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 that Solution necessary and sufhcient (An integer is divisible by 2 if and only if the integer isn't inside circle q can't be inside circlep either. Since the statement and is even.) its contrapositive are both true or else both false, they are called logically b. necessary (If lines and m are parallel, then they are coplanar. Note that the equivalent. The following statementsI are logically equivalent. Iirst statement is not sufficient for the second because two lines may be copla- True statement:nar without If a figurebeing isparallel.) a triangle, then it is a polygon. Tiue contrapositive: If a hgure is not a polygon, then it is not a triangle. First statement Second statement Since a statement and its contrapositive are logically equivalent, we may C prove32. x)4 a statement by proving its contrapositive.x is positive. Sometimes that is easier. 33. AnThere integer is one is moreodd. conditional related toThe "If squarep, then of q" an that integer we will is odd.con- inuerse are not logically equivalent. sider.34. Lines A statement / ar.d m doand not its intersect. Lines / and m are parallel. 3s.Statement:LA is a lfright p, thenangle. q. LABC is a right triangle. Inverse:36. A polygon If isnot equilateral. p, then not q. A polygon is regular. USINGTrue37. Alternate statement: EULER interior DIAGRAMS If a anglesfigure isformed a triangle, thenLines it /is and a polygon. m are parallel. is not a triangle, then it is not a polygon' Falseby inverse: lines I and Ifm a and figure transversal / are congruent.

38. a. Given:Summary dnll oc; ofADll Related BC If-Then Statements Prove: 1A: /-C; LB: LD Givenb. Tell statement: what is given If p,and then what q. is to be proved in the converse of Contrapositive:part (a). Then lfwrite not aq, proof then ofnot the P. converse. Converse:c. Combine what youlf q, have then provedp. in parts (a) and (b) into an Inverse:if-and-only-if statement.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.

EULER DIAGRAMS2-7 Converse, Contrapositive, Inverse The relationships just summarized per- its con- Tomit show us to thebase relationship conclusions between on the contrapos- an if-then statement and verse,itive ofit isa truehelpful if-then to use statement circle diagrams bvt not (also on called Venn diagrams orthe Euler converse diagrams). or inverse. For example, sup- t is p is false. pose, To we represent accept this a statement statement p, as we true: draw a circle named p. If p true, we think of a point inside circle p. If p is false, we think of a p is true. pointAll Olympic outside competitors,arecircle p. athletes. (If a person is an Olympic competitor, then circlep must also that Inperson the diagram is an athlete.) at the left below, a point that lies inside lie inside circle q. In otherwords: If p,then q. Check to see that the middle diagram represents the converse: If q, then p. Check the diagram at the right also. 92 / Chapter@ 2 \f p, then q. OIf thenp. p if and only if q. 4,

Compare the following if-then statements. Parallel Lines and Planes / 9l 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 that isn't inside circle q can't be inside circlep either. Since the statement and its contrapositive are both true or else both false, they are called logically equivalent. 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 may prove 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 Statements Given 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 on the 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, then that person is an athlete.)

92 / Chapter 2 MORE EULER DIAGRAMS

Ex. If competitors are Olympians then they are athletes

This statement is paired with four different statements below.

l. Giuen: lf p, then q. All Olympic competitors are athletes. p Ozzie is an Olympian. @ Conclude: q Ozzie is an athlete. \ athletes

2. Giuen: lf p, then q. All Olympic competitors are athletes. rlot q Ned is not an athlete. Conclude: not p Ned is not an Olympic com- @ petitor. \ athletes

3. Giuen: lf p, then q. All Olympic competitors are athletes. q Anne is an athlete. No conclusion follows. Anne might be an Olympic competitor or she might not be.

4. Giuen: lf p,lhen q. All Olympic competitors are athletes. Irot p Nancy is not an Olympic competitor. @i No conclusion follows. Nancy might be an athlete or she might not be.

Classroom Exercrses

1. 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 inverse be true? Must its contrapositive be true? 5. A certain conditional is false. Must its converse be false? Must its inverse be false? Must its contrapositive be false?

Parallel Lines and Planes / 93

CLASSWORK (Homework in not completed in class)

1. Given: All senators are at least 30 years old. a. Reword this statement in if-then form.

b. Make a circle diagram to illustrate the statement.

c. If the given statement is true, what can you conclude from each of the following additional statements? If no conclusion is possible, say no.

1. Jose Avila is 48 years old. ______2. Rebecca Castelloe is a senator ______3. Constance Brown is not a senator. ______4. Ling Chen is 29 years old. ______

2. Given: When it is not raining, I am happy a. Reword this statement in if-then form.

b. Make a circle diagram to illustrate the statement.

c. If the given statement is true, what can you conclude from each of the following additional statements? If no conclusion is possible, say no.

1. I am not happy. ______2. It is not raining. ______3. I am overjoyed. ______4. It is raining. ______

3. Given: All my students love geometry a. Reword this statement in if-then form.

b. Make a circle diagram to illustrate the statement.

c. If the given statement is true, what can you conclude from each of the following additional statements? If no conclusion is possible, say no.

1. Stu is my student. ______2. Luis loves geometry. ______3. Stells is not my student. ______4. George does not love geometry. ______