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Current time:0:00Total duration:7:24

Proving the ASA and AAS triangle congruence criteria using transformations

CCSS.Math:

Video transcript

what we're going to do in this video is show that if we have two different triangles that have one pair of sides that have the same length so these blue sides in each of these triangles have the same length and they have two pairs of angles where for each pair the corresponding angles have the same measure so this gray angle here has the same measure as this angle here and then these double orange arcs show that this angle ACB has the same measure as angle D F E and so we're going to show that if you have two of your angles and a side that have the same measure or length that we can always create a series of rigid transformations that map's one triangle onto the other or another way to say it they must be congruent by the rigid transformation definition of congruence and the reason why I wrote angle side angle here and angle angle side is to realize that these are equivalent because if you have two angles then you know what the third angle is going to be so for example in this case right over here if we know that we have two pairs of angles that have the same measure then that means that the third pair must have the same measure as well so we'll know this as well so if you really think about it if you have the side between the two angles that's equivalent to having an angle and angle and a side because as long as you have two angles the third angle is also going to have the same measure as the corresponding third angle on the other triangle so let's just show a series of rigid rigid transformations that can get us from ABC to D EF so the first step you might imagine we've already shown that if you have two segments of equal length that they are congruent you can have a series of rigid transformations that map's one onto the other so what I want to do is map segment a C on to DF and the way that I could do that is I could translate point A to B on top of point D so then I'll call this a prime and then when I do that this segment AC is going to look something like this I'm just sketching it right now it's going to be in that direction but then and the whole the rest of the triangle is going to come with it so let's see the rest of that orange side set aside a B is going to look something like that but then we could do another rigid transformation which is rotate about point D or point a-prime they're the same point now so that point C coincides with point F and so just like that you would have two rigid transformations that get us that map AC onto D F and so a prime where a is mapped is now equal to D and F is now equal to C Prime but the question is where does point B now sit and the realization here is that angle measures are preserved and since angle measures are preserved we are either going to have a situation where this angle let's see this angle is angle C a B gets preserved so then it would be C prime a prime and then B prime would have to sit someplace on this ra4 if we're gonna preserve the measure of angle C a B B prime is going to sit someplace along that Ray because an angle is defined by two rays that intersect at the vertex or start at the vertex and because this angle is preserved that's the angle that is formed by these two rays you could say r AC a and r a c b we know that v prime also has to sit someplace on this Ray as well so B prime also has to sit someplace on this Ray and I think you see where this is going if V prime because these two angles are preserved because this angle and this angle are preserved have to sit someplace on both of these rays they intersect at one point this point right over here that coincides with point so this is where B prime would be so that's one scenario in which case we've shown that you can get a series of rigid transformations from this triangle to this triangle but there's another one there is a circumstance where the angles get preserved but instead of being on instead of the angles being on the I guess you could say the bottom right side of this blue line you could imagine the angles get preserved such that they are on the other side so the angles get preserved so that they are on the other side of that blue line and then the question is in that situation where would be prime end up well actually let me draw this a little bit let me do this a little bit more exact let me replicate these angles so I'm going to draw an arc like this an arc like this and then I'll measure this distance it's light so it's just like this we've done this in other videos when we're trying to replicate angles so it's like that far and so let me draw that on this point right over here this far so if the angles are on that side of line I guess we could say DF or a prime C prime we know that B prime would have to sit someplace on this Ray so let me draw that as neatly as I can someplace on this Ray and it would have to sit someplace on the Ray formed by the other angle so let me see if I can draw that as neatly as possible so let me make a make a arc like this if I did that a little bit bigger than I need to but hopefully we it serves our purposes I measure this distance right over here if I measure that distance over here it would get us right over there so B prime either sits on this Ray or it could sit or and it has to sit I should really say on this Ray that goes to this point at this point and it has to sit on this Ray and you can see where these two rays intersect is right over there so the other scenario is if the angles get preserved in a way that they're on the other side of that blue line well then B prime is there and then we could just add one more rigid transformation to our series of rigid transformations which is essentially or is a reflection across line D F or a prime C prime why will that work - map B prime on - e well because reflection is also a rigid transformation so angles are preserved and so as this angle gets flipped over it's preserved as this angle gets flipped over the measure of it I should say is preserved and so that means we'll go to that first case where then these rays would be flipped onto these rays and B prime would have to sit on that intersection and there you have it if you have two angles and if you have two angles you're gonna know the third if you have two angles and a side that have the same measure or length if we're talking about an angle or side well that means that they are going to be congruent triangles