Holt McDougal Geometry 2-1 Using Inductive Reasoning to Make Conjectures 2-1 Using Inductive Reasoning to Make Conjectures Holt Geometry Warm Up Warm Up.

Holt McDougal Geometry

2-1 Using Inductive Reasoning to Make Conjectures2-1 Using Inductive Reasoning to Make

Conjectures

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2-1 Using Inductive Reasoning to Make Conjectures

Find the next item in the pattern.

Example 1A: Identifying a Pattern

January, March, May, ...

The next month is July.

Alternating months of the year make up the pattern.



Find the next item in the pattern.

Example 1B: Identifying a Pattern

7, 14, 21, 28, …

The next multiple is 35.

Multiples of 7 make up the pattern.



When several examples form a pattern and you assume the pattern will continue, you are applying inductive reasoning. Inductive reasoning is the process of reasoning that a rule or statement is true because specific cases are true. You may use inductive reasoning to draw a conclusion from a pattern. A statement you believe to be true based on inductive reasoning is called a conjecture.



Complete the conjecture.

Example 2A: Making a Conjecture

The sum of two positive numbers is ? .

The sum of two positive numbers is positive.

List some examples and look for a pattern.1 + 1 = 2 3.14 + 0.01 = 3.153,900 + 1,000,017 = 1,003,917



Check It Out! Example 2

The product of two odd numbers is ? .

Complete the conjecture.

The product of two odd numbers is odd.

List some examples and look for a pattern.1 1 = 1 3 3 = 9 5 7 = 35



To show that a conjecture is false, you have to find only one example in which the conjecture is not true. This case is called a counterexample.

To show that a conjecture is always true, you must prove it.

A counterexample can be a drawing, a statement, or a number.



Inductive Reasoning

1. Look for a pattern.

2. Make a conjecture.

3. Prove the conjecture or find a counterexample.



Show that the conjecture is false by finding a counterexample.

Example 4A: Finding a Counterexample

For every integer n, n3 is positive.

Pick integers and substitute them into the expression to see if the conjecture holds.

Let n = 1. Since n3 = 1 and 1 > 0, the conjecture holds.

Let n = –3. Since n3 = –27 and –27 0, the conjecture is false.

n = –3 is a counterexample.




Example 4B: Finding a Counterexample

Two complementary angles are not congruent.

If the two congruent angles both measure 45°, the conjecture is false.

45° + 45° = 90°



Check It Out! Example 4b

Supplementary angles are adjacent.


The supplementary angles are not adjacent, so the conjecture is false.

23° 157°


2-2 Conditional Statements2-2 Conditional Statements

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2-2 Conditional Statements

By phrasing a conjecture as an if-then statement, you can quickly identify its hypothesis and conclusion.



Identify the hypothesis and conclusion of each conditional.

Example 1: Identifying the Parts of a Conditional Statement

A. If today is Thanksgiving Day, then today is Thursday.

B. A number is a rational number if it is an integer.

Hypothesis: Today is Thanksgiving Day.

Conclusion: Today is Thursday.

Hypothesis: A number is an integer.

Conclusion: The number is a rational number.




"A number is divisible by 3 if it is divisible by 6."

Identify the hypothesis and conclusion of the statement.

Hypothesis: A number is divisible by 6.

Conclusion: A number is divisible by 3.



Many sentences without the words if and then can be written as conditionals. To do so, identify the sentence’s hypothesis and conclusion by figuring out which part of the statement depends on the other.



Write a conditional statement from the following.

Example 2A: Writing a Conditional Statement

An obtuse triangle has exactly one obtuse angle.

If a triangle is obtuse, then it has exactly one obtuse angle.

Identify the hypothesis and the conclusion.

An obtuse triangle

has exactly one obtuse angle.




Write a conditional statement from the sentence “Two angles that are complementary are acute.”

If two angles are complementary, then they are acute.

Identify the hypothesis and the conclusion.

Two angles that are complementary

are acute.



A conditional statement has a truth value of either true (T) or false (F). It is false only when the hypothesis is true and the conclusion is false.

To show that a conditional statement is false, you need to find only one counterexample where the hypothesis is true and the conclusion is false.



Determine if the conditional is true. If false, give a counterexample.

Example 3A: Analyzing the Truth Value of a Conditional Statement

If this month is August, then next month is September.

When the hypothesis is true, the conclusion is also true because September follows August. So the conditional is true.



Determine if the conditional is true. If false, give a counterexample.

Example 3B: Analyzing the Truth Value of a Conditional Statement

You can have acute angles with measures of 80° and 30°. In this case, the hypothesis is true, but the conclusion is false.

If two angles are acute, then they are congruent.

Since you can find a counterexample, the conditional is false.



Definition Symbols

A conditional is a statement that can be written in the form "If p, then q."

p q

Related Conditionals



Definition Symbols

The converse is the statement formed by exchanging the hypothesis and conclusion.

q p




Definition Symbols

The inverse is the statement formed by negating the hypothesis and conclusion.

~p ~q




Definition Symbols

The contrapositive is the statement formed by both exchanging and negating the hypothesis and conclusion.

~q ~p




Write the converse, inverse, and contrapostive of the conditional statement “If an animal is a cat, then it has four paws.” Find the truth value of each.


If an animal is a cat, then it has four paws.




Inverse: If an animal is not a cat, then it does not have 4 paws.

Converse: If an animal has 4 paws, then it is a cat.

Contrapositive: If an animal does not have 4 paws, then it is not a cat; True.

If an animal is a cat, then it has four paws.

There are other animals that have 4 paws that are not cats, so the converse is false.

There are animals that are not cats that have 4 paws, so the inverse is false.

Cats have 4 paws, so the contrapositive is true.



Related conditional statements that have the same truth value are called logically equivalent statements. A conditional and its contrapositive are logically equivalent, and so are the converse and inverse.


2-3 Using Deductive Reasoning to Verify Conjectures2-3 Using Deductive Reasoning

to Verify Conjectures

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2-3 Using Deductive Reasoning to Verify Conjectures

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.



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.



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.

Holt McDougal Geometry 2-1 Using Inductive Reasoning to Make Conjectures 2-1 Using Inductive Reasoning to Make Conjectures Holt Geometry Warm Up Warm Up.

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