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CCSS Math: 8.G.A.5

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