
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    the exact same
    thing would happen.
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    One side would be on
    this parallel line,
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    and the other side would
    point at the exact same point.
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    So given that, we
    know that not only is
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    this side equivalent
    to this side,
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    it is also equivalent
    to this side over here.
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    And that tells us
    that that's also
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    equivalent to that
    side over there.
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    So all of these things
    in green are equivalent.
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    And by the same exact
    argument, this angle
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    is going to have the same
    measure as this angle.
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    And that's going to be
    the same as this angle,
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    because they are opposite,
    or they're vertical angles.
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    Now the important
    thing to realize
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    is just what we've deduced here.
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    The vertical angles are equal
    and the corresponding angles
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    at the same points of
    intersection are also equal.
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    And so that's a
    new word that I'm
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    introducing right over here.
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    This angle and this
    angle are corresponding.
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    They represent kind of
    the top right corner,
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    in this example, of
    where we intersected.
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    Here they represent still, I
    guess, the top or the top right
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    corner of the intersection.
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    This would be the
    top left corner.
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    They're always going to be
    equal, corresponding angles.
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    And once again,
    really, it's, I guess,
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    for lack of a better
    word, it is a bit obvious.
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    Now on top of that,
    there are other words
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    that people will see.
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    We've essentially just proven
    that not only is this angle
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    equivalent to this
    angle, but it's also
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    equivalent to this
    angle right over here.
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    And these two angles--
    let me label them
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    so that we can make
    some headway here.
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    So I'm going to use
    lowercase letters
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    for the angles themselves.
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    So let's call this lowercase
    a, lowercase b, lowercase c.
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    So lowercase c for the
    angle, lowercase d,
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    and then let me call
    this e, f, g, h.
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    So we know from vertical
    angles that b is equal to c.
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    But we also know
    that b is equal to f
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    because they are
    corresponding angles.
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    And that f is equal to g.
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    So vertical angles
    are equivalent,
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    corresponding angles
    are equivalent,
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    and so we also know, obviously,
    that b is equal to g.
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    And so we say that alternate
    interior angles are equivalent.
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    So you see that they're
    kind of on the interior
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    of the intersection.
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    They're between the
    two lines, but they're
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    on all opposite sides
    of the transversal.
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    Now you don't have to
    know that fancy word,
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    alternate interior
    angles, you really just
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    have to deduce what
    we just saw over here.
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    Know that vertical angles
    are going to be equal
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    and corresponding angles
    are going to be equal.
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    And you see it with
    the other ones, too.
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    We know that a is going
    to be equal to d, which
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    is going to be equal to h,
    which is going to be equal to e.
Title:
Angles Formed by Parallel Lines and Transversals
Description:

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Video Language:
English
Team:
Khan Academy
Duration:
07:07

English subtitles

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