Transcript Chapter 2 Connecting Reasoning and Proof

Chapter 2
Connecting Reasoning and Proof
In this chapter, you will:
Make conjectures
 Use the laws of logic to make
conclusions
 Solve problems by looking for a pattern
 Write algebraic proofs
 Write proofs
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Why It’s Important
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Law – The job of a lawyer is to present
the client’s for guilt or innocence so that
jurors can use logical reasoning to
determine whether the client is guilty or
not. In this chapter, you will learn about
two basic types of logic that can be used
to help a person determine whether
something is true or false.
Inductive reasoning relies on patterns in
past occurrences to reach a conclusion.
 Deductive reasoning uses a rule to reach
a conclusion.
 Lawyers may use both types of logic as
they present their cases to juries.
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2-1 Inductive Reasoning and
Conjecturing
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Inductive Reasoning – Reasoning that
uses a number of specific examples to
arrive at a plausible generalization or
prediction. When you observe the same
thing happening again and again and
form a conclusion from those
observations, you are using inductive
reasoning.
Deductive Reasoning – A system of reasoning
used to reach conclusions that must be true
whenever the assumptions on which the
reasoning is based are true.
 When you use laws of logic and statements
that are known to be true to reach a
conclusion, you are using deductive
reasoning.
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Conjecture – An educated guess.
Looking at several specific situations to
arrive at a conjecture is called inductive
reasoning.
 Conjectures are based on observations
of a particular situation.
 Conjecture based on several
observations may be true or false.
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Counterexample – An example used to
show that a given general statement is
not always true.
Example 1 – Page 70
Some conjectures are:
 The ball will strike the long side of the
table at its midpoint.
 The ball will then bounce off the rail at
the same angle.
 The ball will continue on a path and
touch the opposite corner.
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Example 2
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For points A, B, and C, AB = 10, BC = 8,
and AC = 5. Make a conjecture and draw
a figure to illustrate your conjecture.
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Conjecture: A, B, and C are noncollinear
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Conjectures are made based on
observations of a particular situation.
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A conjecture based on several
observations may be true or false.
Example 3
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Eric Pham was driving his friends to
school when his car suddenly stopped
two blocks away from school. Make a list
of conjectures that Eric can make and
investigate as to why his car stopped.
Some conjectures are:
The car ran out of gas.
 The battery cable lost its contact with the
battery.
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Example 4
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Given that points P, Q, and R are
collinear, Joel made a conjecture that Q
is between P and R. Determine if his
conjecture is true or false.
Page 72
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Explain the meaning of conjecture.
Why three points on a circle could never be
collinear.
Determine if the conjecture is true or false.
Given: <1 and <2 are supplementary angles.
<1 and <3 are supplementary angles.
Conjecture: <2 = <3
Give a conjecture
Lines l and m are perpendicular.
 If l and m are perpendicular, then they
form a right angle.
 Points H, I, and J are each located on
different sides of a triangle, make a
conjecture about points H, I, and J.
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Determine if the conjecture is true or
false. Explain your answer and give a
counterexample if the conjecture is false
 Given : FG = GH
 Conjecture: G is the midpoint of FH.
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