Transcript 2.1-2.3: Reasoning in Geometry

2.1-2.3:
Reasoning in Geometry
Helena Seminati
Stephanie Weinstein
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2.1: An Intro to Proofs
 A proof
is a convincing argument that
something is true.
 Start
 Can
with givens: postulates or axioms.
be formal or informal.
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Types of Proofs
m<1
m<2
m<3
m<4
20°
?
?
?
30°
?
?
?
40°
?
?
?
x°
?
?
?
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2.2 An Intro to Logic





“If-then” statements
are conditionals.
Formed as “if p,
then q” or “p implies
q.”
Conditionals are
broken into two parts:
Hypothesis is p.
Conclusion is q.
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Reversing Conditionals
If a car is a Corvette, then it is a Cheverolet.

A converse is created
when you
interchange p and q
(hypothesis and
conclusion).
 A counterexample
proves a converse
false.
ex: If a car is a
Cheverolet, then it is a
Corvette.
ex: A Silverado.
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Logical Chains
 A logical
chain is a set of linked
conditionals.
 If
cats freak, then mice frisk.
 If sirens shriek, then dogs howl.
 If dogs howl, then cats freak.
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Conditionals from Logical Chains
First, identify the
hypothesis and
 If cats freak, then mice frisk. conclusions.
 If sirens shriek, then dogs
Strike out any repeats.
howl.
 If dogs howl, then cats freak.
String them together to
form a conditional.
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If-Then Transitive Property
 An
extension of logical chains, the If-Then
 Transitive Property is:
 Given:
 “If A then
B, and
 if B then C.”
One can conclude:
“If A then C.”
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2.3 Definitions
 A definition
is a type of conditional, written
in
 a different form.
 A definition
can apply to made-up
polygons
 or traditional ones.
 A definition
has a property that the original
 conditional and the converse are both9
Definition of a Vehicle
“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!

Vehicles

Planes
Cars
Wheelbarrows
Bicycle
Roller-coaster
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


Not vehicles
Books
Computers
DSL
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Biconditionals
 Two
true conditionals (of a definition) can
be
 combined into a compact form by joining
the
 hypothesis and the conclusion with the
 phrase “if and only if.”
 Statements
using “if and only if” are
 biconditionals.
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Helpful Websites

An introduction to proofs:
 http://library.thinkquest.org/16284/g_intro_2.htm

Conditional statements and their converses:
 http://www.slideshare.net/rfant/hypothesisconclusion-geometry-14

More on conditionals:
 http://library.thinkquest.org/2647/geometry/cond/con
d.htm
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A Quick Review
 What
are some types of proofs?
 What two parts form a conditional
statement?
 What is the If-Then Transitive Property
 What is the essential phrase in a
biconditional?
 What is the converse of this statement:

If bob is old, then his bones are
frail.
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