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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 he began by assuming so the way you prove by contradiction you're like okay well what if this weren't true let me prove that that can't happen well let's see what he does 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 okay so he's he's saying let me draw this what eduardo is trying to do the way I'm drawing it it's 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 then let's say that's angle a and this is angle B and it's also greater than itthat's what obtuse means right which theorem will a draw to reach to reach a contradiction well you know 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 let's say that let's call this angle C all right we know a plus B plus C have to be equal to 180 degrees right or another way to view it is C well let me think of it this way C is equal to 180 minus a minus B or another way you could 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 great the 90 then 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 right there that is the contradiction now let's see which which and then you would say okay then therefore you cannot have two angles that are more than ninety degrees or two angles that are obtuse and that would be your proof by contradiction but let's see what what Eduardo let's see if two angle let's see if we 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 ZAR is angle if two supplementary angles are equal the angle each measure 90 well let me I bet we didn't use that the largest angle is triangle is opposite the longest side note the sum of the measures of the angles of a triangle honey yep that's the first thing we wrote down right there so it's choice D that's the theorem Eduardo used to reach the contradiction next question problem 5 problem five okay this one use the proof okay so this is let me it's a big question let me see if I can copy and paste the whole thing I've copied it alright 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 a B side a B is congruent to side B C so like we could say you know that side is equal to that side that's given D is the midpoint of AC so that means D is equidistant to an AC so that means that ad and AC are of equal length let me write that so ad and AC are f equal length or ad and EC prove the triangle abd abd is congruent to triangle C B C B D all right and just so you know congruent triangles are triangles that are the same in every way except they might have been rotated and skewed and well not skewed 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 and they could be expanded or contracted in some way if you're congruent you have similar try you have a similar triangles but they also have the same side lengths but even though they have the same side lengths that could be flipped over like you can just look at this one abd looks like it's a mirror image of D be a or DBC right so just eyeballing it already feels like they're congruent triangles but let's see how they go about proving it so statement one a B is congruent to BC they give us that D is the midpoint of AC right that was given fair enough ad is congruent to CD ad is congruent to CD right 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 right anything is congruent to itself so we could just put three that just says you know the BD for that triangle is the same length as a BD for this triangle or enough reflexive property fancy word for a very simple idea and then finally they say triangle abd is congruent to CBD okay well I mean from the get-go we've already using these statements we've already shown that they have the same exact three side measures right 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 and that's that's what we know after the first three steps so what reason can we be what reason can be used to prove that the triangles are congruent what we just said these three steps we show that all the sides are the same so this SSS that you see what reason that means side side side 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 side and then the angle and then the other side that those are congruent we'll probably run into those in the next couple of questions but anyway this showed that all three sides of those both triangles are equal and then so we can say by the side side side I guess we could say by the side side side reasoning I'm not that good with with terminology by the side side side reasoning these are both congruent triangles and I said that's you know that's kind of one of the ways of thinking about it congruent triangle is that all the sides are going to be the same length next question next question all right all right in the figure below a B is greater than BC okay 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 that measure of angle a is equal to measure of angle C measure of angle a does this a measure of angle a is equal to measure of angle C okay okay it follows that a B is equal to BC right a B is equal to BC and I don't know if you've run into this already but you learn 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 and that's what the definition of congruence is is that the measures of the angle 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 angles that are 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 a B is equal to BC fair enough then they say this contradicts the state the given statement that a B is greater than BC right right I mean you know it says it follows that a B is equal to B seen it contradicts this statement okay where are they going with this what conclusion can be drawn from this country from this contradiction from this contradiction okay let's see measure of angle a is equal to measure of angle B no I can't measure of angle a is equal to measure of angle B no that's not the case I mean you know I can think of an example these could both be 30 degree angles if these are both 30 degree angles add up to 60 then we'd have to get this would have to be 120 for them to all 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 measure of angle a does not equal the measure of angle B well they could right I mean all of these angles could be 60 degrees you know that we'd 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 then 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 measure of angle a is oh I see what they're saying here so they're saying sorry and this is my bad they're saying a B is definitely greater than BC right a B is definitely greater than BC now they said if if we assume that measure of angle a is equal to measure of angle C it follows that a B is equal to BC they didn't say that this is definitely true they just said that if we assume that this is true right but they didn't say that you know this is you know a definitely case and that's where the contradiction came because if we assumed it then a B could not be greater than BC right because then a B would equal BC so now I see what they're asking so this is this is an assumption this isn't actual proven to be true so this contradicts the given statement that a B is greater than BC right that's true what conclusion can be drawn from this contradiction so when we made the assumption that the measure of angle a is equal to the measure of angle C that follow 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 right because if they 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