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Geometry 2.2 Big Idea: Analyze Conditional Statements

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Conditional Statement: A logical statement with 2 parts, a hypothesis and a conclusion. IF... (hypothesis) THEN... (conclusion)

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IF-THEN Form Statements of fact can be rewritten in IF-THEN Form. Ex.1) Ants are insects. If it is an ant, then it is an insect.

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Ex. 2) When x = 6, x 2 = 36. If x = 6, then x 2 = 36.

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If False, you need only provide one counterexample. Just like conjectures, a conditional statement can be True or False. If True, you would have to prove all examples are True. If False, you need only provide one counterexample.

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Converse: Switch the hypothesis and conclusion. Converses can be True or False, as well.

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Converse: Ex. If it is an insect, then it is an ant. Ex. If it is an insect, then it is an ant. (True/False ?) (Counterexample of Converse: A mosquito is an insect but it’s not an ant.)

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Conditional Statement: If 2 rays are opposite rays, then they have a common endpoint. (True/False ?) Converse: If 2 rays have a common endpoint, then they are opposite rays. If 2 rays have a common endpoint, then they are opposite rays. (True/False ?)

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Conditional statements and their converses can both be true, both be false or have only one be true. No assumptions can be made.

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Inverse: Negate both Negate (say it’s not true) both the hypothesis and the conclusion. If it is not an ant, then it is not an insect. If it is not an ant, then it is not an insect. (True/False ?)

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Contrapositive: Negate both in the converse Negate both the hypothesis and conclusion in the converse of the conditional statement.

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Ex. If it not an insect, then it is not an ant. Ex. If it not an insect, then it is not an ant. (True/False ?)

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Summary C.S (T) C.S.: If it is an ant, then it is an insect. (T) Conv (F) Conv.: If it is an insect, then it is an ant. (F) Inv.: (F) Inv.: If it is not an ant, then it is not an insect. (F) Contra.: (T) Contra.: If it is not an insect, then it is not an ant. (T)

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conditional statement and its contrapositive always either both False or both True converse and the inverse. A conditional statement and its contrapositive (the negation of the converse) are always either both False or both True. This is also true for the converse and the inverse.

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Equivalent Statements Equivalent Statements : If two statements are both true or both false. If two statements are both true or both false. Ex.1) C.S. and its contrapositive Ex.2) converse and inverse

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Biconditional Statement: Contains phrase “If and only If” (can be written only when the C.S. and its converse are true) Any good definition can be written as a biconditional statement.

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C.S.: C.S.: If 2 rays are opposite rays, then they share a common endpoint and lie on the same line. Biconditional Statement: Two rays are opposite if and only if they share a common endpoint and lie on the same line.

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