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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.

  • 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.

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

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C1 angle congruent measure def rigid mapped

Showing angle congruence equivalent to having same measure

  • 7 0
    林宜悉 posted on 2020/04/06
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