Transcript Geometry Inductive Reasoning and Conjecturing Section 2.1 Inductive Reasoning and Conjecturing Conjecture - An educated guess. Inductive Reasoning - Reasoning that uses a number.

Geometry
Inductive Reasoning and Conjecturing
Section 2.1
Inductive Reasoning and Conjecturing
Conjecture - An educated guess.
Inductive Reasoning - Reasoning that uses a number of specific
examples to arrive at a plausible generalization or prediction.
You use inductive reasoning when you “see” or “notice” a
Forpattern
example:
the sequence
1, 3,
7, … weobject,
can make
in a given
sequence
of numbers,
in5,a painted
or inathe
conjecture that behavior
the next three
numbers
the sequence is
of someone
or in
something.
9, we
11, can
13, make
… a conjecture related to
Once we recognize a pattern
the pattern behavior and determine if a conclusion is possible.
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Inductive Reasoning and Conjecturing
Conclusions arrived at by inductive reasoning lack the logical
certainty (which means it may not be true all the time) as those
arrived at by deductive reasoning. Sherlock Holmes uses deductive
reasoning when trying to determine “who done it?”
Mathematicians use inductive reasoning when they notice
patterns in numbers, nature, or something’s behavior. They set up
specific examples that explore those patterns. Once they have a
good idea how the pattern works, they then make a conjecture and
try to generalize that conjecture.
This generalization uses deductive reasoning to arrive at a
mathematical proof of the patterns they explored.
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Inductive Reasoning and Conjecturing
Given points A, B, and C, AB = 10, BC = 8, AC = 5, is there a
conjecture we can make?
First, draw the figure and make a conjecture.
A
10
10
8
8
C
5
A
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C
B
B
Is it possible for C to be
So what
is your
conjecture?
between
A and
B?
C
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Inductive Reasoning and Conjecturing
Given that points P, Q, and R are collinear. What kind of conjecture
The answer
is no.
Ourabout
second
figure,
which
contradicts
thepoints?
first figure,
could you
make
which
point
is between
which
is called a counterexample. A counterexample is a false or
contradictory
example
disproves
a conjecture,
How about
Q isthat
between
P and
R?
P
Q
R
Is this conjecture true or false?
Suppose it is true, could the following also be true?
P
R
Q
Have I violated any of the given with either of the examples?
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Inductive Reasoning and Conjecturing
Given:
D
(3p + 24)o
(5p - 4)o
O
F
Find  DOF
and  DOG.
E
(3p + 24)o + (5p - 4)o = 180o
3p + 24 + 5p - 4 = 180
8p + 20 = 180
8p + 20 - 20 = 180 - 20
8p = 160
G
8p = 160
2
2
p = 20
 DOF = (3p + 24)o
= (3 x 20 + 24)o
= (60 + 24) o
= 84o
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 DOG = (5p - 4)o
= (5 x 20 - 4)o
= (100 - 4) o
= 96o
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Inductive Reasoning and Conjecturing
Suppose  MON is a right angle and L is in the interior of  MON.
If m MOL is 5 times m LON, find m LON.
90 =  MON +  LON
= 5( LON) +  LON
= 6 ( LON)
90
=  LON
6
M
L
15 =  LON
O
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Inductive Reasoning and Conjecturing
Summary
Through inductive reasoning, we take specific examples of some
process and use it to find general patterns within the boundaries of the
process. From these general patterns, we can declare a conjecture
based on the patterns.
Then we can test the conjecture to determine if the process is true
or false.
Conjectures derived from inductive reasoning are not always
true, but they can be the basis for a strategy to use in deductive
reasoning.
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Inductive Reasoning and Conjecturing
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