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## Geometry (all content)

### Course: Geometry (all content) > Unit 17

Lesson 1: Worked examples- Challenge problems: perimeter & area
- Challenging perimeter problem
- CA Geometry: Deductive reasoning
- CA Geometry: Proof by contradiction
- CA Geometry: More proofs
- CA Geometry: Similar triangles 1
- CA Geometry: More on congruent and similar triangles
- CA Geometry: Triangles and parallelograms
- CA Geometry: Area, pythagorean theorem
- CA Geometry: Area, circumference, volume
- CA Geometry: Pythagorean theorem, area
- CA Geometry: Exterior angles
- CA Geometry: Pythagorean theorem, compass constructions
- CA Geometry: Compass construction
- CA Geometry: Basic trigonometry
- CA Geometry: More trig
- CA Geometry: Circle area chords tangent
- Speed translation

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# CA Geometry: Deductive reasoning

1-3, deductive reasoning and congruent angles. Created by Sal Khan.

## Want to join the conversation?

- Why does he say that he's not a huge fan of definitional questions, like deductive reasoning?(46 votes)
- Because it's a math test! Math isn't only about knowledge of theory, it's about UNDERSTANDING the theory. You shouldn't see a definitional question in a math test, but rather there should be a question testing the candidate's understanding of the topic asked by having to APPLY the theory and use it to solve the problem. That way, you kill two birds with one stone; you test one's theoretical knowledge as well as his/her computational skills which is what math is all about :)(96 votes)

- On question 3, statement II, Sal comes to the conclusion that it indeed is sound deductive reasoning. My thought process was, however, that 1/4 for instance cannot be written as a repeating decimal, yet it is rational. The fact that pi cannot be written as a repeating decimal does not necessarily make it irrational, and therefore I went by A. Where am I wrong?(2 votes)
- Valid deductive reasoning does not test whether the premises are True. Instead, it only states what would be the case IF the premises are true. IF the premises are True, then the conclusion must also be True, in a properly constructed deductive argument. This is known as being logically valid.

The issue of whether the premises are actually True is covered under testing whether the argument is logically sound. In order to be logically sound, the argument must be both logically valid and it must have premises that actually are True.

Sal misused the terminology at the end of the video, where he said "this is sound deductive reasoning". He should have stated "this is valid deductive reasoning".

Thus, if it is the case that all rational numbers can be written as a repeating decimal, then π cannot be rational because it cannot be written as a repeating decimal. This is an example of use of deductive reasoning. This is logically valid, but it is not logically sound.

Whether a number is a terminating or repeating decimal depends on the number base you use. We use base 10 numbers, under which ⅓ is a repeating decimal and ¼ is a terminating decimal. However, in base 9, one third is a terminating "decimal" and one fourth is a repeating "decimal" (we don't actually call them "decimals" in non-base 10 number systems).

However, 0 is rational but cannot be written as a repeating decimal in any number base. Therefore, the premise is false: not all rational numbers can be written as repeating decimals. Thus the argument is a logically unsound argument even though it is logically valid.

Therefore, you are correct: the argument was logically unsound. It was, however, logically valid and should have been included as an example of valid deductive reasoning (which just so happened to be unsound).

Note: By definition, a rational number is a number that can be expressed as the ratio of two integers.(3 votes)

- I'm not sure if Sal was correct on number 3. I didn't get how he explained it.(4 votes)
- Think about it like this: in both arguments, there is a statement, followed by another statement that corresponds or relates to the first statement somehow. Then there is the key word 'therefore', prior to the conclusion made that was based on the two statements. Deductive reasoning is a little tricky, but just remember this basic outline and you'll be fine: statement, statement, 'therefore', conclusion. I hope this helps!(1 vote)

- Can we change the statement "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.” to “A boy can fit into the red booth if he is 100 lbs. Jimmy cannot fit into the red booth. Therefore, Jimmy is not 100 lbs.”?(2 votes)
- Yes, you can logically make that assumption given the first two statements. Good job on figuring that out!(2 votes)

- isn't this considered cheating? these are questions for an official test(2 votes)
- No. These questions are released questions meaning that they are no longer in use. These can be used for study before the actual CST.(2 votes)

- could you show an example of inductive reasoning?(2 votes)
- On question 2, The angles 3 and 4 are obviously not 90 degrees, So can't you just decipher the answer from that observation alone?(1 vote)
- You CANNOT go by observation alone because the diagram might not be drawn to scale. After all, what looks like parallel lines might actually be angled at 0.1° and thus intersect at some point.

Thus, You can only use what is clearly established. The information you use must either be given or logically proven to be true.

In Problem 2, ∠3 and ∠4 are congruent. However, if and only if the transversal (t) intersects ℓ and m at a right angle, then ∠3 and ∠4 are also supplementary. But, since we don't know the angle, we cannot say one way or the other.

But we know that ℓ and m MUST be parallel because ∠3 and ∠4 are congruent.

SO, you**cannot**rely on what the diagram looks like, but only upon what has been clearly proved or given.(4 votes)

- i had a question whether a statement can be both deductive and inductive reasoning?(1 vote)
- I can't think of any statement to say yes to your answer, so I think that it is not possible for a statement to be both deductive and inductive reasoning.

Deductive reasoning is making a specific conclusion based a general statement.

(just an example) Most students in grade 7 love to read books. Ian is a 7th grade student. From this, we can derive that Ian loves to read books.

Inductive reasoning is generalizing an idea based on specific statements.

(just an example) Ian, Tanya, and Tiffany love to read books. They are in 7th grade. From this, we generalize that 7th grade students love to read books.(3 votes)

- What is deductive reasoning in more simple terms? If anyone can answer this question it would be great thanks!! :)(2 votes)
- This confuses me, you cant assume something is 100% true based off of one or two examples... Do they still use this in geometry today?(2 votes)

## 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.