Harry

PotterChapter 2 Review: Reasoning and Proof

No Spells Allowed

This Chapter Covers:

2-1: Inductive Reasoning and Conjecture

2-2: Logic

2-3: Conditional Statements

2-4: Deductive Reasoning

2-5: Postulates and Paragraph Proofs

2-6: Algebraic Proof

2-7: Proving Segment Relationships

2-8: Proving Angle Relationships

Sophie HeflinMaile MurphyErin Dennis

2-1: Inductive Reasoning and Conjecture

Topics Covered:

Inductive reasoning: reasoning that uses a number of specific examples to arrive at a conclusion.

Conjecture: A concluding statement reached using inductive reasoning.

Counterexample: A false example, such as a number, a drawing, or a statement that disproves a conjecture.

2-2: LogicTopics Covered:

Statement: A sentence that is either true or false.

Truth value: The truth value of a statement is either true (T) or false (F).

Negation: The negation of a statement has the opposite meaning, as well as an opposite truth value.

Compound statement: Two or more statements joined by the word and or or.

Conjunction: A compound statement using the word and.

Disjunction: A compound statement that uses the word or.

Truth table: A convenient method for organizing the truth values of statements.

P Q P ^Q T T T T F F F T F F F F

Conjunction Truth Table

2.3-Conditional Statements

Conditional Statement-A statement that can be written in if-then form.

if-then statement-is the form if p, then q.

The Hypothesis-of a conditional statement is the phrase immediately following the word if.

The Conclusion-of a conditional statement is the phrase immediately following the word then.

Related Conditional-statements based on a given conditional statement.

The Converse-is formed by exchanging the hypothesis and conclusion of the conditional.

The Inverse-is formed by negating both the hypothesis and conclusion of the conditional.

The Contrapositive-is formed by negating both the hypothesis and the conclusion of the conclusion of the converse of the conditional.

Logically Equivalent- Statements with the same truth values

2-4: Deductive Reasoning

Deductive Reasoning- A conjecture made using facts, rules, definitions, or properties that reach logical conclusions from given statements.

Valid- A conjecture that is logically correct.

Law of Detachment- A valid form of deductive reasoning:

If p --> q is a true statement and p is true, then q is true.

Law of Syllogism- Another valid form of deductive reasoning:

If p--> q and q--> r are true statements, then p-->r is a true statement.

2-5: Postulates

Postulate (Axiom): A statement that is accepted as true without proof.

Proof: A logical argument in which each statement you make is supported by a statement that is accepted as true.

Theorem: A statement or conjecture that has been proven.

Deductive argument: a logical chain of statements linking the given to what you are trying to prove.

2-6: Algebraic Proof

Algebraic Proof: A proof that is made up of a series of algebraic statements.

Two-column Proof (Formal Proof): A proof that contains statements and reasons organized in two columns.

Statements Reasons 1. C = 5/9(F- 32)

2. 9/5C=9/5 x 5/9(F-32)

3. 9/5C= F - 32

4. 9/5C + 32= F - 32 + 32

5. 9/5C + 32= F

6. F= 9/5C + C

1. Given

2. Mult. Prop. of Equality

3. Subst. Prop. of Equality

4. Add. Prop. of Equality

5. Subst. Prop. of Equality

6. Symmetric Prop. of Equality

2-7: Proving Segment Relationships

Segment Addition Postulate: If A, B, and C are collinear, then point B is between A and C if and only if AB + BC = AC

Properties of Segment Congruence:Reflexive Property of Congruence: LineAB is congruent to lineABSymmetric Property of Congruence: If lineAB is congruent to lineCD,

then lineCD is congruent to lineABTransitive Property of Congruence: If lineAB is congruent to lineCD

and lineCD is congruent to lineEF, then lineAB is congruent to lineEF.

2-8: Proving Angle RelationshipsTopics to know:

Angle Addition PostulateSupplement TheoremComplement TheoremReflexive Property of CongruenceSymmetric Property of CongruenceTransitive Property of CongruenceCongruent Supplements TheoremCongruent Complements TheoremVertical Angles TheoremRight Angle Theorems