![]() ![]() And so since all of these rigid transformations The distance between AĪnd C is just like that. Where would C now sit? Well, we can see theĭistance between A and C. Second rigid transformation, point A will now coincide with D. But then we could doĪnother rigid transformation that rotates about point E, or B prime, that rotates that orange side, and the whole triangle with it, onto DE. Transformation to do that, if we just translated like that, then side, woops, then side B A would, that orange side wouldīe something like that. The way that we could do that in this case is we could map point B onto point E. Segment onto the other with a series of rigid transformations. With the same length that they are congruent. ![]() That have the same length, like segment AB and segment DE. So the first thing that we could do is we could reference back to where we saw that if we have two segments Transformation definition the two triangles are congruent. That allow us to do it, then by the rigid Because if there is a series of rigid transformations So to be able to prove this, in order to make this deduction, we just have to say that there's always a rigid transformation if we have a side, angle, side in common that will allow us to map 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. Lengths or measures, then we can deduce that these two triangles must be congruent by the rigid motionĭefinition of congruency. We have a side, an angle, a side, a side, an angle and a side. And the angle that isįormed between those sides, so we have two correspondingĪngles right over here, that they also have the equal measure. Has the same length side as this orange side here. Side has the same length as this blue side here, and this orange side Of corresponding sides that have the same length, for example this blue Two different triangles, and we have two sets And we're done.Going to do in this video is see that if we have Saying these are my statements, statement, and this is my The two-column proofs, I can make this look a little bit more like a two column-proof by In previous videos, and just to be clear, sometimes people like So we now know that triangleĭCA is indeed congruent to triangle BAC because of angle-angle-side congruency, which we've talked about ![]() And so now, we have two angles and a side, two angles and a side, that are congruent, so we can now deduce byĪngle-angle-side postulate that the triangles are indeed congruent. So, just to be clear, this angle, which is CAB, is congruent to this angle, which is ACD. Part of a transversal, so we can deduce that angle CAB, lemme write this down, I shouldīe doing different color, we can deduce that angle CAB, CAB, is congruent to angle ACD, angle ACD, because they are alternate,Īlternate interior, interior, angles, where a transversal Parallel to DC just like before, and AC can be viewed as Saying that something is going to be congruent to itself. We know that segment AC is congruent to segment AC, it sits in both triangles,Īnd this is by reflexivity, which is a fancy way of Well we know that AC is in both triangles, so it's going to be congruent to itself, and let me write that down. Triangle DCA is congruent to triangle BAC? So let's see what we can deduce now. Over here is 31 degrees, and the measure of this angle Let's say we told you that the measure of this angle right The information given, we actually can't prove congruency. Looks congruent that they are, and so based on just Information that we have, we can't just assume thatīecause something looks parallel, that, or because something Make some other assumptions about some other angles hereĪnd maybe prove congruency. If you did know that, then you would be able to 'cause it looks parallel, but you can't make thatĪssumption just based on how it looks. Side that are congruent, but can we figure out anything else? Well you might be tempted to make a similar argument thinking that this is parallel to that ![]() To be congruent to itself, so in both triangles, we have an angle and a We also know that both of these triangles, both triangle DCA and triangleīAC, they share this side, which by reflexivity is going Parallel to segment AB, that's what these little arrows tell us, and so you can view this segment AC as something of a transversalĪcross those parallel lines, and we know that alternate interior angles would be congruent, so we know for example that the measure of this angle is the same as the measure of this angle, or that those angles are congruent. Pause this video and see if you can figure Like to do in this video is to see if we can prove that triangle DCA is congruent to triangle BAC. ![]()
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