Angle congruence equivalent to having same measure | Congruence | Geometry | Khan Academy

What we're going to do in this video is demonstrate that angles are congruent if and only if they have the same measure. 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 that 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 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 that map angle ABC onto angle DEF.

By definition, rigid transformations 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 the measure of angle ABC is equal to the measure of angle DEF. 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.

To help us there, let's just visualize these angles. So, draw this really fast: angle ABC. An angle is defined by two rays that start at a point, and that point is the vertex. So, that's ABC. And then let me draw angle DEF. So it might look something like this: DEF.

What we will now do is let's do our first rigid transformation. Let's translate angle ABC so that B maps to point E. If we did that, so we're going to translate it like that, then ABC is going to 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 be 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 coincides with ray EF.

Now, you're just going to 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," because 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 the end ray BC coincides with ray EF, now those two rays would be equivalent because the measure of angle ABC is equal to the measure of angle DEF. That will also tell us that ray BA now 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 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.

Because of that, the angles now completely coincide. So we know that angle ABC is congruent to angle DEF, and we're now done. We've proven both sides of the statement: if they're congruent, they have the same measure; if they have the same measure, then they are congruent.