Geometry 2.2Geometry 2.2Big Idea: Analyze

Conditional Statements

Conditional Statement: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 FormIF-THEN Form.

Ex.1) Ants are insects.Ex.1) Ants are insects.

If it is an ant, then it is If it is an ant, then it is an an insect.insect.

Ex. 2) When x = 6, xEx. 2) When x = 6, x2 2 = 36.= 36.

If x = 6, then xIf x = 6, then x22 = 36. = 36.

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

Converse:Converse:

Switch the hypothesis and Switch the hypothesis and conclusion.conclusion.

Converses can be True or False, as well.

ConverseConverse: :

Ex. If it is an insect, then it is an Ex. If it is an insect, then it is an ant. ant. (True/False ?)

(Counterexample of Converse:

A mosquito is an insect but it ’s not an ant.)

Conditional Statement: Conditional Statement: If 2 If 2 rays are opposite rays, then rays are opposite rays, then they have a common they have a common endpoint.endpoint.

(True/False ?)

ConverseConverse::

If 2 rays have a common If 2 rays have a common endpoint, then they are endpoint, then they are opposite rays. 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:Inverse:

NegateNegate (say it’s not true) bothboth the hypothesis and the conclusion.

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

Contrapositive:Contrapositive:

Negate bothNegate both the hypothesis and conclusion in the conversein the converse of the conditional statement.

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

SummaryC.SC.S.: If it is an ant, then it is

an insect. (T)(T)

ConvConv.: If it is an insect, then it is an ant. (F)(F)

Inv.Inv.:: If it is not an ant, then it is not an insect. (F)(F)

Contra.Contra.:: If it is not an insect, then it is not an ant. (T)(T)

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

Equivalent StatementsEquivalent Statements:

If two statements are both If two statements are both true or both false.true or both false.

Ex.1) C.S. and its Ex.1) C.S. and its contrapositivecontrapositive

Ex.2) converse and inverseEx.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.: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.