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GEOMETRY CHAPTER 2 JOURNAL. VALERIA IBARGUEN 9-1. CONDITIONAL STATEMENT. This is a type of statement that can be written in a form of “ if p , then q ” P= Hypothesis Q= conclusion EXAMPLES: If m<A =195°, then <A is obtuse

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GEOMETRY

CHAPTER 2

JOURNAL

VALERIA IBARGUEN 9-1


Conditional statement
CONDITIONAL STATEMENT

  • Thisis a typeofstatementthat can be written in a formof “ifp, thenq”

  • P=Hypothesis

  • Q=conclusion

  • EXAMPLES:

  • Ifm<A=195°, then <A isobtuse

  • Ifaninsectis a butterfly, thenit has fourwings

  • Ifanangleisobtuse, thenit has a measureof 100°


Counter examples
COUNTER-EXAMPLES

  • A counter-exampleis a typeofexamplethatprovesif a conjectureorstatementisfalse. Thiscould be a drawing, a statementor a number.

  • EXAMPLES:

  • Forany real numberx, x2 >x

    5, 52 > 5

    5, 25 > 5

  • Supplementaryangles are adjecent

  • Theradiusofeveryplanet in the solar systemislessthan 50,000 km.


Definition
DEFINITION

  • Thisis a statementthattellsordiscribes a mathematicalobjectand can be written as a truebiconditionalstatement. A definitionincludes“ifandonlyif”

  • EXAMPLES:

  • A figure is a triangleifandonlyifitis a three-sidedpolygon.

  • A ray, segmentorlineis a segment bisector ifandonlyifit divides a segmentintotwocongruentsegments.

  • A traingleisstraightifandonlyifitmeasures 180°.


Bi conditional statements
BI-CONDITIONAL STATEMENTS

  • Thisis a statmentthatiswritten in theform“pifandonlyifq”. They are important. Thisisusedwhen a conditionalstatementandits converse are combinedtogether.

  • EXAMPLES:

  • Converse: Ifx=3, then 2x+5=11 Biconditional: 2x+5=11 ifandonlyifx=3

  • Converse: If a point divides a segmentintotwocongruentsegments, thenthepointis a midpoint. Biconditional: A pointis a midpointifandonlyifit divides thesegmentsintotwocongruentsegments.

  • Converse: Ifthe dates is July 40th, thenitIndependenceday. Biconditional: ItisIndependencedayifandonlyifitis July 40th.


Deductive reasoning
DEDUCTIVE REASONING

  • Thisisthetypeofprocess in whichwe use logictodrawconclusionsofsomething.

  • EXAMPLES:

  • If a team wins 10 games, thethey play in thefinals. If a team plays in thefinalstheytheytravelto Boston. TheReavens won 10 games. CONCLUSION:TheReavenswilltravelto Boston.

  • Iftwoanglesform a linear pair, thenthey are adjecent. Iftwoangles are adjecent, thentheyshare a side. <1 and <2 form a linear pair. CONCLUSION: <1 and<2 share a side.

  • If a polygonis a triangle, thenit has threesides. If a polygon has threesidesthenitisnot a quadrilateral. Polygonis a P triangle. CONCLUSION: A polygonisnot a quadrilateralbecauseithasthreesides.


Laws of logic
LAWS OF LOGIC

  • Lawofdetachment:

  • Ifp-qistrueweshouldassumeif P istruethen Q mustalso be true

  • LawofSyllogism:

  • If P-Q istrueand Q then R istruethenif P istrue are must be true P and R istrue.


Law of detachment
LAW OF DETACHMENT

  • Given:Iftwosegments are congruentthentheyhavethesamelength. AB≅XY

    Conjecture:AB=XY

    hypothesis: twosegments are congruent

    conclusion: theyhavethesamelenght

    ThegivenAB≅XYstatementsdoes match thehypothesis so theconjecture IS true.

  • Given: Ifyou are 3 times tardy, youmustgotodetention. John is in detention.

    Conjecture: John wastardy at least 3 times.

    hypothesis: you are tardy 3 times

    conclusion: youmustgotodetention.

    Thestatementgiventousmatchestheconclusionof a trueconditiona, butthehypothesisisnottruesince John can be in detentionforanotherreason so theconjectureis NOT valid.


Law of syllogism
LAW OF SYLLOGISM

  • GIVEN: Ifm<A 90°, then <A isacute. If <A isacutethenitisnot a rightangle.

    p= themeasureofanangleislessthen 90°

    q= theangleisacute

    r= theangleisnot a rightangle.

    -Thisistryingtoexplainusthatpqandqristheconclusionofthefirstconditionalandthehypothesisofthesecondconditionalyou can tellthat at the en pr. So IT IS VALID

  • Given: If a numberis divisible by 4 thenitis divisible by 2. If a numberis even, thenitis divisible by 2.

    Conjecture: If a numberis divisible by 4, thenitis even.

    p= A numberis divisible by 4

    q= A numberis divisible by 2

    r= A numberis even

    -Whatthismeansisthatpqandrq. TheLawofSyllogismcannot be usedtodrawconclusionssinceqistheconlcusionofbothconditionalstatements, even thoughpristruethelogicusedtodratheconclusionisNOT VALID.


Algebraic proof
ALGEBRAIC PROOF

  • Analgebraicproofisanargumentthat uses logic, definitions, properties. To do one, youhaveto do a 2 columproof.

  • EXAMPLES:

    a)Prove: x=2 if

    Given: 2x-6=4x-10


Algebraic proof1
ALGEBRAIC PROOF

b)-5=3n+1c)sr=3.6




Two colum proofs
TWO-COLUM PROOFS

  • To do a twocolumproofsyouhavetolisteachstepof how youfoundyouranswer.

  • EXAMPLES:



Linear pair postulate lpp
LINEAR PAIR POSTULATE (LPP)

  • Thisiswhenall linear pairs are linear postulates, SUPPLEMENTARY

  • EXAMPLES:

    Given: angle<1 and < 2 are linear pair

    Prove: <1 and <2 supplementary.


Linear pair postulate
LINEAR PAIR POSTULATE

Given: <1 and <2 are supplementary <3 and <4 are supplementary.

Prove:<1≅<4


Linear pair postulate1
LINEAR PAIR POSTULATE

Given: BE ≅ CE, DE ≅ AE

Prove: AB ≅ CD


Congruent complements and supplements theorems
CONGRUENT COMPLEMENTS AND SUPPLEMENTS THEOREMS

  • CONGRUENT COMPLEMENT THEOREM:


Congruent complements and supplements theorems1
CONGRUENT COMPLEMENTS AND SUPPLEMENTS THEOREMS

  • CONGRUENT SUPPLEMENT THEOREM:


Vertical angles theorem
VERTICAL ANGLES THEOREM

  • VERTICAL ANGLE THEOREM:


Common segments theorem
COMMON SEGMENTS THEOREM

  • COMMON SEGMENTS THEOREM:


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