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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.