2.1 Inductive Reasoning and Conjecture

Friday, 2/3/12Dress for Success for Extra CreditChapter 2 Student Notes2.1Inductive Reasoning and ConjectureConjecture -Make a conjecture from the given statement.

Given: The toast is burnt.Conjecture: ___________________________

Given: It is winter.Conjecture: ___________________________

Given: Angle A is a right angle.Conjecture: ___________________________Counterexample -Write a counterexample for each conjecture.

Conjecture: The sky is blue.Counterexample: ________________________________

Conjecture: Angle 1 and Angle 2 are congruent.Counterexample: ________________________________

Conjecture: l and m are parallelCounterexample: ________________________________

Determine if each conjecture is true or false. Give a counterexample for any false conjecture.1.Given: A, B, C are collinear.Conjecture: A, B, C are on the same line.

2.Given: 1 is a right angle.Conjecture: m1 = 90.

3.Given: AB = BC.Conjecture: B is the midpoint of AC. T / FT / FT / FDetermine if each conjecture is true or false. Give a counterexample for any false conjecture.Given: The dog is brown.Conjecture: It is a chocolate lab.

Given: 3 and 4 form a linear pair.Conjecture: 3 4

Given: 1 and 2 are complementaryConjecture: m1 = 45, m2 = 45.T / FT / FT / Fm1 = 48,m2 = 422.2LogicStatement -Truth Value

Negation -

Compound Statement

Conjunction -Symbol for And:Disjunction -Symbol for Or:Circle the statement that is true.p: Angle A is a right angle.r: Angle A is an obtuse angle.r: Angle A is an acute angle.

1.pqr

2.pqr>>>>Truth Table -Negationp~pTFFTExamples of Truth Tables.Conjunctionpqp qTTTFFTFF>Disjunctionpqp qTTTFFTFF>2.3Conditional StatementsConditional Statement -Statement: A right angle has a measure of 90 degrees.If-then:

Statement: A car has four wheels.If-then:

Statement: A triangle has 3 sides.If-then: Parts of a Conditional StatementHypothesis

Conclusion

If it is a car, then it has four wheels.Converse -Conditional: If it is a car then it has 4 wheels.Converse:

Conditional: If it is a pig, then it can fly.Converse:

Conditional: If it is a right angle, then it measure 90.Converse: Inverse -Conditional: If it is a car then it has 4 wheels.Inverse:

Conditional: If it is a pig, then it can fly.Inverse:

Conditional: If it is a right angle, then it measure 90.Inverse:Contrapositive -Conditional: If it is a car, then it has 4 wheels.Contrapositive:

Conditional: If it is a pig, then it can fly.Contrapositive:

Conditional: If it is a right angle, then it measure 90.Contrapositive:

Identify the converse, inverse and contrapositive of each conditional statement. Determine if each statement is true or false.T / FIf you go to WMHS, then you are a hornet.T / FConverse: __________________________ __________________________T / FInverse: __________________________ __________________________T / FContrapositive: ______________________ ______________________Identify the converse, inverse and contrapositive of each conditional statement. Determine if each statement is true or false.T / FIf it is a right angle, then it measures 90.T / FConverse: __________________________ __________________________T / FInverse: __________________________ __________________________T / FContrapositive: ______________________ ______________________2.4Deductive ReasoningDeductive Reasoning -

Law of Detachment

Law of Syllogism

Examples of the Laws of Detachment and Syllogism.Detachment

If it is a triangle, then it has 3 sides.

Syllogism

If it is a Jeep, then it has 4 wheel drive.Determine whether the 3rd statement is valid based on the given information. If not, write invalid. If it is a dog, then it has 4 legs.

Rover is a dog.

Rover has 4 legs.Is it valid?

Does it follow one of our Laws?Determine whether the 3rd statement is valid based on the given information. If not, write invalid. If you are 18 or older, then you are an adult.

If you are an adult, then you can vote.

If you are 18 or older, then you can vote.Is it valid?

Does it follow one of our Laws?Use the Law of Detachment or the Law of Syllogism to determine if a valid conclusion can be reached. If it can, state it and the law used. If not, write no conclusion.If it is a car, then it has 4 wheels.A Ferrari is a car. __________________________If you go to the store, then you will go to the post office.If you go to the post office, then you will buy stamps. _________________________________________Use the Law of Detachment or the Law of Syllogism to determine if a valid conclusion can be reached. If it can, state it and the law used. If not, write no conclusion.If you are in college, then you are at least 18.Pete is in college. ______________________________Right angles are congruent.Angle 1 and Angle 2 are congruent._______________________________Postulate 2.5Postulates Statement that is accepted without proof.Postulate 2.1 - ABPostulate 2.2 - PACBPlane P Plane ABC Postulate 2.3______________________________________________________________Postulate 2.4_________________________ __________________________________________________________________________Postulate 2.5____________________________________________________________________________________________________________________________________Postulate 2.6______________________________________________________________Postulate 2.7______________________________________________________________PRMidpoint Theorem -AMBDetermine if each statement is always, sometimes or never true.A, B, and C are collinear.

A, B, and C, are coplanar.

RST is a right angle.

Two planes intersect to form a line.

If AB = BC, the B is the midpoint of AC.

Vertical angles are adjacent.

If B is the midpoint of AC, then AB = BC.Determine the number of segments that can be drawn connecting each pair of points.1.2.2.6Algebraic ProofPropertiesReflexive:SymmetricTransitiveSubstitutionPropertiesDistribution

Addition / Subtraction

Multiplication / DivisionIdentify each property that justifies each statement.If 7 = x, then x = 7.

If x + 5 = 7, then x = 2

If x = 7 and 7 = y, then x = y.

If m1 + m2 = 180 and m2 = m3, then m1 + m3 = 180.Identify each property that justifies each statement.2x + 1 = 2x + 1

If x 6 = 7, then x = 13

If 2(x + 3) = 7, then 2x + 6 = 7.

If 2x = 16, then x = 8.

Given: 2x 5 = 13Prove: x = 9Statements

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_____________ Reasons

1.___________

2.___________

3.___________Given: 8 n = 4 n Prove: n = 12Statements

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___________________Reasons

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______________23Given: 2x + 1 3Statements

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_________________Reasons

1.Given

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_________________Prove: x = 10= 7 2x + 1 3= 72.7Proving Segment RelationshipsRuler PostulateThe points on any line or line segment can be _____________________________________________________________________________________________Betweenness of Points A point can only be between two __________________________________________CABSegment Addition PostulateIf B is between A and C, ACBSegment Addition Postulate StatementsMake a statement using the previous postulate about each figure.

XYMJKLRSTXY =JK = RS = Theorem 2.2Reflexive

Symmetric

TransitiveGiven: AB XY, AC XZProve: BC YZA B C

Z Y XStatements

1.AB XY, AC XZ____________________________

________________________________________________________

Reasons

__________________________________________

________________________________________________________

Given: MO PO, MN PRProve: NO ROStatements

1.MO PO, MN PR________________________________

________________________________________________________________ Reasons

1. Given____________________________

_______________________________________________________

MNPRO2.8Proving Angle RelationshipsAngle Addition PostulateIf R is in the interior of PQS, then mPQS = ORPRQS12Make an angle addition postulate statement about each figure.mMNO = MPONLMJKmJKL = Supplement TheoremIf the noncommon sides of two adjacent angles form a right angle, then the angles are complementary.1 2m1 + m2 = ____o 3 4m3 + m4 = ____oComplement TheoremTheorem 2.5Reflexive

Symmetric

TransitiveTheorem 2.6312If m1 + m2 = 180o, Theorem 2.7312If m1 + m2 = 90o, Vertical Angles Theorem 1

42 3

Abbreviation:Theorem 2.9Theorem 2.10Theorem 2.11Theorem 2.12Theorem 2.13Given: MNO RST, MNQ RSP Prove: QNO PSTStatements1. MNQ RSP, MNQ RSP __________________________________________________________________

____________________________________________________________________________________________________________________________________Reasons1. Given______________________________

____________________________________________________________NQOMSPTR69Find the measure of each numbered angle.1.m1 = 72

21m2 = 2.3 and 4 are complementary, m4 = 48.

534m3 = m5 =

Find the measure of each numbered angle.3.m6 = x 5, m7 = 2x 476m6 = m7 = Find the measure of each numbered angle.4.m8 = 52.98m9 = Find the measure of each numbered angle.5.10 and 11 are complementary. 13 11, m12= 38.1011m13 = 1213m11 = m10 =