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## Transformations & congruence

# Angle congruence equivalent to having same measure

CCSS.Math:

## Video transcript

- [Instructor] What we're
going to do in this video 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.