Videos > CA Geometry: Deductive reasoning
CA Geometry: Deductive reasoning Video Lecture
FAQs on CA Geometry: Deductive reasoning Video Lecture
| 1. What is deductive reasoning in geometry? |  |
Ans. Deductive reasoning in geometry is a process of logical reasoning where conclusions are drawn based on previously known facts and principles. It involves using established rules and properties to make logical deductions and arrive at new conclusions.
| 2. How is deductive reasoning used in geometry proofs? | |
Ans. Deductive reasoning is an essential tool in geometry proofs. It is used to justify each step in a proof by providing logical reasoning and evidence. By applying deductive reasoning, one can connect known facts, definitions, postulates, and theorems to prove the desired statement or theorem.
| 3. What are some common methods of deductive reasoning used in geometry? | |
Ans. In geometry, some common methods of deductive reasoning include the law of detachment, the law of syllogism, and the rules of transitivity and symmetry. These methods help establish logical connections between given information and the desired conclusion in a proof.
| 4. Can deductive reasoning be used to prove all geometric statements? | |
Ans. No, deductive reasoning alone cannot prove all geometric statements. There are certain statements in geometry, known as axioms or postulates, that are accepted as true without proof. These axioms serve as the foundation of deductive reasoning and cannot be proven using deductive reasoning itself.
| 5. How can one improve their skills in deductive reasoning in geometry? | |
Ans. Improving deductive reasoning skills in geometry can be achieved through practice and familiarity with geometric concepts, definitions, postulates, and theorems. Solving a variety of geometry problems, working on proofs, and studying different geometric properties can help enhance deductive reasoning abilities. Additionally, seeking guidance from teachers or utilizing online resources can provide valuable insights and strategies for mastering deductive reasoning in geometry.
Text Transcript from Video
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.
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.
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Sep 08, 2025
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