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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.