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• - [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.

• 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.

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

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# Showing angle congruence equivalent to having same measure

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林宜悉 posted on 2020/04/06
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