Geometry 201 unit 2.4

Geometry: Deductive reasoning, laws of detachment and syllogism
Geometry: Deductive reasoning, laws of detachment and syllogism

Warm Up
Identify the hypothesis and conclusion of each
conditional.
1. A mapping that is a reflection is a type of
transformation.
2. The quotient of two negative numbers is positive.
3. Determine if the conditional “If x is a number
then |x| > 0” is true. If false, give a
counterexample.
H: A mapping is a reflection.
C: The mapping is a transformation.
H: Two numbers are negative.
C: The quotient is positive.
F; x = 0.

Apply the Law of Detachment and the
Law of Syllogism in logical reasoning.
Objective

Deductive reasoning is the process of using
logic to draw conclusions from given facts,
definitions, and properties.

Is the conclusion a result of inductive or
deductive reasoning?
Example 1A: Media Application
There is a myth that you can balance an egg on
its end only on the spring equinox. A person was
able to balance an egg on July 8, September 21,
and December 19. Therefore this myth is false.
Since the conclusion is based on a pattern of
observations, it is a result of inductive reasoning.

Is the conclusion a result of inductive or
deductive reasoning?
Example 1B: Media Application
There is a myth that the Great Wall of China is
the only man-made object visible from the
Moon. The Great Wall is barely visible in
photographs taken from 180 miles above Earth.
The Moon is about 237,000 miles from Earth.
Therefore, the myth cannot be true.
The conclusion is based on logical reasoning from
scientific research. It is a result of deductive
reasoning.

Check It Out! Example 1
There is a myth that an eelskin wallet will
demagnetize credit cards because the skin of
the electric eels used to make the wallet holds
an electric charge. However, eelskin products
are not made from electric eels. Therefore, the
myth cannot be true. Is this conclusion a result
of inductive or deductive reasoning?
The conclusion is based on logical reasoning from
scientific research. It is a result of deductive
reasoning.

In deductive reasoning, if the given facts are
true and you apply the correct logic, then the
conclusion must be true. The Law of
Detachment is one valid form of deductive
reasoning.
Law of Detachment
If p q is a true statement and p is true, then
q is true.

Determine if the conjecture is valid by the Law
of Detachment.
Example 2A: Verifying Conjectures by Using the Law
of Detachment
Given: If the side lengths of a triangle are 5 cm,
12 cm, and 13 cm, then the area of the triangle
is 30 cm2
. The area of ∆PQR is 30 cm2
.
Conjecture: The side lengths of ∆PQR are 5cm,
12 cm, and 13 cm.

The given statement “The area of ∆PQR is 30 cm2

matches the conclusion of a true conditional. But this
does not mean the hypothesis is true. The dimensions
of the triangle could be different. So the conjecture is
not valid.
Example 2A: Verifying Conjectures by Using the Law
of Detachment Continued
Identify the hypothesis and conclusion in the
given conditional.
If the side lengths of a triangle are 5 cm, 12 cm,
and 13 cm, then the area of the triangle is 30 cm2
.

Determine if the conjecture is valid by the Law
of Detachment.
Example 2B: Verifying Conjectures by Using the Law
of Detachment
Given: In the World Series, if a team wins four
games, then the team wins the series. The Red
Sox won four games in the 2004 World Series.
Conjecture: The Red Sox won the 2004 World
Series.

Example 2B: Verifying Conjectures by Using the Law
of Detachment Continued
Identify the hypothesis and conclusion in the given
conditional.
In the World Series, if a team wins four games,
then the team wins the series.
The statement “The Red Sox won four games in the
2004 World Series” matches the hypothesis of a true
conditional. By the Law of Detachment, the Red Sox
won the 2004 World Series. The conjecture is valid.

Check It Out! Example 2
Determine if the conjecture is valid by the Law
of Detachment.
Given: If a student passes his classes, the
student is eligible to play sports. Ramon passed
his classes.
Conjecture: Ramon is eligible to play sports.

Identify the hypothesis and conclusion in the given
conditional.
If a student passes his classes, then the
student is eligible to play sports.
The statement “Ramon passed his classes” matches
the hypothesis of a true conditional. By the Law of
Detachment, Ramon is eligible to play sports. The
conjecture is valid.
Check It Out! Example 2 Continued

Another valid form of deductive reasoning is
the Law of Syllogism. It allows you to draw
conclusions from two conditional statements
when the conclusion of one is the hypothesis of
the other.
Law of Syllogism
If p q and q r are true statements, then
p r is a true statement.

Determine if the conjecture is valid by the Law
of Syllogism.
Example 3A: Verifying Conjectures by Using the Law
of Syllogism
Given: If a figure is a kite, then it is a
quadrilateral. If a figure is a quadrilateral, then
it is a polygon.
Conjecture: If a figure is a kite, then it is a
polygon.

Example 3A: Verifying Conjectures by Using the Law
of Syllogism Continued
Let p, q, and r represent the following.
p: A figure is a kite.
q: A figure is a quadrilateral.
r: A figure is a polygon.
You are given that p → q and q → r.
Since q is the conclusion of the first conditional
and the hypothesis of the second conditional, you
can conclude that p → r. The conjecture is valid
by Law of Syllogism.

Determine if the conjecture is valid by the Law
of Syllogism.
Example 3B: Verifying Conjectures by Using the Law
of Syllogism
Given: If a number is divisible by 2, then it is
even. If a number is even, then it is an integer.
Conjecture: If a number is an integer, then it is
divisible by 2.

Example 3B: Verifying Conjectures by Using the Law
of Syllogism Continued
Let x, y, and z represent the following.
x: A number is divisible by 2.
y: A number is even.
z: A number is an integer.
You are given that x → y and y → z. The Law of
Syllogism cannot be used to deduce that z → x.
The conclusion is not valid.

Check It Out! Example 3
Determine if the conjecture is valid by the Law
of Syllogism.
Given: If an animal is a mammal, then it has
hair. If an animal is a dog, then it is a mammal.
Conjecture: If an animal is a dog, then it has
hair.

Let x, y, and z represent the following.
x: An animal is a mammal.
y: An animal has hair.
z: An animal is a dog.
Check It Out! Example 3 Continued
You are given that x → y and z → x.
Since x is the conclusion of the second conditional
and the hypothesis of the first conditional, you can
conclude that z → y. The conjecture is valid by Law
of Syllogism.

Draw a conclusion from the given information.
Example 4: Applying the Laws of Deductive
Reasoning
A. Given: If 2y = 4, then z = –1. If x + 3 = 12,
then 2y = 4. x + 3 = 12
Conclusion: z = –1.
B. If the sum of the measures of two angles is
180°, then the angles are supplementary. If two
angles are supplementary, they are not angles
of a triangle. m∠A= 135°, and m∠B= 45°.
Conclusion: ∠A and ∠B are not angles of a triangle.

Check It Out! Example 4
Draw a conclusion from the given information.
Given: If a polygon is a triangle, then it has
three sides.
If a polygon has three sides, then it is not a
quadrilateral. Polygon P is a triangle.
Conclusion: Polygon P is not a quadrilateral.

Lesson Quiz: Part I
Is the conclusion a result of inductive or
deductive reasoning?
1. At Reagan High School, students must pass
Geometry before they take Algebra 2. Emily is in
Algebra 2, so she must have passed
Geometry.
deductive reasoning

Determine if each conjecture is valid?
2. Given: If n is a natural number, then n is an
integer. If n is an integer, then n is a rational
number. 0.875 is a rational number.
Conjecture: 0.875 is a natural number.
not valid
Lesson Quiz: Part II
3. Given: If an American citizen is at least 18 years
old, then he or she is eligible to vote. Anna is a 20-
year-old American citizen.
Conjecture: Anna is eligible to vote. valid

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