Geometry Drill 10/8/12Geometry Drill 10/8/12Aki, Bard, and Coretta live in Aki, Bard, and Coretta live in

Albany, Biloxi, and Chicago. No Albany, Biloxi, and Chicago. No one lives in a city that begins with one lives in a city that begins with the same letter as her name. Aki the same letter as her name. Aki writes letters to her friend in writes letters to her friend in Chicago. Who lives in what city?Chicago. Who lives in what city?

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Identify, write, and analyze the truth value of conditional statements.Write the inverse, converse, and contrapositive of a conditional statement.

Objectives

conditional statementhypothesisconclusiontruth valuenegationconverseinversecontrapostivelogically equivalent statements

Vocabulary

By phrasing a conjecture as an if-then statement, you can quickly identify its hypothesis and conclusion.

VocabularyVocabularyConditional statementConditional statementA statement written in A statement written in the form IF___,THEN___.the form IF___,THEN___.

If P, then Q.If P, then Q.P –> Q (NOTATION)P –> Q (NOTATION)

VocabularyVocabularyConditional statements:Conditional statements:If P, then QIf P, then QP implies QP implies QQ, if PQ, if PALL MEAN SAME THINGALL MEAN SAME THING

VocabularyVocabularyHypothesis-statement Hypothesis-statement following the word “if”.following the word “if”.

Conclusion-statement Conclusion-statement following the word following the word “then”. “then”.

FACT OR FICTION???FACT OR FICTION???

IF TWO ANGLES ARE IF TWO ANGLES ARE SUPPLEMENTARY, SUPPLEMENTARY, THEN THEY ARE THEN THEY ARE BOTH RIGHT BOTH RIGHT ANGLES.ANGLES.

REVERSE THE REVERSE THE HYPOTHESIS & HYPOTHESIS & CONCLUSIONCONCLUSION

FACT OR FACT OR FICTION???FICTION???

IF TWO ANGLES ARE IF TWO ANGLES ARE RIGHT ANGLES, THEN RIGHT ANGLES, THEN THEY ARE THEY ARE SUPPLEMENTARY SUPPLEMENTARY ANGLES.ANGLES.

VOCABULARYVOCABULARYCONVERSE- A CONVERSE- A conditional statement conditional statement with the hypothesis and with the hypothesis and conclusion conclusion interchanged.interchanged.

If Q, then P. Q –>PIf Q, then P. Q –>P

FACT OR FACT OR FICTION???FICTION???

If x = 4, then xIf x = 4, then x22 = 16 = 16

Is the converse true?Is the converse true?

If xIf x22 = 16, then x = 4. = 16, then x = 4.

Write the converse. Is the Write the converse. Is the converse true?converse true?

1. If two angles are 1. If two angles are vertical , then they are vertical , then they are congruent.congruent.

2. If x > 0, then x2. If x > 0, then x2 2 > 0.> 0.

The negation of statement p is “not p,” written as ~p. The negation of a true statement is false, and the negation of a false statement is true.

Definition Symbols

The converse is the statement formed by exchanging the hypothesis and conclusion.

q p

Definition Symbols

The inverse is the statement formed by negating the hypothesis and conclusion.

~p ~q

Definition SymbolsThe contrapositive is the statement formed by both exchanging and negating the hypothesis and conclusion.

~q ~p

Write the converse, inverse, and contrapositive of the conditional statement. Use the Science Fact to find the truth value of each.

Example 4: Biology Application

If an animal is an adult insect, then it has six legs.

Example 4: Biology Application

Inverse: If an animal is not an adult insect, then it does not have six legs.

Converse: If an animal has six legs, then it is an adult insect.

If an animal is an adult insect, then it has six legs.

No other animals have six legs so the converse is true.

Contrapositive: If an animal does not have six legs, then it is not an adult insect.Adult insects must have six legs. So the contrapositive is true.

No other animals have six legs so the converse is true.

Write the converse, inverse, and contrapostive of the conditional statement “If an animal is a cat, then it has four paws.” Find the truth value of each.

Check It Out! Example 4

If an animal is a cat, then it has four paws.

Check It Out! Example 4

Inverse: If an animal is not a cat, then it does not have 4 paws.

Converse: If an animal has 4 paws, then it is a cat.

Contrapositive: If an animal does not have 4 paws, then it is not a cat; True.

If an animal is a cat, then it has four paws.

There are other animals that have 4 paws that are not cats, so the converse is false.

There are animals that are not cats that have 4 paws, so the inverse is false.

Cats have 4 paws, so the contrapositive is true.

If-Then Transitive Property If-Then Transitive Property (postulate)(postulate)

Given: If A, then B. Given: If A, then B. If If B, then C.B, then C.

Conclusion: Conclusion: If If A, then C.A, then C.

(logic chain)(logic chain)

If yellow is brown, If yellow is brown, then red is blue.then red is blue.

If black is white, then If black is white, then yellow is brown.yellow is brown.

If red is blue, then If red is blue, then green is orange.green is orange.

If black is white, then If black is white, then yellow is brown.yellow is brown.

If yellow is brown, If yellow is brown, then red is blue.then red is blue.

If red is blue, then If red is blue, then green is orange.green is orange.

Write as a Write as a conditionalconditional

ALL MATH ALL MATH TEACHERS TEACHERS ARE MEN.ARE MEN.

WRITE IN IF-THEN WRITE IN IF-THEN FORM.FORM.

IF A PERSON IS A IF A PERSON IS A MATH TEACHER, MATH TEACHER, THEN THEY ARE THEN THEY ARE

A MAN.A MAN.

A A VENN DIAGRAM VENN DIAGRAM is sometimes used is sometimes used in connection with in connection with

conditionalsconditionals

If p , then q.If p , then q.

qp

Make a Venn Make a Venn diagramdiagram

If Ed lives in Texas, If Ed lives in Texas, then he lives south then he lives south

of Canadaof Canada

Venn DiagramVenn DiagramIf Ed lives in Texas, then he lives If Ed lives in Texas, then he lives

south of Canada.south of Canada.

Texas South of Canada

Texas

CounterexampleCounterexampleIf Ed lives south of Canada, then he If Ed lives south of Canada, then he

lives in Texas.lives in Texas.

Texas

South of CanadaEd lives in Maryland