how are rigid transformations used to justify the sas congruence theorem? This is a topic that many people are looking for. cfiva.org is a channel providing useful information about learning, life, digital marketing and online courses …. it will help you have an overview and solid multi-faceted knowledge . Today, cfiva.org would like to introduce to you Proving the SAS triangle congruence criterion using transformations | Geometry | Khan Academy. Following along are instructions in the video below:
What were going to do in this video is see that if we have have two different triangles and we have two sets of corresponding sides that have the length for example this blue side has the same length as this blue side here. And this orange side has the same length side as this orange side here and the angle. That is formed between those sides.
So we have two corresponding angles right over here that they also have the equal measure. So we could think about we have a side an angle. A side a side an angle and a side if those have the same lengths or measures.
Then we can deduce that these two triangles must be congruent by the rigid motion definition of congruency or the short hand is if we have side angle side in common and the angle is between the two sides then the two triangles will be congruent. So to be able to prove this in order to make this deduction. We just have to say that theres always a rigid transformation.
If we have a side angle side in common that will allow us to map one triangle onto the other because if there is a series of rigid transformations that allow us to do it then by the rigid transformation definition. The two triangles are congruent so the first thing that we could do is we could reference back to where we saw that if we have two segments that have the same length like segment. Ab and segment.
De. If we have two segments with the same length that they are congruent you can always map one segment onto the other with a series of rigid transformations. The way that we could do that in this case is we could map point b onto point e.
So this would be now ill put b prime right over here and if we just did a transformation to do that if we just translated like that then side woops. Then side b. A would that orange side would be something like that but then we could do another rigid transformation that rotates about point e or b.
Prime that rotates that orange side and the whole triangle with it onto de in which case. Once we do that second rigid transformation point a will now coincide with d or we could say a prime is equal to d. But the question is where would c now sit.
Well we can see the distance between a and c in fact we can use our compass for it the distance between a and c is just like that and so since all of these rigid transformations preserve distance. We know that c prime the point that c gets mapped to after those first two transformations c. Prime.
Its distance is going to stay the same from a prime. So c. Prime is going to be some place some place along this curve right over here.
We also know that the rigid transformations preserve angle measures and so we also know that as we do the mapping the angle will be preserved. So either side ac will be mapped to this side right over here. And if thats the case then f would be equal to c.
Prime and we would have found our rigid transformation based on sas and so therefore the two triangles would be congruent. But theres another possibility that the angle gets conserved. But side ac is mapped down here.
So theres another possibility that side ac due to our rigid transformations or after our first set of rigid transformations looks something like this it looks something like that in which case c. Prime would be mapped right over there and in that case. We can just do one more rigid transformation.
We can just do a reflection about de or a prime b prime to reflect point c. Prime over that to get right over there. How do we know that c.
Prime would then be mapped to f. Well this angle. Would be preserved due to the rigid transformation so as we flip it over as we do the reflection over de the angle will be preserved and a prime c prime will then map to df and then wed be done.
We have just shown that theres always a series of rigid transformations as long as you meet this sas criteria. That can map one triangle onto the other and therefore they are congruent. .
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