Geometry 2.2 Big Idea: Analyze Conditional Statements

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Presentation transcript:

Geometry 2.2 Big Idea: Analyze Conditional Statements

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

Statements of fact can be rewritten in IF-THEN Form. Ex 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.

Ex. 2) When x = 6, x2 = 36. If x = 6, then x2 = 36.

Just like conjectures, a conditional statement can be True or False 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.

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

Converse: 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.)

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. (True/False ?)

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

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

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

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

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

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.

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

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.

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.