Lesson 2: Biconditional Statements and Logical Reasoning

Perpendicular lines - lines that intersect to form a right angle

Line perpendicular to a plane - a line that intersects a point in the plane and is perpendicular to

Biconditional Statement - a statement that contains the phrase "if and only if '

Same as writing a conditional statement AND its converse

Conditional Statement: If three lines are coplanar, then they lie in the same plane.

converse: æ tltrt. fùr, l¿. irt t7'e Sa*te Pl*t, tl'*' fh'f a'r<ø7la'wr

Biconditional: Three lines are coplanar if and only if they lie in the same plane.

every line in the plane

Ex l) Biconditional: x:3if andonly if x2:9,

Rewrite the biconditional statement as a condtional statement and its converse,

Ex 2) The ceiling fan runs if and only if the light switch is on.

Conditiona l: -+1 +h' el.; ltnJ f-" cv'/t6¡ '11'\þ +l/ú-

Converse: -¿f, -f{n-- l;¡hi s*ikl" t'5 oA t ¡+1"on

Ex 3) You scored a touchdown if and only if the football crossed the goal line,

Conditional: T+ Voa scoçe¿! 4 l-utl^J^-'t t t4^ +1^*

converse: T€ *Àc &Ò+bJ cþs*( fhz go". lin e , *h"^nEx 4) The expression 3x+ 4 is equal to 10 if and only if x is 2.

Conditional: T4 gx+ 'l -- (O ì +l"w Y. = Z '

Converse: -L+ y=?l *he*, 3x+Y=lo.

r€T€

X=3 ¡ fÀøn x¿=1

X¿= ? I -11+e*r- .a--3I roe

lSlnt ¡*i[.htl^- eu lA,y

F-ls <

r3

4^o{l -

lLtvlS.

{ofLu cro>tJrlre- 3on( fl'aE'

vd SLoß¿ qr' ¡<,*t"Ja^"+'

Conditional Statement: p -> q

Converse: L à ?

Inductive Reasoning - examples and patterns

Deductive Reasoning - facts, definitions, and

Two Laws of Deductive Reasoning:

1) Law of Detachment

Inverse: /1/"+ p - rlot-L Contrapositive:

If p+q is a true conditional statement and p is true, then q is true,

2) Law of Syllogism

If p--}q and q-) r are true conditional statements, then p-)r is true.

Determine if the conclusion is valid.

Ex I ) Il the sun is shining, then it is a beautiful day.The sun is shining,

Conclusion: It is a beautiful day.

are used to form a conjecture

postulates are used to write a logical argument

Ex 2) IlChris watches a kalate movie, then he beats up his little blother'.IlChris beats up his little brother, then he gets grounded lor a week.

Conclusion: IlChris watches a karate movie, then he gets grounded lor a week.

Ex 3) IfCheryl becomes a nurse, then she will take care olher father.Cheryl takes care olher father.

Conclusion: Cheryl became a nurse

/uot Lã /"{f

Ex 4) IlEric plays too much Call of Duty, then his girlfriend will dump him.If Eric plays too much Call of Duty, then he will lose his job.

Conclusion: If Eric's girlfriend dumps him, then he will lose his job.

If it is valid, identify which law you used.

V-\i/ r Døto-s l*"^+

Ex 5) If Nicole does the dishes alone, then her husband is in trouble.Nicole's husband is in trouble. , ì /conclusion: Nicole did rhe dishes alone. fuVnt

Ex 6) Iftwo points are collinear, then they are also coplanar.If two points are coplanar, then they lie in the same plane.

Conclusion: Ifltwo points are collinear, then they lie in the same plane

Ex 7) Iftwo lines are perpendicular, then they form right angles.Two lines are perpendicular. I

conctusion: Right angles are f'ormed \/n\¿d (

Ex 8) If Mr. Leman doesn't like you, then you will lail geometry.

t¡¿'l¿J

V^t¡rt s'¡[(1ts'n

You lail geometry.

Conclusion: Mr. Leman doesn't like you.

f^'¿^\iJ

\/^lvl t 5'{lllia"t

í-',¡rt,.l)

Lesson 2 Practice: Logical Reasoning

For each of the following determine if the conclusion is valid. If the conclusion is valid, justiff it witheither the Law of Detachment or the Law of Syllogism.

Conclusion: Nicole graduated from Coal City High School.

Todd chases chickens..-Conclusion: Todd is a fox. . t/vo+ t/ohu)

If tar, he will get a gorgeous date for homecoming.

/t/r/- Vnllol

Conclusion: If Jake plays Rock Band, then he will get a gorgeous date for homecoming.

1/^hJ, | 3V /hsisrvtPZ

If Brandon studies geometry, then he passes the test.

(O"t*"1^^^^'^+)

Conclusion: If Brandon studies geometry, then he will be happy.pî

( Ð.þ.1r"'"^+)

rLIf the Scholastic Bowl team plays lots of video games, then they will have quick thumbs.If the Scholastic Bowl team plays lots of video games, then they will lose sleep at night.rcConclusion: If the Scholastic Bowl team has quick thumbs, then they will lose sleep at night,

LC

rIf the measure of an angle is less than 90o, then it is acute.mZA:60o. L

(Conclusion: ZA is acute.

1-

/"|,vl , sy//"3,5rvt

/.1- lol,J (sypsin)

r/ olil' , 0.1"h*'-'"t

Determine a valid conclusion using the Law of Detachment or Law of Syllogism. If a valid conclusioncannot be reached, state that there is no valid conclusion possible.

t7. If Donald takes a nap the back yard, then Chip and Dale will anger him.

Chip and Dale anger Donald.ct

conclusion' rr/o vnlio( conel,,,sion

If Eric goes to Joliet, then he eats at Taco Bell.rcConclusion:

If Kirk plays o the tennis team, then he knows how to volley.Kirk knows how volley. Z

tt/o

Conclusion:

10, IfBob does not eat school lunch, then he eats gorgonzola cheese,If Bob eats gorgonzola cheese, then his girlfriend won't kiss him.

Conclusion: -.yÇ ß16 /æ, A+ e"t s.h-(

t/n l|J, Cort"lostort

l/o

I I . If three points are noncollinear, then they are contained in a plane.

J^liJ

Three points are noncollinear,

conclusion: Tlv- *-hæ<

Qrt.losø^

12. If a segment is bisected by a line, then the line passes through the midpoint of the segment.If a line passes through the midpoint of a segment, then the two created segments are congruent,

conclusion' -+l A- s<j,uro"t* is List'l¿ LY a- I ine, 14*'t'-1-J

13. If perpendicular lines intersect, then right angles are formed,If perpendicular lines intersect, then four 90o angles are created.

conclusion: fu/o únl,'ol ØAclosioA

¡o fn*s a,r¿- ø'+4q¡ 't"ol 'h a- 7 /*tt '

(,:".h ¡ *lrwt h,z gi r l{'ric^nc/.,.rnn ï þ,ss h)n

*h. t*"SeS rrnr.** S

uYrvc*1.ofz

Rewrite the biconditional statement as a conditional statement and its converse.

14. Two angles are congruent if and only if they have the same measure.

Statement: Tl Z a15les or.- conSro"*t t fu^ +h'{ hq'c- fi'converse: ;-4 Z a^gks hen fhr. &,vtt¿ v4eÁs6î ) -fh'-^ +h'Y aß-

15. A ray bisects an angle if and only if it divides the angle into two congruent angles.

Statement: --L+ o.. f^l b¡set axl otgle, 'fl,'n ;+ ol¡'t¡Jes f{*- AL ùrh l"^converse: î+ a^ #,:^ir,ff'Tn+" 1t¿,o cn'yruø* a'gtes , *l'r'nr '{ r's

b; "...loo(_16. Two lines are perpendicular if and only if they intersect to form right angles.

Statement: -Å ltto lu.s ^tr- fnfr^¿;.u|, 1 #nt +lrry

Converse: T+ 'fuo llnrs inlr¡"e-t i. {o.^ r-,Sh I o,'glrt ,

Determine if each statement is true or false.

F

F

F

F

F

F

17. Perpendicular lines intersect to form right angles.

18. Inductive reasoning uses patterns and observations to make conjectures.

19, The converse of "If this is homecoming week, then we will see all kinds of weird outfits" is "lfwe see all kinds of weird outfits, then this is homecoming week."

20. The inverse of "lf this is homecoming week, we will see all kinds of weird outfits" is "lf this isnot homecoming week, then we will not see all kind of weird outfits."

21. A conditional statement and its contrapositive mean the same thing.

22. If two lines intersect, their intersection is a point.

23. If two planes intersect, their intersection is a line.

24. If two lines intersect, then exactly one plane contains them.

25. A biconditional statement is considered true if the converse is true.

26. DC is perpendicular to line m.

27. Line n bisects ZJCH.

28. /.ABJ and IDCH are supplementary.

29. /7B is perpendicular to linep.

30. Points A, F, and G are collinear.

ìnlc.rs"&

'l-l^^^ +AæV arzP*Púicolar.

{" âr-,r-ùhf "',t1lcs,

3 L Find the midpoint of the segment that goes from (4, - I ) to (-2,7),

'l+4 . -J_r7z)Zf14

32. lf (3, -2) is the midpoint of a line segment and ( I , 4) is one endpoint, what is the other endpoint.

/+x = J ,lrr -- -z (x' v)

33. For A(3, l) and B(-2,6), f,rnd AB.

34. x: IL(l2x+10)i

z2

(Lx+/o) r(zx+z)=(îollx +t2 = t7o

t4x = (G7

l+r,-b 4+y =-r¡x=5 v -- -s

2x+2)"

(t , s)

WX36. IVWX is a right angle

35. x:

x=12

¡tl +m¿2'1Ôzx-t+Xtsó=1O

mZVV/Y: ¡2x-8)o = z(t¡.) _l = JL-E = 2,!nZXWY: (x + 50)'= (r.) ¡So , Oc

I(llx-4)

mZVWY:nZXWY:

3x + 42 =1o3x=48x =lu

38. An angle is 2 more than 3 times its complement. Find the measure of the angle.

llx-q = lxrî9x+6)'

Z'TGG

Zx -4 = G

zx -- loY--l

A= /"st'1o 'A = e-'?le"^"'d

37, KM bisects ZJKL ZX = 12

*ZJKM: (Sx-6)o= Ç(t ) -c Y = c¿

m.lMKL: ({x+6)o

G

K+

n+Í4,¡r¡ o øl mklGx-ø = (X+t"

lr:,ZJKI|l= 3ô

A' 3(çþ-n) +zA= 27ô-A4 +L

qA = 272A 'G3