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GEOMETRY CHAPTER 2 JOURNALPowerPoint Presentation

GEOMETRY CHAPTER 2 JOURNAL

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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

- 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

- 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

- 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

- 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

- 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

- 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

- GIVEN: Ifm<A 90°, then <A isacute. If <A isacutethenitisnot a rightangle.
p= themeasureofanangleislessthen 90°

q= theangleisacute

r= theangleisnot a rightangle.

-Thisistryingtoexplainusthatpqandqristheconclusionofthefirstconditionalandthehypothesisofthesecondconditionalyou can tellthat at the en pr. 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

-Whatthismeansisthatpqandrq. TheLawofSyllogismcannot be usedtodrawconclusionssinceqistheconlcusionofbothconditionalstatements, even thoughpristruethelogicusedtodratheconclusionisNOT VALID.

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 PROOF

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

SEGMENT AND ANGLE PROPERTIES OF CONGRUENCE AND EQUALITY

- PROPERTY OF EQUALITY:

SEGMENT AND ANGLE PROPERTIES OF CONGRUENCE AND EQUALITY

- PROPERTIES OF CONGRUENCE:

TWO-COLUM PROOFS

- To do a twocolumproofsyouhavetolisteachstepof how youfoundyouranswer.
- EXAMPLES:

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.

CONGRUENT COMPLEMENTS AND SUPPLEMENTS THEOREMS

- CONGRUENT COMPLEMENT THEOREM:

CONGRUENT COMPLEMENTS AND SUPPLEMENTS THEOREMS

- CONGRUENT SUPPLEMENT THEOREM:

VERTICAL ANGLES THEOREM

- VERTICAL ANGLE THEOREM:

COMMON SEGMENTS THEOREM

- COMMON SEGMENTS THEOREM:

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