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LOGIC, REASONING, & PROOF
CH 1-1 USING PATTERNS & INDUCTIVE REASONING
Inductive reasoning is a type of reasoning that allows you to reach conclusions based on pattern of
specific examples or past events.
Example 1
a. Find the next two terms in this sequence: 2, 4, 6, 8,…
Describe the pattern.
b. Find the next two terms in this sequence: 3, 6, 12, 24,…
Describe the pattern.
c. Find the next two terms in this sequence: 1, 2, 4, 5, 10, 11, 22,…
Describe the pattern.
A conclusion reached by using inductive reasoning is sometimes called a conjecture.
Example 2
Use inductive reasoning to find the sum of the first 20 odd numbers.
Ch 1-7 Using Deductive Reasoning
Deductive reasoning is a process of reasoning logically from given facts to a conclusion.
Properties of congruence:
Reflexive property: Any geometric object is
congruent to itself.
Symmetric property: If one geometric object is
congruent to a second, then the second is
congruent to the first.
Transitive property: If one geometric object is
congruent to a second, and the second is congruent
to a third, then the first object is congruent to
the third object.
Properties of equality
Addition Property
Subtraction Property
Multiplication Property
Division Property
Substitution Property
Distributive Property
Example 1
Give a reason for each step
4x – 6 = 94
4x = 100
x = 25
A convincing argument that uses deductive reasoning is also called a proof. A conjecture that is
proven is a theorem.
Theorems
Vertical angles theorem: Vertical angles
are congruent.
Congruent Supplements Theorem: If two
angles are supplements of congruent angles
(or the same angle), then the two angles are
congruent.
Congruent Complements Theorem: If two
angles are complements of congruent angles
(or the same angle), then the two angles are
congruent.
Example 2
Use a 2-Column proof (Statement & Reason) to prove the Congruent Supplements Theorem.
Ch 4-1 Using Logical Reasoning
Another name for an if-then statement is a conditional. Every conditional has two parts. The part
following the if is the hypothesis, and the part following the then is the conclusion.
Example 1
Identify the hypothesis and the conclusion in this statement.
If it is February, then there are only 28 days in the month.
When you determine whether a conditional is true or false, you determine its truth value. To show that
a conditional is false, you need to find only one counterexample for which the hypothesis is true and the
conclusion is false.
Example 2
Is the conditional from example 1 true or false? Explain.
Example 3
Find a counterexample for this conditional: If the name of a state contains the word New, then the state
border ocean.
The converse of a conditional interchanges the hypothesis and conclusions.
Example 4
Write the converse of this statement: If a polygon is a quadrilateral, then it has four sides.
When a conditional and its converse are true, you can combine them as biconditional (if and only if).
Example 5
Write a biconditional statement for the example 4 problem.
Example 6
Write the two statements that make up this definition: A right angle has a measure of 90o. Then write
the definition as a biconditional.
Conditional:
Converse:
Biconditional:
The negation of a statement has the opposite meaning.
Example 7
Write the negation of each statement.
a.) An angle is acute
b.) two lines are not parallel
The inverse of a conditional negates both the hypothesis and the conclusion.
The contrapositive of a conditional interchanges and negates both the hypothesis and the conclusion.
Example 8
a. Write the following for this statement: If it is raining, then it is cloudy.
Conditional:
Converse:
Inverse:
Contrapositive:
Biconditional:
b. What are the truth values for each statement.
Conditional:
Converse:
Inverse:
Contrapositive:
Biconditional: