Honors Geometry Chapter 2 Guided Notes

(2-1) Inductive Reasoning and Conjecture: Inductive Reasoning: Conjecture: Counterexample:

12.1 Enrichment (1-12) Read the definitions of No, All and Some. The rules to negate statements are in the box on the handout.

(2-2) Logic: An “and” statement in logic is called a _____________________. An “and” statement is only true if _________________________. An “or” statement is called a __________________________. An “or” statement is true if _____________________________. Two statements are __________________________ if they have the exact same ______________________.

p q ~p ~q

T T

T F

F T

F F

__________________________ (D.S.): If one part of a true “or” statement is ____________, then the other part must be ____________. Given: p or q, ~q Conclusion:

12.2 Enrichment (1-8)

(2-3) Conditional Statements: A ________________ statement is a statement in ______________ form. The “if” part is the __________________.

The “then” part is the ________________. Ex: If you live in Northville, then you live in Michigan. The _________________ of a statement is formed by _____________ the hypothesis and conclusion. (backwards) Ex: The ___________________ of statement is the __________________ of the statement. (negative) Ex: The ___________________of a statement is the ___________________ (backwards and negative). Ex: The contrapositive of a _________ statement is always __________, so

we call them _______________ statements. (C.T.T.)

An if-then statement is only false if the hypothesis is ___________ and the conclusion is ___________.

p q ~p ~q

T T

T F

F T

F F

Notice again that 𝑝 → 𝑞 is logically equivalent to ~𝑞 → ~𝑝, because they have the same truth tables. A ________________________ statement is a statement that contains

the phrase ________________________.

Saying “I am working if and only if it is Saturday,” means… ________________________________________ AND ________________________________________

A good __________________________ can be written as a biconditional

statement.

Ex: Two angles are ________________ if and only if they share a

__________________________, but no __________________.

Ex: A biconditional statement is only true if it’s ________________________.

(2-4) Deductive Reasoning:

_______________________________ is drawing logically ___________ conclusions by using an argument. (This is the type of reasoning we use in proofs.) Law of Detachment: Given: If p then q, p Conclusion: Law of Syllogism: Given: If A then B, If B then C. Conclusion:

Examples Law of Detachment (L.O.D) Premises: If Liam forgets his lunch, then he will be hungry. Liam forgot his lunch.

Conclusion: Law of Syllogism (L.O.S.) Premises: If Liam forgets his lunch, then he will be hungry. If Liam is hungry, then he will be in a bad mood.

Conclusion: Contrapositive of a True statement is True (C.T.T.) Premise: If Liam forgets his lunch, then he will be hungry.

Conclusion:

(C.T.T./L.O.D.) Premises: If Liam forgets his lunch, then he will be hungry. Liam wasn’t hungry. Conclusion:

12.3 Enrichment (Complete All) Apply the concepts in this section and the logic proofs done in class to this enrichment.

(2-5) Postulates and Paragraph Proofs:

Postulate: Proof: Theorem:

(2-6/2-7) Algebraic Proof/Proving Segment Relationships:

Addition (A.P.O.E.): If 𝑎 = 𝑏, then

Subtraction (S.P.O.E.): If 𝑎 = 𝑏, then

Multiplication (M.P.O.E.): If 𝑎 = 𝑏, then

Division (D.P.O.E.): If 𝑎 = 𝑏, then Ex: Solve the following equation and write reasons next to each step.

Statements Reasons

3(𝑥 − 2) + 2𝑥 = 19

Reflexive: Any measure or shape is congruent to ________________: Transitive: If two things are equal/congruent to the same thing, then they

are equal/congruent to _______________________.

Symmetric: The order in which things are equal/congruent doesn’t matter.

Substitution: If a = b, then b can be substituted in for a in any equation. Note: Substitution can only be used with numbers/measures, not shapes.

Segment Addition Postulate: Overlapping Segments Theorem

Given: 𝐴𝐵̅̅ ̅̅ ≅ 𝐶𝐷̅̅ ̅̅

Prove: 𝐴𝐶̅̅ ̅̅ ≅ 𝐵𝐷̅̅ ̅̅

Statements Reasons

(2-8) Proving Angle Relationships: Angle Addition Postulate: Linear Pair Postulate: If two angles form a linear pair, then they are

________________________.

Congruent Supplements/Complements Theorem: Two angles that are supplementary/complementary to the same angle are ______________.

Given: 𝑚∠1 +𝑚∠2 = 180° 𝑚∠2 +𝑚∠3 = 180°

Prove: ∠1 ≅ ∠3

Vertical Angles Theorem: If two angles are vertical angles, then they are

___________________.

Given: Picture

Prove: ∠1 ≅ ∠2

All right angles are ______________________.

2.4 Enrichment (1-7)

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