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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.