Angles at the intersection of two lines

Let's say I have two
intersecting line segments.

So let's call that segment AB.

And then I have segment CD.

So that is C and that is D. And
they intersect right over here

at point E. And let's
say we know, we're given,

that this angle right over here,
that the measure of angle--

That B is kind of, I don't know
why I wrote it so far away.

So let me make that
a little bit closer.

Let me make that B
a little bit closer.

So let's say-- I'll
do that in yellow.

Let's say that we know that
the measure of this angle

right over here,
angle BED, let's

say that we know that
measure is 70 degrees.

Given that information,
what I want to do,

based only on what
we know so far

and not using a protractor,
what I want to do

is figure out what the other
angles in this picture are.

So what's the measure of angle
CEB, the measure of angle AEC,

and the measure of angle AED?

So the first thing
that you might

notice when you
look at this, I've

already told you that
this is a line segment

and that this is a line segment.

You see that angle BED and
angle CEB are adjacent.

And we also see that if
you take the outer sides

of those angles, it
forms a straight angle.

And we also see that angle
CED is a straight angle.

So we know that these two angles
must also be supplementary.

They're next to each other
and they form a straight angle

when you take their outer sides.

So we know that angle BED and
angle CEB are supplementary,

which means they add up to,
or that their measures add up

to 180 degrees.

Supplementary angles.

Which tells us that
the measure of angle

BED plus the measure
of angle CEB--

and I keep writing measure here.

Sometimes you'll just
see people write,

angle BED plus angle CEB
is equal to 180 degrees.

Now we already know the measure
of angle BED is 70 degrees.

So we already know that
this thing right over here

is 70 degrees.

And so 70 degrees plus
the measure of angle CEB

is 180 degrees.

You subtract 70 from
both sides, and we

get the measure of angle
CEB is equal to 110 degrees.

I just subtracted 70
from both sides of that.

So we figured out that this
right over here is 110 degrees.

Well, that's interesting.

And I went through
more steps than you

would if you were doing
this problem quickly.

If you did this problem quickly
in your head, you'd say,

look this is 70 degrees,
this angle plus this angle

would be 180 degrees, so
this has to be 110 degrees.

So now let's use the
same logic to figure out

what angle CEA is.

So now we care about the
measure of angle CEA.

And we can use the exact same
logic that we used over here.

Angle CEA and angle
CEB, they are adjacent.

They form a straight angle,
if you look at their outsides,

so they must be supplementary.

They form a straight
angle right over here.

So they're supplementary.

So they must add
up to 180 degrees.

So the measure of angle CEA
plus the measure of angle CEB,

which is 110 degrees, must
be equal to 180 degrees.

So once again, subtract
110 from both sides.

You get the measure of angle
CEA is equal to 70 degrees.

So this one right over
here is also 70 degrees.

And what we'll learn
in the next video

is that this is no coincidence.

These two angles, angle
CEA and angle BED,

sometimes they're called
opposite angles-- well,

I have often called
them opposite angles,

but the more correct term
for them is vertical angles.

And we haven't proved it.

We've just seen a
special case here

where these vertical
angles are equal.

But it actually turns out that
vertical angles are always

equal.

But we haven't proved it to
ourselves for the general case.

But let me just
write down this word

since it's a nice new word.

So angle CEA and angle
BED are vertical.

And you might say, wait, they
look like they're horizontal,

they're next to each other.

And the vertical
really just means

that they're across
from each other,

across an intersection
from each other.

Angle CEB and angle
AED are also vertical.

So let me write that down.

Angle CEB and angle
AED are also vertical.

And that might even make
a little bit more sense,

because it literally is, one
is on top and one is on bottom.

They're kind of vertically
opposite from each other.

But these horizontally
opposite angles

are also called vertical angles.

So now we have one angle left
to figure out, angle AED.

And based on what
I already told you,

vertical angles tend to be,
or they are always, equal.

But we haven't proven that to
ourselves yet so we can't just

use that property to say
that this is 110 degrees.

So what we're going to do
is use the exact same logic.

CEA and AED are
clearly supplementary.

Their outsides form
a straight angle.

They're clearly
supplementary, so CEA and AED

must add up to 180 degrees.

Or we could say the
measure of angle AED

plus the measure of angle CEA
must be equal to 180 degrees.

We know the measure
of CEA is 70 degrees.

We know it is 70 degrees.

So you subtract 70
from both sides.

You get the measure of angle
AED is equal to 110 degrees.

So we got the exact
result that we expected.

So this angle right over
here is 110 degrees.

And so if you take any
of the adjacent angles

that their outer sides
form a straight angle,

you see they add up to 180.

This one and that
one add up to 180.

This one and that
one add up to 180.

This one and that
one add up to 180.

And this one and that
one add up to 180.

If you go all the way
around the circle,

you'll see that they
add up to 360 degrees.

Because you literally are
going all the way around.

So 70 plus 110 is
180, plus 70 is 250,

plus 110 is 360 degrees.

I'll leave you there.

This is the first
time that we've

kind of found some interesting
results using the tool

kit that we've built up so far.

In the next video,
we'll actually

prove to ourselves using pretty
much the exact same logic here,

but we'll just do it with
generalized numbers--

we won't use 70
degrees-- to prove

that the measure of
vertical angles are equal.