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Current time:0:00Total duration:10:17

CA Geometry: Deductive reasoning

Video transcript

all right we're doing the California Standards released questions in geometry now and here's 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 in math class but we'll do it and hopefully I'll help you understand what deductive reasoning is although I do think that you've 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 the definitions let me just 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 conclusions that you know must be true like if I said that you know all 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 oh 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 these two other statements that you knew or two and this one has to be true if those are tortures that bill must be tall that is deductive reasoning deductive reasoning not must be tall right not bill must be deductive so anyway you you have some statements and you deduce other statements that are actually must be true given those and you often 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 we 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 if you 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 let's see if we could find the definition of deductive reasoning in in the California Standards language so use a logic use logic to draw conclusions based on accepted statements yeah well actually that sounds about right that's what we did here we use logic to draw conclusions based on acceptance statements which were those two so I'm gonna 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 oh no that's that's not really anything related to deductive reasoning D inferring a general truth by examining a number of specific examples well this is actually this is more of what what I just talked about inductive reasoning so they want to know what the duct of reasoning is so I'm going to go with a use use logic to draw conclusions based on accepted statements next problem next problem okay let me copy and paste the whole thing as copy and paste is essential with these geometry problems I won't have to redraw everything okay in the diagram below angle 1 and this right here will just you should learn means congruent and when you say to congruence are well 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 why you know there's a difference tree incongruent and being equal is that you know congruence as well they can have the same angle measure but they could be in different directions and they can you know they though the Rays that come out from them could be of different lengths and all that although I'd often say that that's equal as well but if we're dealing with congruence 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 we're measuring them in degrees or radians all right 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 & 4 supplementary angles so what a supplementary mean that means that angle 3 plus angle 4 have to be equal to 180 this is this is a definition of a supplementary angle supplementary if you like I added a supplementary right supplementary so angles 3 & 4 they're actually opposite angles and you could play with these if you if you had these two lines and you kind of change the angle which they intersect you would see that angles 3 & 4 are actually going to be congruent angles they're always going to be equal to each other and measure right so they're equal to each other and if angle 4 is you know let's say angle 4 we don't know what it is if angle 4 is 95 degrees and 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 that's 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 we're 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 will only be true if both of those angles are 90 degrees let's see line line L is parallel to line m yep 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 go we do this quite a bit but opposite angles are equal and that should be intuitive to you at this point because you can you can imagine that if these these two lines if I were to change the angle 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 be congruent to angle 2 and then if these two lines are parallel if L and M are parallel then 2 & 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 a 4 & 2 are the same then that means that these two lines are parallel so these this is definitely true angle 1 is congruent to angle 3 angle 1 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 there are opposite angles or you know instead of staying congruent I could say equal then 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 alright and then the last one 2 is congruent to 3 2 is congruent to 3 well by the same logic if 1 if 1 is equal is congruent to 4 and since 1 & 2 are opposites it's also the same as 2 & 4 is because it's opposite of 3 it's congruent to that all of these angles have to be the same thing so 2 & 3 would also be congruent angles so all of the other ones must be true BC and D so a is definitely our choice next problem the next problem let me copy and paste it okay okay 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 1 every multiple of 4 is even 300 676 is a multiple of 4 so that so this is deductive reasoning right because you know that they say every multiple of 4 is even so you pick any multiple of 4 is going to be even 376 is a multiple 4 therefore it has to be even so this is correct logic so statement 1 is definitely a 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 if you're rational that means then you can write it as a repeating decimal repeating decimal that's like point three three three three two three right that's one third that's all they mean by repeating decimal right but notice this statement right here number can be written as a repeating decimal if it is rational that doesn't say that it 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 you are rational 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 can pi be rational well if pi was rational if pi was in this set if pi were it rational then we could it would be you could write it as a repeating decimal but it says 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 1/n to use deductive reasoning as far as I know let's see next problem actually I'm out of time see you in the next video