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CA Geometry: Proof by contradiction

Video transcript
We're on problem number four, and they give us a theorem. It says a triangle has, at most, one obtuse angle. Fair enough. Eduardo is proving the theorem above by contradiction. So the way you prove by a contradiction, you're like, well what if this weren't true. Let me prove that that can't happen. Well let's see what he did anyway. He began by assuming, that in triangle ABC, angle A and B are both obtuse. Which theorem will Eduardo use to reach a contradiction? OK, let me draw this, what Eduardo is trying to do. The way I'm drawing it is actually very hard. So this is actually not drawn at all to scale. So he's saying that angle A and angle B are both obtuse. So this means that this angle is greater than 90. Let's say that's angle A. And this is angle B. And it's also greater than 90. That's what obtuse means. Which theorem will Eduardo use to reach a contradiction? Well, before even reading the choices, think about it. What do we know about triangles? That all of the angles add up to 180 degrees, right? So if this is angle A, this is angle B, and then let's call this angle C. We know A plus B plus C have to be equal to 180 degrees, right? Or another way to view it is, C is equal to 180 minus A minus B. Or another way you can think of it, I'm just writing it a bunch of different ways. C is equal to 180 minus A plus B, right? Now, let me ask you a question. If we assume from the get-go, as Eduardo did, if we assume that both A and B are greater than 90 degrees, what's A plus B going to be at least greater than? If this is greater than 90 and that's greater than 90, then A plus B is going to be greater than 90 plus 90. So this has to be greater than 180. So if this is greater than 180, and we're subtracting it from 180, so this essentially says if angle A is greater than 90, and angle B is greater than 90, than what we can deduce is, from this statement right here. From this equation right here. If this and this is greater than 90 then this whole term is greater than 180. So then the deduction would be that C has to be less than zero, and we can't have negative angles. So right there, that is the contradiction. And then you would say, OK, therefore you cannot have two angles that are more than 90 degrees or two angles that are obtuse. And that would be your proof by contradiction. Let's see if what we did can be phrased in one of these choices. If two angles of a triangle are equal, the sides opposite the angles are equal. No. If two supplementary angles are equal, the angles each measure 90. Well, we didn't use that. The largest angle of a triangle is opposite the longest side. No. The sum of the measures of the angles of a triangle is 180. That's the first thing we wrote down right there. So it's choice D. That's the theorem Eduardo used to reach a contradiction. Next question. Problem five. OK, this one. OK, it's a big question. Let me see if I can copy and paste the whole thing. I've copied it. All right. I think it all fits in the window. Let's see, it says use the proof to answer the question below. So given that side AB is congruent to side BC. So we could say that side is equal to that side. That's given. D is the midpoint of AC. So that means D is equidistant between AC. So that means that AD and DC are equal length. Let me write that. Prove that triangle ABD is congruent to to triangle CBD. All right, and just so you know, congruent triangles are triangles that are the same in every way, except they might have been rotated. They could have been rotated in some way. If you had similar triangles, then you could also have different side measures. They're just kind of the same shape, but they could be expanded or contracted in some way. If you're congruent, you have similar triangles but they also have the same side lengths. But even though they have the same side lengths, they could be flipped over. Like, you can just look at this one. ABD looks like it's a mirror image of DBC. So, just eyeballing it, it already feels like they're congruent triangles. Let's see how they go about proving it. So statement one, AB is congruent to BC, they give us that. D is the midpoint of AC. That was given, fair enough. AD is congruent to CD. That's because D is the midpoint of AC. We did that part right there, definition of midpoint. Fair enough. BD is congruent to BD, of course. Anything is congruent to itself. So that just says the BD for that triangle is the same length as the BD for this triangle. Fair enough, reflexive property. Fancy word for a very simple idea. And then finally, they say triangle ABD is congruent to CBD. OK, well from the get-go, using these statments, we've already shown that they have the same exact three side measures. Both triangles have a side of length BD. Both triangles have a side of length AD or DC. And both triangles have a side of length BA. So all of their sides are the same length. That's what we know after the first three steps. So what reason can be used to prove that the triangles are congruent? Well we just said, these three steps showed that all the sides are the same. So this SSS that you see. What reason? That means side, side, side. And that's just the argument that you use in your geometry class to say that all three sides of both triangles are congruent. This means that you have an angle, an angle, and a side. This means that you have an angle, and then the side between the two angles. And then the next angle that all of those are congruent. And this says that one of the sides and the angle, and the other side, that those are congruent. We'll probably run into those in the next couple of questions. But anyway, this shows that all three sides of both triangles are equal. And then so, we could say by the side, side, side reasoning, I'm not that good with terminology. By the side, side, side reasoning, these are both congruent triangles. And I said, that's one of the ways of thinking about a congruent triangle, is that all the sides are going to be the same length. Next question. All right. In the figure below, AB is greater than BC. OK, so this side is greater than that side. Although the way they drew it, they all look the same. So let's see what we can do. If we assume that measure of angle A is equal to measure of angle C, it follows that AB is equal to BC. AB is equal to BC. And I don't know if you've run into this already, but you learned that if you have two angles that are congruent, or if the measures are the same. This is essentially saying that angle A is congruent to angle C. They instead just wrote it as that the measures of the angles are equal. That's what the definition of congruence is, is that the measures of the angles are equal. You could have written angle A is congruent to angle C. But anyway, if you have two angles that are equal, then the sides that are opposite those angles are also going to be equal. So this side right here is going to be equal to that side. And that's what they wrote here. It follows that AB is equal to BC. Fair enough. Then they say this contradicts the given statement that AB is greater than BC. Right, it says, it follows that AB is equal to BC and it contradicts this statement. Where are they going with this? What conclusion can be drawn from this contradiction? Let's see, measure of angle A is equal to measure of angle B. No, that's not the case. I can think of an example. These can both be 30 degree angles. If these are both 30 degree angles, add up to 60, then this would have to be 120 for them to all add up to 180. And it would completely gel with everything else we've learned. So, A is definitely not right. That A does not have to be equal to B. Measure of A does not equal the measure of angle B. Well, they could, right? All of these angles could be 60 degrees. We haven't said that B definitely does not equal A. This could be 60, that could be 60, and so could this be 60. And we'd be dealing with an equilateral triangle. So I don't think that's right either. Measure of angle A is equal to measure of angle C. I see what they're saying here. Sorry, and this is my bad. They're saying, AB is definitely greater than BC. Now, they said if we assume that measure of angle A is equal to measure of angle C, it follows that AB is equal to BC. They didn't say that this is definitely true. They just said that if we assume that this is true. But they didn't say this is a definite case. And that's where the contradiction came. Because if we assumed it, then AB could not be greater than BC. Because then AB would equal BC. So now I see what they're asking. So this is an assumption. This isn't actually proven to be true. So this contradicts the given statement that AB is greater than BC. Right, that's true. What conclusion can be drawn from this contradiction? So we made the assumption that the measure of angle A is equal to the measure of angle C. That follows that these two sides are equal, which contradicted the given statement. Therefore, we know that the measures of these two angles cannot be equal to each other. Because if they were, then we would contradict the given assumption. So, we know from the contradiction that the measure of angle A cannot equal the measure of angle C. And we can't make that assumption because it leads to a contradiction. So the correct answer is D. All right, I'll see you in the next video.