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

All rights belong to their

respective owners.

Copyright Disclaimer Under

Section 107 of the

Copyright Act 1976,

allowance is made for “fair

use” for purposes such as

criticism, comment, news

reporting, TEACHING,

scholarship, and research.

Fair use is a use permitted

by copyright statute that

might otherwise be

infringing.

Non-profit, EDUCATIONAL

or personal use tips the

balance in favor of fair use.