INDUCTIVE REASONING AND CONJECTURE. DEFINITIONS Conjecture: a best guess based on known information. Inductive Reasoning: using specific examples to arrive.

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Presentation transcript:

INDUCTIVE REASONING AND CONJECTURE

DEFINITIONS Conjecture: a best guess based on known information. Inductive Reasoning: using specific examples to arrive at a generalization or prediction. Counterexample: an example that demonstrates that a conjecture is not true.

EXAMPLES 2121

PRACTICE Pg. 64 #11-20, 29-36

CONDITIONAL STATEMENTS

DEFINITIONS Conditional statement: a statement that can be written in if-then form. If-then statement: written in the form "if p, then q" If I study, then I will get good grades Hypothesis: the "if" part. "If I study" Conclusion: the "then" part. "Then I will get good grades"

DEFINITIONS Related conditionals: other statements based on a conditional statement Converse: if there are clouds in the sky, then it is raining. If it is raining, then there are clouds in the sky Contrapositive: if there are no clouds in the sky, then it is not raining Inverse: if it is not raining, then there are no clouds in the sky. The original statement and the contrapositive are always logically equivalent.

EXAMPLES Hypothesis Conclusion Hypothesis Conclusion

EXAMPLES Tru e Fals e

EXAMPLES Write the converse, inverse, and contrapositive of the following statement: If there is a lot of snow, then school is cancelled. Converse: If school is cancelled, then there is a lot of snow. Inverse: If there is not a lot of snow,then school is not cancelled. Contrapositive: If school is not cancelled, then there is not a lot of snow.

PRACTICE Pg. 78 #16-27, 34-39

POSTULATES AND PARAGRAPH PROOFS

VOCABULARY Postulate (or Axiom): A statement that describes a fundamental relationship between the basic terms of Geometry. It is accepted as true. Theorem: a statement that can be proven true. Proof: a logical argument in which each statement is supported by a postulate, theorem, or logic. Paragraph Proof: an informal proof to prove that a conjecture is true.

POSTULATES 2.1--Through any two points, there is exactly one line 2.2--Through any three points not on the same line, there is exactly one plane 2.3--A line contains at least 2 points A plane contains at least 3 non-collinear points If 2 points are in a plane, then the line containing those points are also in the same plane If 2 lines intersect, they intersect at exactly one point If 2 planes intersect, they intersect at exactly one line If M is the midpoint of AB, then AM=MB.

EXAMPLE Never Always Sometime s Always

PRACTICE Pg. 92 #16-27

ALGEBRAIC PROOF

PROPERTIES

EXAMPLE Simplif y

EXAMPLE

PRACTICE Pg. 97 #14-25

PROVING SEGMENT RELATIONSHIPS

POSTULATE

EXAMPLE Substitution

PRACTICE PG. 104 #12-21

ANGLE RELATIONSHIPS

POSTULATES

THEOREMS

EXAMPLES

PRACTICE PG. 112 #16-24, 27-32