“IF-THEN STATEMENTS” DAY 17 DEDUCTIVE REASONING: The process of using orderly statements to make logical conclusions. IF-THEN STATEMENTS: CONDITIONAL STATEMENTS _________________________________________________________________________ CONDITIONAL CONVERSE (FLIPPED CONDITIONAL) IF P THEN Q IF Q THEN P P is the Hypothesis; Q is the Conclusion Q is the Hypothesis; P is the Conclusion If a figure is a triangle, then it’s a polygon If a figure is a polygon, then it’s a triangle TRUE FALSE INVERSE (CONDITIONAL with NOT) CONTRAPOSITIVE (FLIPPED CONDITIONAL WITH NOT) If NOT P, then NOT Q. If NOT Q, then NOT P. If a figure it’s NOT a triangle, If a figure it’s NOT a polygon, Then it’s NOT a polygon then it’s NOT a triangle FALSE TRUE CONDITIONAL and CONTRAPOSITIVE are LOGICALLY EQUIVALENT (either both True or both False) CONVERSE and INVERSE are both LOGICALLY EQUIVALENT (either both True or both False) _______________________________________________________________________________ GOAL: ONLY USE GOOD DEFINITIONS TO PROVE SHORTCUTS….. A GOOD DEFINTION WHEN THE CONDITIONAL AND CONVERSE ARE BOTH TRUE…. STATEMENT : Perpendicular lines form right angles. (A GOOD DEFINITION) CONDITIONAL CONVERSE IF two lines are perpendicular, IF two lines form right angles, THEN they form right angles. THEN the two lines are perpendicular “TRUE” “TRUE” “IF AND ONLY IF” STATEMENTS: When Conditional and its Converse are both “TRUE” BICONDITIONAL: Two lines are perpendicular “IF AND ONLY IF” they form right angles. STATEMENT: Two lines are perpendicular if they form right angles (A GOOD DEFINITION) STATEMENT: A triangle is a polygon (NOT A GOOD DEFINITON) CONDITIONAL IF a figure is a triangle, THEN it is a polygon. TRUE CONVERSE IF a figure is a polygon, THEN it is a triangle. FALSE IF AND ONLY IF: NOT POSSIBLE

PAIR WORK: Express as a Conditional and a Converse Statement. True or False. 1. All equilateral triangles are isosceles (NOT A GOOD DEFINITION) CONDITIONAL: If________________________________, THEN__________________________ TRUE/FALSE CONVERSE: If____________________________________, THEN__________________________ TRUE/FALSE IF AND ONLY IF STATEMENT: NOT POSSIBLE 2. Obtuse triangles have an obtuse angle. (A GOOD DEFINITION) CONDITIONAL: If________________________________, THEN__________________________ TRUE / FALSE CONVERSE: If____________________________________, THEN__________________________ TRUE / FALSE IF AND ONLY IF STATEMENT: A triangle is obtuse “IF AND ONLY IF” it has an obtuse angle.

USING EULER DIAGRAMS TO MAKE CONCLUSIONS DAY18

EULERDIAGRAMS

Compare the following if-then statements.

Statement: lf p,lhen q.Contrapositive: If not q, then not P.

You already know that the diagram at the right represents "lf p, then q."The diagram also represents "If not Q, then not pi' because a point thatisn't inside circle q can't be inside circlep either. Since the statement andits contrapositive are both true or else both false, they are called logicallyequivalent. The following statements are logically equivalent.

True statement: If a figure is a triangle, then it is a polygon.Tiue contrapositive: If a hgure is not a polygon, then it is not a triangle.

Since a statement and its contrapositive are logically equivalent, we mayprove a statement by proving its contrapositive. Sometimes that is easier.

There is one more conditional related to "If p, then q" that we will con-sider. A statement and its inuerse are not logically equivalent.

Statement: lf p, then q.Inverse: If not p, then not q.

True statement: If a figure is a triangle, then it is a polygon.False inverse: If a figure is not a triangle, then it is not a polygon'

Summary of Related If-Then StatementsGiven statement: If p, then q.Contrapositive: lf not q, then not P.Converse: lf q, then p.Inverse: If not p, then not q.

A statement and its contrapositive are logically equivalent.A statement is not logically equivalent to its converse or to its inverse.

The relationships just summarized per-mit us to base conclusions on the contrapos-itive of a true if-then statement bvt not onthe converse or inverse. For example, sup-pose we accept this statement as true:

All Olympic competitors,are athletes.(If a person is an Olympic competitor, thenthat person is an athlete.)

92 / Chapter 2

Solution

First statementx)4An integer is odd.Lines / ar.d m do not intersect.LA is a right angle.A polygon is equilateral.Alternate interior angles formedby lines I and m and transversal/ are congruent.

a. Given: dnll oc; ADll BCProve: 1A: /-C; LB: LD

necessary and sufhcient (An integer is divisible by 2 if and only if the integeris even.)necessary (If lines I and m are parallel, then they are coplanar. Note that theIirst statement is not sufficient for the second because two lines may be copla-nar without being parallel.)

b.

C 32.33.34.3s.36.37.

Second statementx is positive.The square of an integer is odd.Lines / and m are parallel.LABC is a right triangle.A polygon is regular.Lines / and m are parallel.

38.

b. Tell what is given and what is to be proved in the converse ofpart (a). Then write a proof of the converse.

c. Combine what you have proved in parts (a) and (b) into anif-and-only-if statement.

2-7 Converse, Contrapositive, InverseTo show the relationship between an if-then statement and its con-verse, it is helpful to use circle diagrams (also called Venn diagramsor Euler diagrams).

, To represent a statement p, we draw a circle named p. If p istrue, we think of a point inside circle p. If p is false, we think of apoint outside circle p.

In the diagram at the left below, a point that lies inside circlep must alsolie inside circle q. In otherwords: If p,then q. Check to see that the middlediagram represents the converse: If q, then p. Check the diagram at the rightalso.

p if and only if q.

tp is false.

OIf 4, thenp.

@\f p, then q.

p is true.

Parallel Lines and Planes / 9lCompare the following if-then statements.

Statement: lf p,lhen q.Contrapositive: If not q, then not P.

You already know that the diagram at the right represents "lf p, then q."The diagram also represents "If not Q, then not pi' because a point thatisn't inside circle q can't be inside circlep either. Since the statement andits contrapositive are both true or else both false, they are called logicallyequivalent. The following statements are logically equivalent.

True statement: If a figure is a triangle, then it is a polygon.Tiue contrapositive: If a hgure is not a polygon, then it is not a triangle.

Since a statement and its contrapositive are logically equivalent, we mayprove a statement by proving its contrapositive. Sometimes that is easier.

There is one more conditional related to "If p, then q" that we will con-sider. A statement and its inuerse are not logically equivalent.

Statement: lf p, then q.Inverse: If not p, then not q.

True statement: If a figure is a triangle, then it is a polygon.False inverse: If a figure is not a triangle, then it is not a polygon'

Summary of Related If-Then StatementsGiven statement: If p, then q.Contrapositive: lf not q, then not P.Converse: lf q, then p.Inverse: If not p, then not q.

A statement and its contrapositive are logically equivalent.A statement is not logically equivalent to its converse or to its inverse.

The relationships just summarized per-mit us to base conclusions on the contrapos-itive of a true if-then statement bvt not onthe converse or inverse. For example, sup-pose we accept this statement as true:

All Olympic competitors,are athletes.(If a person is an Olympic competitor, thenthat person is an athlete.)

92 / Chapter 2

MOREEULERDIAGRAMSEx.IfcompetitorsareOlympiansthentheyareathletes

This statement is paired with

l. Giuen: lf p, then q.

pConclude: q

four different statements below.

All Olympic competitorsare athletes.Ozzie is an Olympian.Ozzie is an athlete.

2. Giuen: lf p, then q.

rlot qConclude: not p

3. Giuen: lf p, then q.

qNo conclusion follows.

4. Giuen: lf p,lhen q.

Irot p

No conclusion follows.

All Olympic competitorsare athletes.Ned is not an athlete.Ned is not an Olympic com-petitor.

All Olympic competitorsare athletes.Anne is an athlete.Anne might be an Olympiccompetitor or she might notbe.

All Olympic competitorsare athletes.Nancy is not an Olympiccompetitor.Nancy might be an athleteor she might not be.

Classroom Exercrses1. State the contrapositive of each statement.

a.Ifx=3,thenx2+l:10..b. lfy(5,theny+6.c. If a polygon is a triangle, then the sum of the measures of its angles is

180.d. If you can't do it, then I can't do it.

2. State the converse of each statement in Exercise l.3. State the inverse of each statement in Exercise 1.

4. A certain conditional is true. Must its converse be true? Must its inversebe true? Must its contrapositive be true?

5. A certain conditional is false. Must its converse be false? Must its inversebe false? Must its contrapositive be false?

Parallel Lines and Planes / 93

@\ athletes

@\ athletes

@i

CW#18/HW#181.Given:Allsenatorsareatleast30yearsold.

a. Rewordthisstatementinif-thenform.Conditional:Ifsomeoneisasenatorthenhe/sheisatleast30yrsoldContrapositive:Ifsomeoneisyoungerthan30yrsoldthenhe/sheisnotasenator

b. MakeacirclediagramtoillustratetheConditionalstatement.

c. Ifthegivenstatementistrue,whatcanyouconcludefromeachofthefollowingadditionalstatements?Ifnoconclusionispossible,sayno.(Hint;OnlymakeconclusionspertheConditionalortheContrapositiveStatements)

1. JoseAvilais48yearsold. ________________________2. RebeccaCastelloeisasenator ________________________3. ConstanceBrownisnotasenator. ________________________4. LingChenis29yearsold. ________________________

2.Given:Whenitisnotraining,Iamhappy

a. Rewordthisstatementinif-thenform.

b. Makeacirclediagramtoillustratethestatement.

c. Ifthegivenstatementistrue,whatcanyouconcludefromeachofthefollowingadditionalstatements?Ifnoconclusionispossible,sayno.

1. Iamnothappy. __________________________2. Itisnotraining. __________________________3. Iamoverjoyed. __________________________4. Itisraining. _________________________

3.Given:Allmystudentslovegeometry

a.Rewordthisstatementinif-thenform.b. Makeacirclediagramtoillustratethestatement.

c. Ifthegivenstatementistrue,whatcanyouconcludefromeachofthefollowingadditional

statements?Ifnoconclusionispossible,sayno.1. Stuismystudent. __________________________2. Luislovesgeometry. __________________________3. Stellsisnotmystudent. __________________________4. Georgedoesnotlovegeometry. _________________________

INDIRECT REASONING /PROOFS DAY 19 INDIRECT REASONING:

1. Uses the idea that if a CONDITIONAL is TRUE, then its CONTRAPOSITIVE is also TRUE. CONDITIONAL: IF P THEN Q CONTRAPOSITIVE: IF NOT Q THEN NOT P

2. Uses the CONTRAPOSITIVE as the INDIRECT REASONING ___________________________________________________________________________ USING INDIRECT REASONING Explain how you would know if a driver applied the brakes. STATEMENT: A car leaves skid marks when it applies the brakes. CONDITIONAL: If a car leaves skid marks then it has applied the brakes CONTRAPOSITIVE: If a car does not apply the brakes, then it will not leave skid marks. INDIRECT REASONING: If a car does not apply the brakes, then it will not leave skid marks. Skid marks were left by the car. Therefore, the car must have applied the brakes. ____________________________________________________________________________ USING INDIRECT REASONING: Explain why ice is forming on the sidewalk in front of Toni’s house. STATEMENT: Ice forms when it is 32F or below. CONDITIONAL: If ice forms then the temperature is 32F or below. CONTRAPOSITIVE: If the temperature is more than 32F, then ice will not form on the sidewalk. INDIRECT REASONING: If the temperature is more than 32F, then ice will not form on the sidewalk.. Ice is forming on the sidewalk. Therefore, the temperature must be 32F or less.

PAIR WORK: USING INDIRECT REASONING Johnnie is too lazy to create flash cards. Explain how you know he isn’t going to get an A STATEMENT: Every student who gets an A in Geometry creates and uses flash cards. CONDITIONAL: If ____________________________ THEN _______________________ CONTRAPOSITIVE: IF ___________________________ THEN_________________________ INDIRECT REASONING:_______________________________________________________ ______________________________________________________________________ INDIRECT PROOFS: PROVING BY CONTRADICTION

1. Assume temporarily that the conclusion is not true.

2. Reason logically until you reach a contradiction of a known fact

3. Therefore, the temporary assumptions must be false and what needs to be proven must be true ____________________________________________________________________ Given (Hypothesis): n is an integer and n2 is even Prove (Conclusion): n is even Indirect Proof:

1. Assume temporarily that n is not even.

2. Then n is odd, and n X n = odd. This contradicts the given information that n2 is even.

3. Therefore, that n is not even must be false. _________________________________________________________________

CW#19 / HW#19 What is the first sentence of an indirect proof of the statement shown?

1. Triangle ABC is equilateral. ______________________________

2. Doug is Canadian. ______________________________

3. ______________________________

4. Kim isn’t a violinist. ______________________________

5. Write an Indirect Proof Given (Hypothesis): A triangle Prove (Conclusion): There can be at most 1 right angle

a. Assume temporarily that ______________________________________________

b. Then _____________________________________________________________ ___________________________________________________________________

c. Therefore __________________________________________________________

6. Write an Indirect Proof Given (Hypothesis): Fresh skid marks appear behind a green car at the scene Prove (Conclude); The car must have applied the brakes.

a. Assume temporarily that ______________________________________________

b. Then _____________________________________________________________ ___________________________________________________________________

c. Therefore __________________________________________________________

a ≥ b

7. Write an Indirect Proof Given (Hypothesis): Ice is forming on the side walk. Prove (Conclude); The temperature outside must be 32F or less.

a. Assume temporarily that ______________________________________________________

b. Then _____________________________________________________________________ _________________________________________________________________________

c. Therefore _________________________________________________________________ What conclusions, if any, can you make from each pair of statements?

8. There are three types of drawbridges; bascule, lift and swing. This drawbridge doesn’t swing or lift. Conclusion: __________________________________________________________________

9. If this were the day of the party, our friends would be home. No one is home. Conclusion: __________________________________________________________________

10. Every traffic controller in the world speaks English on the job. Sumiko does not speak English Conclusion: __________________________________________________________________

11. If non-vertical lines are perpendicular, then the product of their slops is -1. The product of the slopes of non-vertical lines is not -1. Conclusion: ___________________________________________________________________