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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.