Angles Formed by Parallel Lines and Transversals

Parallel lines, transversal lines, corresponding angles

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.

Now 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 we're looking at right over here.

And they never intersect! They never intersect. So they're on the same plane but they never intersect each other.

If those two things are true, their not the same line, they never intersect, they can be on the same

plane, then we say that these lines are parallel.

They are moving in the same general direction, in fact the exact same general direction, if we are looking at it from

an algebraic point of view, we would say they have the same slope,

but they have different intersect, they involve different points.

If we do accorded axes here they would intersect that in a different point but they would have the same exact slope.

What I want to do is think how angles relate to paralel lines.

So right over here we have this two paralel lines.

We can say that line AB is paralel to line CD

Sometimes you'll see it specified on geometric drawings like this.

I'll put a little arrow here to show that this two lines are paralel

and if you already used the single arrow then you might put a double arrow

to show that this line is paralel to that line right over there.

Now what I want to do is draw a line that intersects both of this paralel lines.

So here's a line that intersects both of them. Let me draw it a bit near than that.

And I'll just call that line "L".

And this line that intersects both of this paralel lines

we call that a transversal. This is a transversal line.

It's transversing both of this paralel lines.

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 intersection between this transversal line

and the two paralel lines.

So we can first of all start of this angle right over here.

That angle right over there we can call that angle...

well if we make some labels here that would be D, this point and then something else

but I'll just call of this angle right over here.

We know that this is going to be equal to it's vertical angle.

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 here is going to be equal to the angle that is it's vertical angle

or the angle that is opposite the intersection so it's going to be equal to that.

And sometimes you will see it specified like this, you will see a double angle mark like that

or sometimes you will see someone write this

to show that this two are equal and this two are equal right over here.

Now the other thing we know is we can do the exact same exersise up here -

this two are gonna be equal to each other and this two are gonna be equal to each other.

They are all vertical angles.

What's interesting here is thinking about the relationship between this angle

right over here and this angle right up over there.

And if you just look at it it's actually obvious what that relationship is.

They are going to be the same exact angle.

And if you put a protractor here and measure it you'd 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 ore obvious

so if I assume that these two lines are parallel and I have a transversal here

what I'm saying is 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 its 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 intuitively obvious.

That if you look at it, as you tilt these lines, 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 would have one side here of the angle at the 0 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 not that not only this side is equivalent to this side

it is also equivalent to this side over here, and that tells us 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 side over here or 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're opposite, or they're vertical angles.

Now the important thing to realize is the vertical angles are equal and the corresponding angles at the same point of intersection are also equal.

So that's a new word I'm introducing right over here. This angle and this angle are corresponding.

They represent kind of the top right corner in this example where we intersected. Here they represent the top right corner of the intersection.

This would be the top left corner. They're always gonna be equal, corresponding angles.

And once again, really it is a bit obvious.

Now on top of that, there are other words that people would see, 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, maybe if I call this

Let me label them so that we can make some headway here. So I'm gonna use lower case letters

for the angles themselves. 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're corresponding angles.

And then f is equal to g. So vertical angles are equivalent. Corresponding angles are equivalent

and so we also know that obviously that b is equal to g.

And so we say that alternate interior angles are equivalent.

So you see that there are kind of on the interior of the intersection.

They're between the two lines but they're all on the 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, that vertical angles are gonna be equal and corresponding angles are gonna be equal.

And you see it with the other ones too. We know that a is gonna be equal to d, which is going to be equal to h, which is going to be equal to e.