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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

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GEOMETRY

CHAPTER 2

JOURNAL

VALERIA IBARGUEN 9-1

### 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

• 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.

-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

• 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.

### LINEAR PAIR POSTULATE

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

Prove:<1≅<4

### LINEAR PAIR POSTULATE

Given: BE ≅ CE, DE ≅ AE

Prove: AB ≅ CD

### 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: