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Angles formed between transversals and parallel lines

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    In this video we're going to
    think a little bit about
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    parallel lines, and other lines
    that intersect the parallel
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    lines, and we call
    those transversals.
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    So first let's think about
    what a parallel or what
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    parallel lines are.
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    So one definition we could use,
    and I think that'll work well
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    for the purposes of this video,
    are they're two lines that
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    sit in the same plane.
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    And when I talk about a plane,
    I'm talking about a, you can
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    imagine a flat two-dimensional
    surface like this screen --
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    this screen is a plane.
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    So two lines that sit in a
    plane that never intersect.
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    So this line -- I'll try my
    best to draw it -- and imagine
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    the line just keeps going in
    that direction and that
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    direction -- let me do another
    one in a different color --
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    and this line right
    here are parallel.
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    They will never intersect.
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    If you assume that I drew it
    straight enough and that
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    they're going in the exact
    same direction, they
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    will never intersect.
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    And so if you think about what
    types of lines are not
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    parallel, well, this green line
    and this pink line
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    are not parallel.
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    They clearly intersect
    at some point.
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    So these two guys are parallel
    right over here, and sometimes
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    it's specified, sometimes
    people will draw an arrow going
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    in the same direction to show
    that those two lines
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    are parallel.
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    If there are multiple parallel
    lines, they might do two arrows
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    and two arrows or whatever.
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    But you just have to say
    OK, these lines will
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    never intersect.
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    What we want to think about
    is what happens when
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    these parallel lines are
    intersected by a third line.
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    Let me draw the
    third line here.
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    So third line like this.
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    And we call that, right there,
    the third line that intersects
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    the parallel lines we
    call a transversal line.
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    Because it tranverses
    the two parallel lines.
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    Now whenever you have a
    transversal crossing parallel
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    lines, you have an interesting
    relationship between
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    the angles form.
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    Now this shows up on a lot
    of standardized tests.
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    It's kind of a core type
    of a geometry problem.
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    So it's a good thing to really
    get clear in our heads.
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    So the first thing to realize
    is if these lines are parallel,
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    we're going to assume these
    lines are parallel, then we
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    have corresponding angles
    are going to be the same.
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    What I mean by corresponding
    angles are I guess you could
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    think there are four angles
    that get formed when this
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    purple line or this
    magenta line intersects
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    this yellow line.
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    You have this angle up here
    that I've specified in green,
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    you have -- let me do another
    one in orange -- you have this
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    angle right here in orange, you
    have this angle right here in
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    this other shade of green, and
    then you have this angle
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    right here -- right there
    that I've made in that
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    bluish-purplish color.
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    So those are the four angles.
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    So when we talk about
    corresponding angles, we're
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    talking about, for example,
    this top right angle in green
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    up here, that corresponds to
    this top right angle in, what
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    I can draw it in that same
    green, right over here.
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    These two angles
    are corresponding.
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    These two are corresponding
    angles and they're
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    going to be equal.
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    These are equal angles.
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    If this is -- I'll make up
    a number -- if this is 70
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    degrees, then this angle
    right here is also
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    going to be 70 degrees.
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    And if you just think about it,
    or if you even play with
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    toothpicks or something, and
    you keep changing the direction
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    of this transversal line,
    you'll see that it actually
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    looks like they should
    always be equal.
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    If I were to take -- let me
    draw two other parallel
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    lines, let me show maybe
    a more extreme example.
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    So if I have two other parallel
    lines like that, and then let
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    me make a transversal that
    forms a smaller -- it's even a
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    smaller angle here -- you see
    that this angle right here
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    looks the same as that angle.
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    Those are corresponding angles
    and they will be equivalent.
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    From this perspective it's kind
    of the top right angle and each
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    intersection is the same.
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    Now the same is true of the
    other corresponding angles.
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    This angle right here in this
    example, it's the top left
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    angle will be the same as the
    top left angle right over here.
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    This bottom left angle will
    be the same down here.
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    If this right here is 70
    degrees, then this down here
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    will also be 70 degrees.
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    And then finally, of course,
    this angle and this angle
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    will also be the same.
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    So corresponding angles -- let
    me write these -- these are
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    corresponding angles
    are congruent.
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    Corresponding angles are equal.
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    And that and that are
    corresponding, that and
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    that, that and that,
    and that and that.
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    Now, the next set of equal
    angles to realize are sometimes
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    they're called vertical angles,
    sometimes they're called
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    opposite angles.
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    But if you take this angle
    right here, the angle that is
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    vertical to it or is opposite
    as you go right across the
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    point of intersection is this
    angle right here, and that is
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    going to be the same thing.
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    So we could say opposite -- I
    like opposite because it's not
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    always in the vertical
    direction, sometimes it's in
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    the horizontal direction, but
    sometimes they're referred
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    to as vertical angles.
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    Opposite or vertical
    angles are also equal.
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    So if that's 70 degrees, then
    this is also 70 degrees.
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    And if this is 70 degrees,
    then this right here
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    is also 70 degrees.
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    So it's interesting, if that's
    70 degrees and that's 70
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    degrees, and if this is 70
    degrees and that is also 70
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    degrees, so no matter what this
    is, this will also be the same
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    thing because this is
    the same as that, that
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    is the same as that.
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    Now, the last one that you need
    to I guess kind of realize are
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    the relationship between
    this orange angle and this
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    green angle right there.
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    You can see that when you add
    up the angles, you go halfway
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    around a circle, right?
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    If you start here you do
    the green angle, then
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    you do the orange angle.
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    You go halfway around the
    circle, and that'll give you,
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    it'll get you to 180 degrees.
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    So this green and orange angle
    have to add up to 180 degrees
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    or they are supplementary.
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    And we've done other videos on
    supplementary, but you just
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    have to realize they form the
    same line or a half circle.
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    So if this right here is 70
    degrees, then this orange angle
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    right here is 110 degrees,
    because they add up to 180.
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    Now, if this character right
    here is 110 degrees, what
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    do we know about this
    character right here?
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    Well, this character is
    opposite or vertical
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    to the 110 degrees so
    it's also 110 degrees.
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    We also know since this angle
    corresponds with this angle,
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    this angle will also
    be 110 degrees.
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    Or we could have said that
    look, because this is 70 and
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    this guy is supplementary,
    these guys have to add up to
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    180 so you could have
    gotten it that way.
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    And you could also figure out
    that since this is 110, this
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    is a corresponding angle,
    it is also going to be 110.
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    Or you could have said
    this is opposite to
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    that so they're equal.
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    Or you could have said that
    this is supplementary with
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    that angle, so 70 plus
    110 have to be 180.
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    Or you could have said 70
    plus this angle are 180.
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    So there's a bunch of ways
    to come to figure out
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    which angle is which.
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    In the next video I'm just
    going to do a bunch of examples
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    just to show that if you know
    one of these angles, you
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    can really figure out
    all of the angles.
Title:
Angles formed between transversals and parallel lines
Description:

Angles of parallel lines

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Video Language:
English
Duration:
07:53

English subtitles

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