Reasoning and Proof

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Reasoning and Proof

• Unit 2

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Inductive reasoning
Statements Logic
Reasoning and Proof
Proofs (Algebra, segments, angles)
Deductive reasoning
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Inductive Reasoning

• Looking at specific situations to arrive at a
  conjecture
• Examples ? generalization

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Deductive Reasoning

• Applying a generalization to specific
  situations to arrive at a conjecture
• Generalization ? examples

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Conjecture

• An educated guess
• aka hypothesis
• a conclusion based on observations

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Counterexample

• a false example
• only takes one to make a conjecture false

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Conditional statement

• a statement in
• If-then form
• If- hypothesis
• -then conclusion

p ? q
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Converse

• a statement in which the hypothesis and
  conclusion are interchanged

q ? p
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Negation

• the denial of a statement
• Usually includes a not in it
• Has the opposite truth value from the original

p ? q
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Inverse

• a statement negating both the hypothesis and
  conclusion of a statement

p ? q
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Contrapositive

• a statement negating both the hypothesis and
  conclusion of the converse

q ? p
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Postulate

• a principle that is accepted without proof
• 6 basic postulates concerning points, lines, and
  planes

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Postulate 2-1

• Through any 2 points, there is exactly one line

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Postulate 2-2

• Through any 3 non-collinear points, there is
  exactly one plane

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Postulate 2-3

• A line contains at least 2 points

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Postulate 2-4

• A plane contains at least 3 non-collinear points

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Postulate 2-5

• If 2 points lie in a plane, then the entire
  line containing them lies in the plane

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Postulate 2-6

• If 2 planes intersect, then their intersection
  is a line

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Deductive Reasoning

• 2 laws of logic used in proofs are
• Law of Detachment
• Law of Syllogism

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Law of Detachment

• If a conditional and its hypothesis is true,
  then the conclusion is true

If p ? q is T p is T, then q is T
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Law of Syllogism

• If 2 conditionals in which the conclusion of
  the 1st is the hypothesis of the 2nd are true,
  then the conditional with the hypothesis of the
  1st and the conclusion of the 2nd is also true

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Law of Syllogism
If p ? q and q ? r are True, then p ? r is True
Similar to the transitive property in algebra
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Properties of equality

• Reflexive
• Symmetric
• Transitive

• Addition subtraction
• Multiplication division

• Substitution
• Distributive

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Properties of equality

• Reflexive
• a a
• Symmetric
• If a b, then b a
• Transitive
• If a b b c, then a c

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Properties of equality

• Addition subtraction
• If a b, then a c b c
• Multiplication divison
• If a b, then ac bc

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Properties of equality

• Substitution
• If a b, then
• a can be replaced with b
• Distributive
• a (b c) ab ac

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Proof
A logical argument that shows that a certain true
hypothesis guarantees the truth of a certain
conclusion
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Two-column proof
A formal proof with statements listed in one
column and reasons in another
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Two-column proof
Given Figure Prove Statements
Reasons a. a. b. b. etc.
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Theorem 2-1

• Congruence of segments is
• reflexive
• symmetric
• transitive

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Theorem 2-2Supplement Theorem
If 2 angles form a linear pair, then they are
supplementary
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Theorem 2-3

• Congruence of angles is
• reflexive
• symmetric
• transitive

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Theorem 2-4
Angles supplementary to the same angle or two
congruent angles are congruent
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Theorem 2-5
Angles complementary to the same angle or two
congruent angles are congruent
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Theorem 2-6
All right angles are congruent
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Theorem 2-7
Vertical angles are congruent
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Theorem 2-8
Perpendicular lines intersect to form 4 right
angles