name: chapter 2: reasoning and proof lesson 2-1: inductive...

Name: Chapter 2: Reasoning and Proof

Lesson 2-1: Inductive Reasoning and Conjecture Date:

reasoning is reasoning that uses a number of specific examples to arrive at a

conclusion.

A concluding statement reached using inductive reasoning is called a .

Example 1: Patterns and Conjecture

Write a conjecture that describes the pattern 2, 4, 12, 48, 240. Then use your conjecture to find the next

item in the sequence.

Example 2: Algebraic and Geometric Conjectures

Make a conjecture about the sum of an odd number and an even number. List some examples that support

your conjecture.

To show that a conjecture is true for all cases, you must it.

It only takes one false example, or to show that a conjecture is not true.

Real-World Example 3: Find Counterexamples

UNEMPLOYMENT Based on the table showing unemployment rates for various counties in Texas, find a

counterexample for the following statement. The unemployment rate is highest in the cities with the most

people.


Lesson 2-2: Logic Date:

A is a sentence that is either true or false.

The of a statement is either true (T) or false (F).

Statements are often represented using a letter such as 𝑝 or 𝑞.

A convenient method for organizing truth values of statements is to use a .

Negation Conjunction Disjunction

Example 1: Construct Truth Tables

A. Construct a truth table for ~𝑝 ∨ 𝑞. B. Construct a truth table for 𝑝 ∨ (~𝑞 ∧ 𝑟).


Lesson 2-3: Conditional Statements Date:

Example 1: Identify the Hypothesis and Conclusion

A. If a polygon has 6 sides, then it is a hexagon.

B. Tamika will advance to the next level of play if she completes the maze in her computer game.

*Do not try to determine whether the argument makes

sense. Instead, analyze the form of the argument to

determine whether the conclusion follows logically from

the hypothesis.

*When the hypothesis is false, the conditional is always

_____________.

* A conditional is false only when the hypothesis is

__________ and the conclusion is _____________.

Example 2: Truth Values of Conditionals

Determine the truth value of the conditional statement. If true, explain your reasoning. If false, give a

counterexample.

A. If you subtract a whole number from another whole number, the result is also a whole number.

B. If last month was February, then this month is March.

Conditional


C. When a rectangle has an obtuse angle, it is a parallelogram.

Example 3: Related Conditionals

Write the converse, inverse, and contrapositive of the following statement. Determine the truth

value of each statement. If a statement is false, give a counterexample.

The sum of the measures of two complementary angles is 90

Conditional:

Converse:

Inverse:

Contrapositive:


Lesson 2-4: Deductive Reasoning Date:

reasoning uses facts, rules, definitions, or properties to reach logical

conclusions.

Real-World Example 1: Inductive and Deductive Reasoning

Determine whether the conclusion is based on inductive or deductive reasoning.

A. WEATHER In Miguel’s town, the month of April has had the most rain for the past 5 years. He thinks

that April will have the most rain this year.

B. WEATHER Sandra learned that if it is cloudy at night it will not be as cold in the morning than if there

are no clouds at night. Sandra knows it will be cloudy tonight, so she believes it will not be cold tomorrow

morning.

Example 2: Law of Detachment

Determine whether the conclusion is valid based on the given information. If not, write invalid. Explain

your reasoning.

Given: If a figure is a square, then it is a parallelogram.

The figure is a parallelogram.

Conclusion: The figure is a square.


Example 3: Law of Syllogism

Determine which statement follows logically from the given statements.

(1) If Jamal finishes his homework, he will go out with his friends.

(2) If Jamal goes out with his friends, he will go to the movies.

A If Jamal goes out with his friends, then he finishes his homework.

B If Jamal finishes his homework, he will go to the movies.

C If Jamal does not go to the movies, he does not go out with his friends.

D There is no valid conclusion.

Example 4: Apply Laws of Deductive Reasoning

Draw a valid conclusion from the given statements, if possible. Then state whether your conclusion was

drawn using the Law of Detachment or the Law of Syllogism. If no valid conclusion can be drawn, write no

conclusion and explain your reasoning.

Given: If it snows more than 5 inches, school will be closed. It snows 7 inches.


Lesson 2-5: Postulates and Paragraph Proofs Date:

Real-World Example 1: Identifying Postulates

ARCHITECTURE Explain how the picture illustrates that the statement is true.

Then state the postulate that can be used to show the statement is true.

A. Points F and G lie in plane Q and on line m. Line m lies entirely in plane Q.

B. Points A and C determine a line.

C. Plane P contains points E, B, and G.

A

is a statement that is

accepted as true

without proof.


Example 2: Analyze Statements Using Postulates

Determine whether the following statement is always, sometimes, or never true. Explain.

A. If plane T contains 𝐸𝐹 ⃡ and 𝐸𝐹 ⃡ contains point G, then plane T contains point G.

B. 𝐺𝐻 ⃡ contains three noncollinear points.

Once a statement has been proven, it is called a , and it can be used as a reason to

justify statements in other proofs.


Lesson 2-6: Algebraic Proof Date:

Example 1: Justify Each Step When Solving an Equation

Solve


Real-World Example 2: Write an Algebraic Proof

Complete the proof: If the formula for the area of a trapezoid is 𝐴 =1

2(𝑏1 + 𝑏2)ℎ, then the height h of the

trapezoid is given by ℎ =2𝐴

(𝑏1+𝑏2).

Given: 𝐴 =1

2(𝑏1 + 𝑏2)ℎ

Prove: ℎ =2𝐴

(𝑏1+𝑏2)

Proof:

Statements Reasons

1. 𝐴 =1

2(𝑏1 + 𝑏2)ℎ 1. Given

2. 2. Multiplication Property of Equality

3. 2𝐴

(𝑏1+𝑏2)= ℎ 3. Division Property of Equality

4. ℎ =2𝐴

(𝑏1+𝑏2) 4. Symmetric Property of Equality

Example 3: Write a Geometric Proof

If 𝐴𝐵̅̅ ̅̅ ≅ 𝐶𝐷̅̅ ̅̅ , 𝐶𝐷̅̅ ̅̅ ≅ 𝑅𝑆̅̅̅̅ , and 𝐴𝐵 = 12, then 𝑅𝑆 = 12.

Given: 𝐴𝐵̅̅ ̅̅ ≅ 𝐶𝐷̅̅ ̅̅ , 𝐶𝐷̅̅ ̅̅ ≅ 𝑅𝑆̅̅̅̅ , and 𝐴𝐵 = 12

Prove: 𝑅𝑆 = 12

Proof:

Statements Reasons

1. 𝐴𝐵̅̅ ̅̅ ≅ 𝐶𝐷̅̅ ̅̅ , 𝐶𝐷̅̅ ̅̅ ≅ 𝑅𝑆̅̅̅̅ 1. Given

2. 𝐴𝐵̅̅ ̅̅ ≅ 𝑅𝑆̅̅̅̅ 2.

3. 𝐴𝐵 = 𝑅𝑆 3. Definition of Congruent Segments

4. 𝐴𝐵 = 12 4.

5. 𝑅𝑆 = 12 5. Substitution


Lesson 2-7: Proving Segment Relationships Date:

Example 1: Use the Segment Addition Postulate

Prove that if 𝐴𝐵̅̅ ̅̅ ≅ 𝐶𝐷̅̅ ̅̅ , then 𝐴𝐶̅̅ ̅̅ ≅ 𝐵𝐷̅̅ ̅̅ .

Given:

Prove:

Statements Reasons

1. 1. Given

2. 𝐴𝐵 = 𝐶𝐷 2.

3. 𝐵𝐶 = 𝐵𝐶 3.

4. 𝐴𝐵 + 𝐵𝐶 = 𝐴𝐶 4.

5. 5. Substitution Property of Equality

6. 6. 2.9 Segment Addition Postulate

7. 𝐴𝐶 = 𝐵𝐷 7.

8. 𝐴𝐶̅̅ ̅̅ ≅ 𝐵𝐷̅̅ ̅̅ 8.


Real-World Example 2: Proof Using Segment Congruence

Prove the following.

Given: 𝐺𝐷̅̅ ̅̅ ≅ 𝐵𝐶̅̅ ̅̅ , 𝐵𝐶̅̅ ̅̅ ≅ 𝐹𝐻̅̅ ̅̅ , 𝐹𝐻̅̅ ̅̅ ≅ 𝐴𝐸̅̅ ̅̅

Prove: 𝐴𝐸̅̅ ̅̅ ≅ 𝐺𝐷̅̅ ̅̅

Proof:

Statements: Reasons:

1. 𝐺𝐷̅̅ ̅̅ ≅ 𝐵𝐶̅̅ ̅̅ , 𝐵𝐶̅̅ ̅̅ ≅ 𝐹𝐻̅̅ ̅̅ 1. Given

2. 𝐺𝐷̅̅ ̅̅ ≅ 𝐹𝐻̅̅ ̅̅ 2. Transitive Property

3. 𝐹𝐻̅̅ ̅̅ ≅ 𝐴𝐸̅̅ ̅̅ 3. Given

4. 𝐺𝐷̅̅ ̅̅ ≅ 𝐴𝐸̅̅ ̅̅ 4. Transitive Property

5. 𝐴𝐸̅̅ ̅̅ ≅ 𝐺𝐷̅̅ ̅̅ 5.


Lesson 2-8: Proving Angle Relationships Date:


Example 1: Use the Angle Addition Postulate

CONSTRUCTION Using a protractor, a construction worker measures that the angle a beam makes with a

ceiling is 42°. What is the measure of the angle the beam makes with the wall?

Real-World Example 2: Use Supplement or Complement

TIME At 4 o’clock, the angle between the hour and minute hands of a clock is 120º. When the second

hand bisects the angle between the hour and minute hands, what are the measures of the angles between the

minute and second hands and between the second and hour hands?


Example 3: Proofs Using Congruent Comp. or Suppl. Theorems

In the figure, ∠1 and ∠4 for a linear pair, and 𝑚∠3 + 𝑚∠1 = 180.

Prove that ∠3 and ∠4 are congruent.

Given: ∠1 and ∠4 for a linear pair. 𝑚∠3 + 𝑚∠1 = 180

Prove: ∠3 ≅ ∠4

Proof:

Statements Reasons

1. ∠1 and ∠4 for a linear pair. 𝑚∠3 + 𝑚∠1 = 180 1. Given

2. ∠1 and ∠4 are supplementary 2.

3. 3. Definition of supplementary angles

4. ∠3 ≅ ∠4 4.

Example 4: Use Vertical Angles

If ∠𝐴 and ∠𝑍 are vertical angles and 𝑚∠𝐴 = 3𝑏 − 23 and 𝑚∠𝑍 = 152 − 4𝑏, find 𝑚∠𝐴 and 𝑚∠𝑍.

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