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2.1-2.3:2.1-2.3:Reasoning in GeometryReasoning in Geometry

Helena SeminatiHelena Seminati

Stephanie WeinsteinStephanie Weinstein

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2.1: An Intro to Proofs2.1: An Intro to Proofs

A A proofproof is a convincing argument that is a convincing argument that something is true.something is true.

Start with givens: postulates or axioms.Start with givens: postulates or axioms.

Can be formal or informal.Can be formal or informal.

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Types of ProofsTypes of Proofs

m<1m<1 m<2m<2 m<3m<3 m<4m<4

20°20° ?? ?? ??

30°30° ?? ?? ??

40°40° ?? ?? ??

x°x° ?? ?? ??

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2.2 An Intro to Logic2.2 An Intro to Logic

““If-then” statements are If-then” statements are conditionalsconditionals..

Formed as “if Formed as “if pp, then , then qq” or “” or “pp implies implies qq.”.”

Conditionals are broken Conditionals are broken into two parts:into two parts:

Hypothesis is Hypothesis is pp..

Conclusion is Conclusion is qq..

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Reversing ConditionalsReversing Conditionals

A A converseconverse is created is created when you when you interchange interchange pp and and qq (hypothesis and (hypothesis and conclusion).conclusion).

A A counterexamplecounterexample proves a converse proves a converse false.false.

ex: If a car is a ex: If a car is a Cheverolet, then it is a Cheverolet, then it is a Corvette.Corvette.

ex: A Silverado.ex: A Silverado.

If a car is a Corvette, then it is a Cheverolet.

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Logical ChainsLogical Chains

A A logical chainlogical chain is a set of linked conditionals. is a set of linked conditionals.

If cats freak, then mice frisk.If cats freak, then mice frisk.

If sirens shriek, then dogs howl.If sirens shriek, then dogs howl.

If dogs howl, then cats freak.If dogs howl, then cats freak.

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Conditionals from Logical ChainsConditionals from Logical Chains

If cats freak, then mice frisk.If cats freak, then mice frisk.

If sirens shriek, then dogs howl.If sirens shriek, then dogs howl.

If dogs howl, then cats freak.If dogs howl, then cats freak.

First, identify theFirst, identify the

hypothesishypothesis and and

conclusionsconclusions..

Strike out any repeats.Strike out any repeats.

String them together toString them together to

form a conditional.form a conditional.

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If-Then Transitive PropertyIf-Then Transitive Property

An extension of logical chains, the An extension of logical chains, the If-ThenIf-Then

Transitive PropertyTransitive Property is: is:

Given: Given: One can One can conclude:conclude:

““If A then B, andIf A then B, and “If A then C.”“If A then C.”

if B then C.”if B then C.”

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2.3 Definitions2.3 Definitions

A A definitiondefinition is a type of conditional, written in is a type of conditional, written ina different form.a different form.

A definition can apply to made-up polygonsA definition can apply to made-up polygonsor traditional ones.or traditional ones.

A definition has a property that the originalA definition has a property that the originalconditional and the converse are both trueconditional and the converse are both true..

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Definition of a VehicleDefinition of a Vehicle

VehiclesVehicles

PlanesPlanes

CarsCars

WheelbarrowsWheelbarrows

BicycleBicycle

Roller-coasterRoller-coaster

Not vehiclesNot vehicles

BooksBooks

ComputersComputers

DSLDSL

“Anything that has wheels and moves people from place to place.”Not all definitions may be precise, so when creating or following one, read carefully!

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BiconditionalsBiconditionals

Two true conditionals (of a definition) can beTwo true conditionals (of a definition) can be

combined into a compact form by joining thecombined into a compact form by joining the

hypothesis and the conclusion with thehypothesis and the conclusion with the

phrase “phrase “if and only ifif and only if.”.”

Statements using “if and only if” areStatements using “if and only if” are

biconditionalsbiconditionals..

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Helpful WebsitesHelpful Websites

An introduction to proofs:An introduction to proofs:http://library.thinkquest.org/16284/g_intro_2.htmhttp://library.thinkquest.org/16284/g_intro_2.htm

Conditional statements and their converses:Conditional statements and their converses:http://www.slideshare.net/rfant/hypothesis-conclusion-ghttp://www.slideshare.net/rfant/hypothesis-conclusion-geometry-14eometry-14

More on conditionals:More on conditionals:http://library.thinkquest.org/2647/geometry/cond/http://library.thinkquest.org/2647/geometry/cond/

cond.htmcond.htm

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A Quick ReviewA Quick Review

What are some types of proofs? What are some types of proofs?

What two parts form a conditional What two parts form a conditional statement?statement?

What is the If-Then Transitive PropertyWhat is the If-Then Transitive Property

What is the essential phrase in a What is the essential phrase in a biconditional?biconditional?

What is the converse of this statement:What is the converse of this statement:

If bob is old, then his bones are frail.If bob is old, then his bones are frail.