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GEOMETRY CHAPTER 2 JOURNAL

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

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  1. GEOMETRY CHAPTER 2 JOURNAL VALERIA IBARGUEN 9-1

  2. 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°

  3. 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.

  4. 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°.

  5. 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.

  6. 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.

  7. 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.

  8. 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.

  9. 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.

  10. 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

  11. ALGEBRAIC PROOF b)-5=3n+1c)sr=3.6

  12. SEGMENT AND ANGLE PROPERTIES OF CONGRUENCE AND EQUALITY • PROPERTY OF EQUALITY:

  13. SEGMENT AND ANGLE PROPERTIES OF CONGRUENCE AND EQUALITY • PROPERTIES OF CONGRUENCE:

  14. TWO-COLUM PROOFS • To do a twocolumproofsyouhavetolisteachstepof how youfoundyouranswer. • EXAMPLES:

  15. TWO-COLUM PROOFS

  16. 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.

  17. LINEAR PAIR POSTULATE Given: <1 and <2 are supplementary <3 and <4 are supplementary. Prove:<1≅<4

  18. LINEAR PAIR POSTULATE Given: BE ≅ CE, DE ≅ AE Prove: AB ≅ CD

  19. CONGRUENT COMPLEMENTS AND SUPPLEMENTS THEOREMS • CONGRUENT COMPLEMENT THEOREM:

  20. CONGRUENT COMPLEMENTS AND SUPPLEMENTS THEOREMS • CONGRUENT SUPPLEMENT THEOREM:

  21. VERTICAL ANGLES THEOREM • VERTICAL ANGLE THEOREM:

  22. COMMON SEGMENTS THEOREM • COMMON SEGMENTS THEOREM:

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