Download Accelerated Geometry – Concepts 5-8

Accelerated Geometry – Concepts 5-8
Reasoning and Proof Building
Concept5–ConditionalStatements(Section2.2)
ConditionalStatement:
Hypothesis:
Conclusion:
Converse:
Example:Usethestatementbelowtoanswereachquestion.
Ifanumberisdivisibleby3,thenitisodd.
a)Whatisthehypothesisofthestatement?
b)Whatistheconclusionofthestatement?
c)Whatistheconverseofthestatement?
Concept5–BiconditionalsandDefinitions(Section2.3)
Biconditional:
Example:Writetheconverseofthestatementbelow.Ifitistrue,writeabiconditionalstatement.
Conditional-Iftwoangleshavethesamemeasure,thentheanglesarecongruent.
Converse-
Biconditional–
GoodDefinition:
Counterexample:
Concept5–InductiveReasoning(Section2.1)
InductiveReasoning:
Conjecture:
Concept5–DeductiveReasoning(Section2.4)
DeductiveReasoning:
LawofDetachment
LawofSyllogism
Ifp → q and q → r are true statements, then p → r is also a true statement.
Concept6–ReasoninginAlgebraandGeometry(Section2.5)
AlgebraicPropertiesofEquality
Property
Example
AdditionProperty
If a = b ,then a + c = b + c .
SubtractionProperty
If a = b ,then a − c = b − c .
MultiplicationProperty
If a = b , then ac = bc .
DivisionProperty
If a = b , then
SubstitutionProperty
If a = b , then a can be substituted
for b in any expression or equation.
DistributiveProperty
a(b + c) = ab + ac a b
= c c
PropertiesofEqualityandCongruence
Property
Example
Explanation
ReflexivePropertyofEquality
a=a ReflexivePropertyofCongruence
RT ≅ RT or ∠5 ≅ ∠5 SymmetricPropertyofEquality
If a = b , then b = a .
SymmetricPropertyofCongruence If LM ≅ RT , then RT ≅ LM .
TransitivePropertyofEquality
If a = b and b = c , then a = c .
TransitivePropertyofCongruence
If ∠A ≅ ∠B and ∠B ≅ ∠C , then
∠A ≅ ∠C .
Proof:
2-ColumnProof
Eachstatementmusthaveareasontojustifyit.Properties,postulates,
definitions,andtheoremsareusedasreasonsinaproof.
Postulate:
Theorem:
HelpfulPostulatesforusinginproofs:
Postulate1-6:SegmentAdditionPostulate
IfthreepointsA,B,andCarecollinearandBisbetweenAandC,
thenAB+BC=AC.
Postulate1-8:AngleAdditionPostulate
IfpointBisintheinteriorof ∠ AOC,
then m∠ AOB + m∠ BOC = m∠ AOC.
Postulate1-9:LinearPairPostulate
Iftwoanglesforalinearpair,thentheyaresupplementary.
Concept6–ProvingAnglesCongruent(Section2.6)
ParagraphProof:
Theorem2-1:VerticalAnglesTheorem
Theorem2-2:CongruentSupplementsTheorem
Iftwoanglesaresupplementarytothesameangle
(ortocongruentangles),then________________________________________ .
Theorem2-3:CongruentComplementsTheorem
Iftwoanglesarecomplementarytothesameangle
(ortocongruentangles),then________________________________________.
Theorem2-4
Theorem2-5
Iftwoanglesarecongruentandsupplementary,
theneachis_________________________.
Concept7–LinesandAngles(Section3.1)
PairsofLines/Segments
ParallelLines:
PerpendicularLines:
SkewLines:
ParallelPlanes:
Transversal:
CorrespondingAngles AlternateInteriorAngles
AlternateExteriorAngles
Same-SideInteriorAngles
(ConsecutiveInterior)
Concept7–PropertiesofParallelLines(Section3.2)
AnglePairsandParallelLines
*UsetheseTheoremsasreasonsforwhytwoanglesarecongruentorsupplementary*
Postulate3-1:Same-SideInteriorAnglesPostulate
Ifatransversalintersectstwoparallellines,thensame-sideinterioranglesare____________________.
Theorem3-1:AlternateInteriorAnglesTheorem
Ifatransversalintersectstwoparallellines,thenalternateinterioranglesare___________________.
Theorem3-2:CorrespondingAnglesTheorem
Ifatransversalintersectstwoparallellines,thencorrespondinganglesare___________________.
Theorem3-3:AlternateExteriorAnglesTheorem
Ifatransversalintersectstwoparallellines,thenalternateexterioranglesare___________________.
Concept7–ProvingLinesParallel(Section3.3)
TheoremstoProveLinesareParallel
*UsetheseTheoremsasreasonsforhowyouknowtwolinesareparallel*
Theorem3-4:CorrespondingAnglesConverse
Iftwolinesandatransversalformcorrespondinganglesthatarecongruent,thenlinesareparallel.
Theorem3-5:AlternateInteriorAnglesConverse
Iftwolinesandatransversalformalternateinterioranglesthatarecongruent,thenlinesareparallel
Theorem3-6:Same-SideInteriorAnglesConverse
Iftwolinesandatransversalformsame-sideinterioranglesthataresupplementary,thenlinesare
parallel.
Theorem3-7:AlternateExteriorAnglesConverse
Iftwolinesandatransversalformalternateexterioranglesthatarecongruent,thenlinesareparallel.
Concept7–ParallelandPerpendicularLines(Section3.4)
Theorem3-8
Iftwolinesareparalleltothesameline,
thentheyare________________________________________.
Theorem3-9
Inaplane,iftwolinesareperpendiculartothesameline,
thentheyare_______________________________________.
Theorem3-10
Inaplane,ifalineisperpendiculartooneoftwoparallellines,
thenitisalso_______________________________________.
Concept8–EquationsofLinesintheCoordinatePlane(Section3.7)
rise y2 − y1
=
Slope=
run x2 − x1
m=
b=
m=
(x1, y1) =
Concept8–SlopesofParallelandPerpendicularLines(Section3.8)
Slopes of Parallel Lines
Two nonvertical lines are parallel if and only if they have ___________________________________.
Any two vertical lines are parallel.
Slopes of Perpendicular Lines
Two nonvertical lines are perpendicular if and only if they have _________________________________.
Any horizontal line and vertical line are perpendicular.