Introduction to angles

Hello. In this series of presentations, I'm gonna try

to teach you everything you need to know about triangles and angles and parallel lines

and this is probably the highest-yield information that you could ever learn, especially in terms of the standardized tests.

And then when we've learned all the rules we'll play something I call the Angle Game,

which is essentially what the SAT makes you do over and over again.

So let's start with some basics.You know what an angles is.

Well actually maybe you don't know what an angle is.

If I have two lines...

and they intersect at some point,

the angle is a measure of exactly how wide the intersection is between those two lines.

So this is the angle. An angle is how wide those two lines open up.

And they're measured either in degrees or radiants. And for the sake of most geometry classes we'll use degrees.

When we start doing Trigonometry we'll use radiants.

And you're probably familiar with this. Zero degrees would be two lines on top of each other...

this if I were to just eyeball it looks like 45 degrees.

If I had the lines even wider apart, like that, that's 90 degrees.

And 90 degree lines are also called perpendicular, because

they are, I feel like saying because they are perpendicular,

but because one is going completely vertical while the other is going horizontal.

Wow, it's actually amazingly difficult to find the exact right wording.

But I think you get the idea. By definition, perpendicular lines are 90 degrees apart from each other.

And you've seen this all the time in things like squares or rectangles.

A rectangle is made up of a bunch of perpendicular lines, or lines at 90 degree angles.

The way you draw a 90 degree angle is you draw a little box like that.

That's the same thing as doing this.

And you could even get wider angles. If you go above 90 degrees... this could be, I don't know, 135 degrees

If you ever want to really measure the angles you could use a protractor.

Then if you had it so wide that the two lines are actually almost forming a line...

that's 180 degrees. And then you could keep going.

If this angle is 135 degrees...

There are 360 degrees in a circle. So this magenta angle would be 360 - 135 degrees

that's 225 degrees.

So you know degrees in a circle are 360 degrees, this is important to know.

It's also important to know that if you go halfway around a circle,

that's 180 degrees.

Like if you viewed the
pivot point as like,

let's say, right here.

I mean it looks like just
one line and it really is.

But that's 180 degrees.

And then if you go quarter
way around the circle,

that's 90 degrees.

All right?

Hopefully you're getting
a bit of an intuition

for what an angle is.

So now I will teach you
a bunch of very useful

rules for angles.

Clear this.

So let me redraw.

So if I had a line like this.

I like using the colors, just
so I think it keeps you from

getting completely bored.

And it might not be completely
intuitive what I'm doing, but

let's add an angle like that.

And so, let's just say-- you
know, I'm not measuring these

exactly-- let's say that
this is 30 degrees.

We know that if we go all the
way around the circle, we know

that that's 360 degrees.

Right?

And that's a very ugly
looking around the circle

angle that I drew.

So then we also know
that this angle right

here is 330 degrees.

Right?

Because this angle plus this
magenta angle is going to

equal the whole circle.

So this is equal
to 330 degrees.

So remember that.

The angles in a circle--
or there are 360

degrees in a circle.

I don't know if you remember.

You probably don't.

This was probably
before you were born.

But there used to be a game
called 720, and it was a

skateboarding game--
it was a video game.

And the 720 was essentially
you were trying to jump

your skateboard and
spin around twice.

And that's 720 degrees.

If you go around a circle
twice that's 720 degrees.

If you just jump and
spin around once, you

went 360 degrees.

So you've probably heard this
in just popular culture.

But anyway.

So 360 degrees in a circle.

And you could imagine half
a circle is 180 degrees.

So the other important thing to
realize is, like we said, if

we go halfway around the
circle it's 180 degrees.

But if we have two angles that
add up to that-- so let's say.

I don't know if these lines are
thick enough for you to see.

Let me draw something thicker.

It doesn't look ideal,
but you get the idea.

So if we have this
angle, let's call it x.

And then this angle is y.

What do we know about the
relationship between x and y?

Well, we know that the entire
angle is half of a circle.

Right?

So that's 180 degrees.

That's 180 degrees,
this entire angle.

So what are angles x and
y going to add up to?

I'm trying to stay
color consistent.

x plus y are going to
equal-- I'm color blind,

I think-- 180 degrees.

Or you could write y is
equal to 180 minus x.

Or x is equal to 180 minus y.

But if x plus y are equal to
180 degrees-- and you can see

that it makes sense that they
do-- if you add the two angles

you go halfway around a circle.

Then that tells us that x and y
are-- and this is a fancy word,

and it's just good to commit
this to memory-- they are

supplementary angles.

That's when you add
to 180 degrees.

Now what if we had
this situation.

Oh my God, that was horrible.

Undo.

Let's say I had this situation.

Let's see.

I draw two perpendicular lines.

Right?

So this is going a quarter
way around the circle.

All right.

Let's say this entire angle
here-- I'm drawing it really

big-- that's 90 degrees.

Right?

They're perpendicular.

And now if I had two
angles within that.

So now if I have two angles
here-- so let's say that this

is x and this is y-- what
do x and y add up to?

Well, x plus y is 90.

And we can say that x and
y are complementary.

And it's important to not get
confused between the two.

Just remember complementary
means two angles add up to 90

degrees, supplementary means
that two angles add

up to 180 degrees.