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CA Geometry: Deductive reasoning

1-3, deductive reasoning and congruent angles. Created by Sal Khan.
Video transcript
All right, we're doing the California Standards released questions in geometry now. And here's the first question. It says, which of the following best describes deductive reasoning? And I'm not a huge fan of when they ask essentially definitional questions in math class. But we'll do it and hopefully it will help you understand what deductive reasoning is. Although, I do think that deductive reasoning itself is probably more natural than the definition they'll give here. Well, actually before I even look at the definitions, let me tell you what it is. And then we can see which of these definitions matches it. Deductive reasoning is, if I give you a bunch of statements and then from those statements you deduce, or you come to some conclusion that you know must be true. Like if, I said that all boys are tall. If I told you all boys are tall. And if I told you that Bill is a boy. Bill is a boy. That's two separate words, a and boy. So if you say, OK, if these two statements are true, what can you deduce? Well I say well, Bill is a boy and all boys are tall. Then Bill must be tall. You deduced this last statement from these two other statements that you knew were true. And this one has to be true if those two are true. So Bill must be tall. That is deductive reasoning. Deductive reasoning. Bill must be tall, not Bill must be deductive. So anyway, you have some statements and you deduce other statements that must be true given those. And you often hear another type of reasoning, inductive reasoning. And that's when you're given a couple of examples and you generalize. Well, I don't want to get too complicated here, because this is a question on deductive reasoning. But generalizations often aren't a good thing. But if you see a couple of examples and you see a pattern there you can often extrapolate and get to a kind of broader generalization. That's inductive reasoning. But that's not what they're asking us about this. Let's see if we can find the definition of deductive reasoning in the California Standards language. Use logic to draw conclusions based on accepted statements. Yeah, well actually that sounds about right. That's what we did here. We used logic to draw conclusions based on accepted statements, which were those two. So I'm going to go with A, so far. Accepting the meaning of a term without definition. Well, I don't even know how one can do that. How do you accept the meaning of something without it having being defined? Let's see, so it's not B. I don't think anything is really B. C, defining mathematical terms to correspond with physical objects. No, that's not really anything related to deductive reasoning either. D, inferring a general truth by examining a number of specific examples. Well, this is more of what I had just talked about, inductive reasoning. So, they want to know what deductive reasoning is, so I'm going to go with A. Use logic to draw conclusions based on accepted statements. Next problem. OK, let me copy and paste the whole thing. Copy and paste is essential with these geometry problems, we won't have to redraw everything. OK, in the diagram below, angle 1, and this right here, you should learn, means congruent. When you say two angles are congruent, so in this case they're saying angle 1 is congruent to angle 4, that means that they have the same angle measure. And the only difference between congruent and being equal is that congruent says they can have the same angle measure, but they could be in different directions. And the rays that come out from them could be of different lengths. Although I'd often say that's equal as well. But if we're dealing with congruency, that's what it means. It essentially just means the angle measures are equal. So we could draw that here. Angle 1 is congruent to angle 4. It just mean that these angle measures are the same. Whether we're measuring them in degrees or radians. All right, now what do they want us to come to a conclusion? Which of the following conclusions does not have to be true? Does not have to be true. Not. Angles 3 and 4 are supplementary angles. What does supplementary mean? That means that angle 3 plus angle 4 have to be equal to 180. This is the definition of a supplementary angle. Supplementary. So angles 3 and 4, they're actually opposite angles. And you can play with these. If you had these two lines and you kind of changed the angle at which they instersect, you would see that angles 3 and 4 are actually going to be congruent angles. They're always going to be equal to each other in measure. So they are equal to each other. And if angle 4 is, we don't know what it is. If angle 4 is 95 degrees, then angle 3 is also going to be 95 degrees. If angle 4 is 30 degrees, angle 3 is also going to be 30 degrees. So I can think of a bunch of cases where this will not be equal to 180. The only way that this would be equal to 180, angle 3 plus angle 4, is if angle 4 were a 90 degree angle, and angle 3 were also a 90 degree angle. But they don't tell us that. All they tell us is that angle 4 and angle 1 are the same, at least when you measure the angles. So I would already go with choice A. That does not have to be true. That'll only be true if both of those angles were 90 degrees. Let's see, line L is parallel to line M. That's true. If that angle is equal to this angle, the best way to think about it is, this angle is also equal to, and you could watch the videos on the angle game that I do. We do this quite a bit. But opposite angles are equal. And that should be intuitive to you at this point. Because you can imagine that if these two lines, if I were to change the angle at which they cross, no matter what angle I do it at, that's always going to be equal to this. So angle 1 is going to congruent to angle 2. And if these two lines are parallel, if L and M are parallel, then 2 and 4 are going to be the same. Or you can think of it the other way. If 4 and 1 are the same, and 1 is the same as 2, then that means 4 is the same as 2. And if 4 and 2 are the same, then that means that these two lines are parallel. So this is definitely true. Angle 1 is congruent to angle 3. Once again, if angle 1 is congruent to angle 4, so those two are congruent, and angle 3 is congruent to angle 4, because they're opposite angles. Or instead of saying congruent, I could say equal. If this is equal to this, and this is equal to this, then this is equal to that. All right. And then, the last one, 2 is congruent to 3. 2 is congruent to 3. Well, by the same logic, if 1 is congruent to 4, and since 1 and 2 are opposite, it's also the same as 2. And 4, because it's opposite of 3, it's congruent to that, all these angles have to be the same thing. So 2 and 3 would also be congruent angles. So all of the other ones must be true, B, C and D. So A is definitely our choice. Next problem. Let me copy and paste it. OK. Consider the arguments below. Every multiple of 4 is even, 376 is a multiple of 4. Therefore, 376 is even. Fair enough. A number can be written as a repeating decimal if it is rational. Pi cannot be written as a repeating decimal. Therefore, pi is not rational. Which ones, if any, use deductive reasoning? OK, so statement one, every multiple of 4 is even. 376 is a multiple of 4. So this is deductive reasoning. Because you know that every multiple of 4 is even. So you pick any multiple of 4, it's going to be even. 376 is a multiple of 4. Therefore, it has to be even. So this is correct logic. So statement one is definitely deductive reasoning. Let's see, statement number two. A number can be written as a repeating decimal if it is rational. So if you're rational, that means that you can write it as a repeating decimal. That's like 0.33333. That's 1/3. That's all they mean by a repeating decimal. But notice, this statement right here, a number can be written as a repeating decimal if it is rational. That doesn't say that a repeating decimal means that it's rational. It just means that a rational number can be written as a repeating decimal. This statement doesn't let us go the other way. It doesn't say that a repeating decimal can definitely be written as a rational number. It just says that if it is rational, a number can be written as a repeating decimal. Fair enough. And then it says pi cannot be written as a repeating decimal. Pi cannot be written as a repeating decimal. So if pi cannot be written as a repeating decimal, pi be rational? Well if pi was rational, if pi was in this set, if pi were rational, then you could write it as a repeating decimal. But it said that you cannot write it as a repeating decimal. So therefore, pi cannot be rational. Cannot be in the set of rationals. So therefore this is also sound deductive reasoning. So both one and two use deductive reasoning. As far as I know. I'm out of time. See you in the next video.