 I'm Adam, I'm a student of Dr. Missildine at Southern Utah University, and today we're going to be exploring a theorem in congruence geometry that supplements of congruence angles are congruence. Now, keep in mind that in congruence geometry, we have Hilbert's axioms of incidence, Hilbert's axioms of betweenness, and Hilbert's axioms of congruence. So that's what we have to work with, along with any theorems we've proved along the way. All right, so let's define what supplements are. So if I construct this angle here, we'll call these points A, B, C, D, then the supplement of the angle A, B, C is the angle D, B, C, because A, B, and D are all on the same line. All right, and that's what a supplement is. Okay, so to prove this, let's go ahead and construct this, and label these points A' B', that should be a D' C', okay, and then by segment translation we can say, okay, there's some segment on the ray B' A', such that the segments A, B, and whatever that new segment is are congruent. And without loss of generality, we can just say that A' is that point that makes them congruent, because we can move it to be that point. So we can say that A, B is congruent to A' B', additionally, and similarly, we can say that B, C is congruent to B' C', and B, D is congruent to B' D', just by moving those points around so that the points are in the right place to be congruent, because if they're not, we can just choose another point so they are, all right, it's up to relabeling, we're just going to go ahead and say that those segments are congruent. And by assumption, we know that these angles are congruent, and to prove that supplements are congruent angles are congruent, we need to show then that angle DBC is congruent to angle D' B' C', and we're going to do that with several triangle congruences. All right, the first of which is going to be from connecting the points A and C, and A' and C', and then we know that this triangle ABC is congruent to A' B' C', because the segment AB is congruent to A' B', angle ABC is congruent to the angle A' B' C', and the segment BC is congruent to the segment B' C'. So by side angle side, we know that these triangles are congruent, and since the corresponding parts of congruent triangles are congruent, we can say a couple things. We can say that the angle ABAC is congruent to the angle B' A' C', and we can say that the segment AC is congruent to the segment A' C', and there are more things we can say about that, but that's all we need for this construction, for this theorem, so we're going to stop there. Now what else do we have? So we know that AB is congruent to A' B', and BD is congruent to B' D'. So by segment addition, the segment AD is congruent to the segment A' D'. All right, so now we have several things we can work with. Let's go ahead and connect one more piece in our triangles from C to D, and from C' to D'. And let's investigate what we have. Well we know that AD is congruent to A' D', DAC, the angle, is congruent to the angle D' A' C', and the segment AC is congruent to the segment A' C', so by another side angle side argument, we can say that the triangle ADC, the big triangle here, is congruent to the triangle A' D', C'. So as corresponding parts of congruent triangles are congruent, we can say that CD is congruent to C' D', and angle ADC is congruent to angle A' D' C'. All right, what does that give us? Well let's look now at the triangle, which I'll highlight because it's getting messy in here, CBD, and the triangle C' B' D'. All right, since BD is congruent to the segment D' D', the angle BDC is congruent to the angle B' D' C', and the segment DC is congruent to the segment D' C', then by another side angle side argument, we know that the triangle BDC is congruent to the triangle B' D' C'. And once again, as corresponding parts of congruent triangles are congruent, the angle CBD is congruent to the angle C' B' D'. And if you'll notice, that's the angle we were trying to prove that was congruent, right? Because we're trying to show the supplements, the angles are congruent, and the supplements we started with were ABC and A' B' C', or those were the angles we started with that were congruent. So their supplements are CBD and C' B' D', therefore, supplements of congruent angles are congruent. Thank you for watching.