is demonstrate that angles are congruent if and only

if they have the same measure,

and for our definition of congruence,

we will use the rigid transformation definition,

which tells us two figures are congruent if and only

if there exists a series of rigid transformations

which will map one figure onto the other.

And then, what are rigid transformations?

Those are transformations that preserve distance

between points and angle measures.

So, let's get to it.

So, let's start with two angles that are congruent,

and I'm going to show that they have the same measure.

I'm going to demonstrate that, so they start congruent,

so these two angles are congruent to each other.

Now, this means by the rigid transformation

definition of congruence, there is a series

of rigid transformations,

transformations that map

angle ABC onto angle,

I'll do it here, onto angle DEF.

By definition, by definition of rigid transformations,

they preserve angle measure, preserve angle measure.

So, if you're able to map the left angle

onto the right angle, and in doing so, you did it

with transformations that preserved angle measure,

they must now have the same angle measure.

We now know that the measure of angle ABC is equal

to the measure of angle DEF.

So, we've demonstrated this green statement the first way,

that if things are congruent,

they will have the same measure.

Now, let's prove it the other way around.

So now, let's start with the idea that measure of angle ABC

is equal to the measure of angle DEF,

and to demonstrate that these are going to be congruent,

we just have to show that there's always a series

of rigid transformations that will map angle ABC

onto angle DEF, and to help us there,

let's just visualize these angles,

so, draw this really fast, angle ABC,

and angle is defined by two rays that start at a point.

That point is the vertex, so that's ABC,

and then let me draw angle DEF.

So, that might look something like this, DEF,

and what we will now do is let's do

our first rigid transformation.

Let's translate, translate angle ABC

so that B mapped to point E,

and if we did that, so we're gonna translate it like that,

then ABC is going to look something like,

ABC is gonna look something like this.

It's going to look something like this.

B is now mapped onto E.

This would be where A would get mapped to.

This would where C would get mapped to.

Sometimes you might see a notation A prime, C prime,

and this is where B would get mapped to,

and then the next thing I would do

is I would rotate angle ABC about its vertex,

about B, so that ray BC,

ray BC, coincides, coincides with ray EF.

Now, you're just gonna rotate the whole angle that way

so that now, ray BC coincides with ray EF.

Well, you might be saying, "Hey, C doesn't necessarily have

"to sit on F 'cause they might be different distances

"from their vertices," but that's all right.

The ray can be defined by any point that sits on that ray,

so now, if you do this rotation, and ray BC coincides

with ray EF, now those two rays would be equivalent

because measure of angle ABC is equal to the measure

of angle DEF.

That will also tell us that ray BA, ray BA now coincides,

coincides with ray ED, and just like that,

I've given you a series of rigid transformations

that will always work.

If you translate so that the vertices are mapped

onto each other and then you rotate it

so that the bottom ray of one angle coincides

with the bottom ray of the other angle,

then you could say the top ray of the two angles

will now coincide because the angles have the same measure,

and because of that, the angles now completely coincide,

and so we know that angle ABC is congruent to angle DEF,

and we're now done.

We've proven both sides of this statement.

If they're congruent, they have the same measure.

If they have the same measure, then they are congruent.