Let's do a couple of examples dealing with angles between
parallel lines and transversals.
So let's say that these two lines are a parallel, so I can
a label them as being parallel.
That tells us that they will never intersect; that they're
sitting in the same plane.
And let's say I have a transversal right here, which
is just a line that will intersect both of those
parallel lines, and I were to tell you that this angle right
there is 60 degrees and then I were to ask you what is this
angle right over there?
You might say, oh, that's very difficult; that's
on a different line.
But you just have to remember, and the one thing I always
remember, is that corresponding angles are always equivalent.
And so if you look at this angle up here on this top line
where the transversal intersects the top line, what
is the corresponding angle to where the transversal
intersects this bottom line?
Well this is kind of the bottom right angle; you could see
that there's one, two, three, four angles.
So this is on the bottom and kind of to the
right a little bit.
Or maybe you could kind of view it as the southeast angle
if we're thinking in directions that way.
And so the corresponding angle is right over here.
And they're going to be equivalent.
So this right here is 60 degrees.
Now if this angle is 60 degrees, what is the
question mark angle?
Well the question mark angle-- let's call it x --the question
mark angle plus the 60 degree angle, they go halfway
around the circle.
They are supplementary; They will add up to 180 degrees.
So we could write x plus 60 degrees is equal
to 180 degrees.
And if you subtract 60 from both sides of this equation you
get x is equal to 120 degrees.
And you could keep going.
You could actually figure out every angle formed between
the transversals and the parallel lines.
If this is 120 degrees, then the angle opposite to
it is also 120 degrees.
If this angle is 60 degrees, then this one right here
is also 60 degrees.
If this is 60, then its opposite angle is 60 degrees.
And then you could either say that, hey, this has to be
supplementary to either this 60 degree or this 60 degree.
Or you could say that this angle corresponds to this 120
degrees, so it is also 120, and make the same exact argument.
This angle is the same as this angle, so it
is also 120 degrees.
Let's do another one.
Let's say I have two lines.
So that's one line.
Let me do that in purple and let me do the other line in a
different shade of purple.
Let me darken that other one a little bit more.
So you have that purple line and the other one
that's another line.
That's blue or something like that.
And then I have a line that intersects both of them; we
draw that a little bit straighter.
And let's say that this angle right here is 50 degrees.
And let's say that I were also to tell you that this angle
right here is 120 degrees.
Now the question I want to ask here is, are these
two lines parallel?
Is this magenta line and this blue line parallel?
So the way to think about is what would have happened
if they were parallel?
If they were parallel, then this and this would be
corresponding angles, and so then this would be 50 degrees.
This would have to be 50 degrees.
We don't know, so maybe I should put a little asterisk
there to say, we're not sure whether that's 50 degrees.
Maybe put a question mark.
This would be 50 degrees if they were parallel, but this
and this would have to be supplementary; they would have
to add up to 180 degrees.
Actually, regardless of whether the lines are parallel, if I
just take any line and I have something intersecting, if this
angle is 50 and whatever this angle would be, they would have
to add up to 180 degrees.
But we see right here that this will not add up to 180 degrees.
50 plus 120 adds up to 170.
So these lines aren't parallel.
Another way you could have thought about it-- I guess this
would have maybe been a more exact way to think about it
--is if this is 120 degrees, this angle right here has to be
supplementary to that; it has to add up to 180.
So this angle-- do it in this screen --this angle right
here has to be 60 degrees.
Now this angle corresponds to that angle, but
they're not equal.
The corresponding angles are not equal, so these
lines are not parallel.