Learn about CA Geometry: deductive reasoning

Khan Academy Presents: CA Geometry: 1-3, deductive reasoning and congruent angles

Learn about CA Geometry: deductive reasoning All right, we’re doing the California’s Standards Released Questions in Geometry now and here is the first question and it says, “Which of the following best describes deductive reasoning?” And I’m not a huge fan of when they ask essentially definitional questions is Math class but we’ll do it and hopefully it will help you understand what deductive reasoning is. Although, I do think that you probably— deductive reasoning itself is probably more natural than the definition they’ll give here. But let’s— well actually, before I even look at that definitions let me just tell you what it is and then we could see which of these definitions matches it. Deductive reasoning is if I gave you a bunch of statements and then from those statements you deduce or you come to some conclusions that you know must be true. Like if I said that you know, all boys are tall. Like 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 okay, if these two statements are true, what can you deduce? Well, I can say well, Bill is a boy and all boys are tall, then Bill must be tall. You deduced this last statement from this two other statements that you knew are true. And this one has to be true if those are true, too. Bill must be tall. That is deductive reasoning. He must be tall, not Bill must be deductive. So anyway, you have some statements and you deduce other statements that are actually—must be true given those. And you ought to hear this, you know, the opposite of it – not the opposite, but another type of reasoning is inductive reasoning. And that’s when you’re given a couple of examples and you generalize. You know, if I said that, well, I don’t want to get too complicated here because this is a question of deductive reasoning. But it’s essentially – I mean you know 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 a broader generalization. That’s inductive reasoning. But that’s not what they’re asking us about this. Let’s see if we could find the definition of deductive reasoning in the California Standards Language. Use logic to draw a conclusion 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 accepting statements, which were those two. So I’m going to go with A so far. Let’s see, 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 the general truth by examining a number of specific examples. Well this is actually more of what I have 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 a conclusion based on accepted statements. Next problem, okay, let me copy and paste the whole thing, because copy and paste is essential with these geometry problems, you draw everything. Okay. In the diagram below, angle 1 – and this right here, well just, you should learn means congruent. And when we say two angles are congruent, so in this case, we’re saying angle 1 is congruent to angle 4 that means that they have the same angle measure. And the only difference why, you know, there’s a difference between congruent and being equal is that you know, congruent has, well, they can have the same angle measure but they could be in different directions and they can, you know, the rays that come out from them could be in different lengths and all that, although I would often say that 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, which just means that these angle measures are the same, whether I’m measuring them in degrees or radiance. Alright, now, what do they want us to come to 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 a definition of a supplementary angle. Feels like I added a— right, supplementary. So angles 3 and 4, they’re actually opposite angles and you could play with these. If you had these two lines and you kind of change the angle which they intersect, 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, right? So they’re equal to each other and if angle 4 is you know, let’s say 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 now be equal to 180. The only way that this would be equal to 180, angle 3 plus angle 4 is if angle 4 were 90 degree angle and angle 3 were also 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 will only be true if both of those angles are 90 degrees. Let’s see. Line L is parallel to line M. Yeah, 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 a t this point because you can imagine that if these two lines, if I were to change the angle which they cross no matter what angle I do at that, that’s always going to be equal to this. So, angle 1 is going to be congruent to angle 2. And then 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. Well, once again, if angle 1 is congruent to angle 4, right, so those two are congruent. And angle 3 is congruent to angle 4 because they’re opposite angles. Where you know, instead of saying congruent, I could say equal. But angle 3 is also going to be congruent. 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 can— 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 is, and because it’s opposite of 3, it’s congruent to that. All of 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. Okay. Consider the arguments below. Every multiple of 4 is even. 376 is a multiple of 4; therefore, 367 is even. Fair enough. A number can be written as a repeating decimal if it is rational. Pi cannot be written decimal; therefore, pi is not rational. Which ones, if any, used deductive reasoning? Okay. So statement 1: every multiple of 4 is even. 376 are a multiple 4. So this is deductive reasoning, right, because you know that, we say that every multiple of 4 is even. So you pick any multiple of 4 is 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 2: 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 repeating decimal. And that’s like 0.333333, right? That’s one-third. That’s all they mean by repeating decimal, right? 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 is 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 you 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. So, if pi cannot be written as a repeating decimal, can 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 says that you cannot write it as repeating decimal. So therefore, pi cannot be rational. It cannot be in the set of rationales. So therefore, this is also sound deductive reasoning. So, both 1 and 2 used deductive reasoning, as far as I know. Let’s see next problem— Oh, actually, I’m out of time. See you in the next video.