Geometry - Unit2 Torgets& InfoThis Unit's theme - Reosoning ond ProofSeptember 12 - October 8 (Approximote Time for Test)Use this sheet os a guide throughout the chopter to see ifin reaching eoch torget listed.

By the end of Unit l, you should know how to...

Identify and use correct vocabulory: Negotion,inductive reasoning, deductive reosoning, converse,inverse, contropositive, bisect, midpoint,perpendicular, complementary, supplementory, rightanqleWrile a conditionol slotement inalong with its converse, inverse,and defermine if the stotementsWrite a biconditional statement as its conditionaland converse statements and determine if thebiconditional stotement is true or folseJustify stolements wilh definitions, postulates,theorems proven in class, or propertiesComplele o two column proof by providing reasonsfhat iustifv eoch oiven statement

Nome: W-

l=

you ore getting the right information

Use the Law of Detachment ond Low of Syllogismto make valid conclusions

Target found in...

IF-THEN formond contrapositive,ore true or false.

Calculate the Surface Areodimensional fiquresAll material covered on the tesf will be based onthese targets. So keep track of your readiness forthe test by updating fhe "Did I reach lhe largel?"column.

Chopter 2

Did I reochthe torget?

Chopter 2 Seclion2, pages 89-95

Key Postulotes, Properties, ond Theorems:

Segmenl Addition Postulote or Angle Addition Postulate

and Volume of three-

Chopter 2 Seclion3, poges 98-lO4

Algebraic Properties (Addition, Subtraction, Multiplication, Division,Substitution, Dislributive)

DIAGRAMS &EXAMPLES!

Reflexive, Symmetric, Transitive Properties of Equality ond Congruence

Chopter 2 Section5, Þaqes ll3-ll9

Verticol angles ore congruent.Linear pairs are supplementary.If two angles ore congruenf, thenIf two angles are congruent, fhenAll right angles are congruent.

Chapler 2Sections 5 & 6Chapter 2 Section4 paqes 106-112

their supplemenfs are congruent.their complemenfs are congruent.

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1-61 ' p^^*Mth. ttr'ror,ffK;5N.tl þe Pt

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Unit 2 Suggested Textbook Problems

Section 2.2p.93 #s-31

Section 2.3p. l0l #7 - 38, 43 - 46

Section 2.4p. 110 #6 - 17

Section 2.5p. lt7 #5 -22Section 2.6p. 124 #6 - 20,23

POSTULATES AND THEOREMS

POSTULATES: Accepted without proof!

Segment Addition PostulateIf B is between A and C, then AB + BC : ACIf AB + BC : AC, then B is between A and C.

Angle Addition PostulateIf P is in the interior of ZRST, then mZRSP + mZPST : mZRST.

Through any two points there exists exactly one line.

If two lines intersect, then their intersection is exactly one point.

Through any three noncollinear points there exists exactly one plane.

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

If two angles form a linear pair, then they are supplementary.

Theorem

THEOREMS: Proven true!

Picture/Rephrase

Lesson 1: Conditional Statements

Conditional Statement: a logical statement beginning with a HYPOTHESIS and ending with a CONCLUSION.

***Normally, conditional statements are written in If-Then form.'¡'r"r'

Example: If Jaiden wakes up at 2i30 am, then Nicole will be grouchy.

HYPOTHESIS: Jaiden wakes up at 2:30 AM

CONCLUSION: Nicole will be grouchy

Underlinethehypothesisandcircle,r'"@inthefollowingconditionalstatements.

1) If I win the lottery, then I will build a racket ball court and indoor tennis facility in Coal City.

2) If Mr. McCleary sees a Sandra Bullock movie, then he will be overexcited and not be able to sleep.

3) If two lines intersect, then their intersection is exactly one point.

4) There will be no more homework in Geometry when Mr. Leman can dunk a basketball,

Rewrite the following conditional statements in If-Then form.

1) Two points are collinear if they lie on the same line.

2) A number divisible by 9 is also divisible by 3.

3) Two planes intersect at a line.

4) Mr. Leman's wife is the most beautiful woman in the world.

Original Statement: lf p, Then q.

Converse: lf q, then p.

Inverse: lf not p, then not q.

Contrapositive: If not q, then notp.

Example l)Conditional: If mlA = 30o, then mZA is acute.

Converse: lf mZA is acute, then mZA = 30o.

Inverse: lf mZA É 30o, thenrnlA is not acute.

Contrapositive: If mZA is not acute, then mZA * 30o.

Equivalent statements: statements that share the same truth value (T or F)

t+{'A conditional statement and its contrapositive ALWAYS share the same truth value.***

++'tThe converse and inverse of a statement ALWAYS share the same truth value.***

Negation: writing the negative, or opposite, of a statement.

Ex: Boys are trouble Negation: Boys are not trouble

A, B, C are collinear Negation: A, B, C are non-collinear

Write the converse, inverse, and contrapositive of each conditional statement.

l) If three points are collinear, then they determine a plane,

2) lf a segment is bisected, then the segment is cut in half.

3) Students do well on tests when they study.

Determine if each statement is true or false.

F 1. Points P and S are collinear.

F 2. Points W, R, and V are coplanar.<> <>F 3. QS and TV are coplanar.

F 4. Points Q, R, S, and T are coplanar.

-T 5. Line QS lies in plane A.

F 6. Lin"fî lies in plane A.

F 7. rine& and lineft intersect only in point R.

Lesson I Practice: Conditional Statements

F 8. The hypothesis of the conditional statement, "If two angles have the same measure, then they arecongruent" is "two angles have the same measure".

F 9. The converse of p - q, is q -, p.

T 10. If a conditional statement is true, its converse is also true.

F I l A conditional statement and its contrapositive mean the same thing.

F 12. If a statement is true, its negation is false.

Write the negation of each statement.

13. Acute angles are not less than 90o.

Ar4. a,gles a-c\- [es' f1"^ ?o''

14. Chocolate is an ideal food.

Write the following conditional statements in the "If.. .., then. ...." form.

15. Adjacent angles share a common side. r

14 an.5tes ¿rre- a-iccerzú¡ +h,,''. flAY sh.f,c- E

16. Perpendicular lines form 4 right angles.

Choøft.\- r's nóf a"^ ìlesl

-4 (ivtes 4¡.c- perTenolt'ru Lr I t.n fu'Y

+"""1.

4,t-

Write the converse, inverse, and contrapositive of the following conditional statement. Determine if eachstatement is true or false.

17 . If four points are collinear, then they are coplanar.

converse fl {o"c p¿,vÌ+s a,r,€- eoplo^r , fu +1*y d'r<' colltnu'r '

Inverse 7'-I {^rc Tolnl.- a.{-€- n"t c-olliaeal , *\n.n lhey 4'ce no'l- co¡løna1'l-t

Contrapositive T+ {orc pornt4s a.ß At coph^*, 11"^r^ tl*V a'f-L nofcpllìnear.

18. Find the midpoint of the segment with endpoints at (5, -2) and (-1, l0).

+, +, (t,q\19. If an endpoint of a segment is at (5, -3), and its midpoint is at (I,2), find the other endpoint of the

segment' f +x r -3 +y - )H=-, l:; (s,7)x=-3 1=-7

20. Find the distance between the following points.

a. (4, -2) and (1, 4) b. (3, 7) and (8, -5)

,l{+sc -- [-qs

21. Solve for x and y.

x: Zf:32

3X+ rl) = Ax-.ss3x -- 75

Y' 2f

(6x - 35)(2y + 1)"

3er) +'þ =ttS

zf+l +llr=t8o2Y =61y z 3Z

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:

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

Ex l) Biconditional: x: 3 if and only if x2:9.

True or False?

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

Ex2) The ceiling fan runs if and only if the light switch is on.

Conditional:

Converse:

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

Conditional:

Converse:

Ex 4) The expression 3x + 4 is equal to l0 if and only if x is 2.

Conditional:

Converse:

every line in the plane

Conditional Statement: p + q

Converse:

Inductive Reasoning - examples and patterns are used to form a conjecture

Deductive Reasoning - facts, definitions, and postulates are used to write a logical argument

Two Laws of Deductive Reasoning:

l) Law of Detachment

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. If it is valid, identiff which law you used.

Ex l) Ifthe sun is shining, then it is a beautiful day.The sun is shining.

Conclusion: It is a beautiful day.

Ex 2) If Chris watches akarate movie, then he beats up his little brother.IfChris beats up his little brother, then he gets grounded for a week.

Conclusion: If Chris \ryatches a karate movie, then he gets grounded for a week.

Ex 3) If Cheryl becomes a nurse, then she will take care of her father.Cheryl takes care of her father.

Conclusion: Cheryl became a nurse.

Ex 4) If Eric 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.

Ex 5) IfNicole does the dishes alone, then her husband is in trouble.Nicole's husband is in trouble.

Conclusion: Nicole did the dishes alone.

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

Conclusion: If two 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.

Conclusion: Right angles are formed

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

Conclusion: Mr. Leman doesn't like you.

lnverse: Contrapositive:

Lesson 2 Practice: Logical Reasoning

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

l. If Nicole graduated from Coal City High School, then she has a diploma.Nicole has a diploma.

Conclusion: Nicole graduated from Coal City High School.

il.| Vol;ul

2. If Todd is a fox, then he chases chickens.Todd chases chickens.

Conclusion: Todd is a fox.

//¿+ v*l¡ J

3. If Jake plays Rock Band, then he will learn to play the real guitar.If Jake plays the real guitar, he will get a gorgeous date for homecoming.

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

V*lú , Sy (qisvn

If Brandon studies geometry, then he passes the test.If Brandon passes the test, then he will be happy.

Conclusion: If Brandon studies geometry, then he will be huppy,

/"1¡Å ¡ syllqisu

If 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.

Conclusion: If the Scholastic Bowl team has quick thumbs, then they will lose sleep at night.

/1/"+ /"1¡'l

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

Conclusion: ZA is acute.

/^t¡'t t0.lrc\'',t*fr

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.

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

Conclusion: ¡1/o /"\ìdl C-onc[us,ovt

If Eric goes to the Joliet, then he shops at Best Buy.If Eric goes to Joliet, then he eats at Taco Bell.

Conclusion t A)o vnlVl ürtjurio^

9. If Kirk plays on the tennis team, then he knows how to volley.Kirk knows how to volley.

Conclusion: 2[ v^lel Øncl,¿sion

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, T+ bL /*t tt;f ec"l scl¡oJ fr*"h¡

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

conclusion : aÁe f{næ. ytvtl s a,re co¡lø;nJ l¡a c' |lo'ae '

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 , T{ ^ s.5wwr* is Líse"l*( by a' line, 1-l'^ ++* *t"o

<¡ce'-.treol ta^k oft- ø*5Nevt{ '

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

Conclusion t A/o J*\¿J &tcl¿s t o

13.

*hu^ hus gtcl#ìenJ*o"'* kt'ss h'w '

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: T1 Jto" a^Jpt atrc c-,rprw^*l +M il"Y f'*' |'ln" sq*t {/te.4S.¡Cc '

Converse: T€ fiuþ o,"rgles t"v"'e- 11"" s'r'ù,,e 'v1€45.,ce , fu 4q tt:

c-ø|,,-J Ne'rr\+,l5. A ray bisects an angle if and only if it divides the angle into two congruent angles.

Statement: fÊ d.- fày b'*<'ls a'-' a'u5l< J fu ;! alstid'es 2;r-j^O\*rrr!'Converse: T?- a- ç^y l¡t,eles a^,\ o43te;r¡1, 1r,h cryrænt a,yles, tLq ;L

L

å[16. Two lines are perpendicular if and only if they intersect to form right angles,

Statement: l+ lv" (i'tes a"rc- ¡eyaÁr-lat, +1/v'^ +1^"y itT+'rsif '1" Êc'ø' n5ht "''"1les '

converse: Tp jwo l¡nes ìnlc.rs.r* l" 4r* n'úhf ouu.6fes , +h^ iln y ø"rc

Determine if each statement is true or false. ?*Púfcol*î '

F

CÐFF

F

F

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 "If 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."

A conditional statement and its contrapositive mean the same thing.

If two lines intersect, their intersection is a point.

If two planes intersect, their intersection is a line.

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

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

DC is perpendicular to line m.

Line n bisects ZJCH.

Z.ABJ and ZDCH are supplementary.

F.B is perpendicular to linep.

Points A, F, and G are collinear.

31. Find the midpoint of the segment that goes from (4, -1) to (-2,7).

32. If (3, -2) is the midpoint of a line segment and (1, 4) is one endpoint, what is the other endpoint.

JJ.

34.

tll-z -l¡7z,t2l

(l2x+10 o

14 -t þL = -zlfl. . tt {+Y o -Y

xof \.-1

lZX vto t Zx¡L -- t 8o

/{xr p= lþI LlX, = lbï

Y. z lZ

ây.-t r Xtlo'?o

36.

35. x: 5

WXZYWX is a right angle

*ZYWY: 12x-8)o

mlXyy=(¡+50)ox: lÇ

3l +42 = 163x=4î

X=f0

(l lx-

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

A+c = 1o

llx-{ = lxtoZv, = lôF'l

K+37. KM bisects ZJKL

^./.IKM:16x-6)o*ZMKL: (4x+6)o

x= Gn./.IK}¿4:

ft= 3c "Zrt= sGo-,1) +zÆ -- 27o -3t4 +z

LIA = 27L

Ç*- ç = {x+t'ZX =tZ

,< --l¿

Lesson 3: Reasoning with Properties

Algebraic Properties of Equality

Let a, b, and c be real numbers:

Addition Property: If a: ó, then at c:b + c

x-3=5Example: + 3

x=8Subtraction Property:

x*6=10Example: -6 -6

-/l-+

+3

Multiplication Property:

Ifq:å,thena-c:b-c

xExampl : 3.-=3.-6

3¡=-18

x-- -$3

Division Property: lf a: b andc *O,then a+c=b+c2x=12

Example: +2 +2x=6

lf a: å, then ac : bc

Substitution Property:x=y

Example: x = 32Therefore, ! = 32

Distributive Property:

. -2(x+ 4)Examole:' 1x-8

If a: b,then a can replace b in any expression or equation

3x*7 x =20yl0x = 20y

a(b+c)=ab+bc

Name the property that justifies each statement.

Ex l) If m2=45o ,then 3(mlA) = 135o.

A) Addition Pro ultiplication Prop C) Subtraction Prop

Ex 2) IfST : 2 and SU : ST + 3, then SU : 5.

Ex 3) If mZT = mlQ, then mZQ= mZ-T

ymmetric Prop B) Transitive Prop C) Substitution Prop

Ex 4) If JK: PQ and PQ : ST, then JK: ST.

A) Addition Prop B) Symmetric Prop ubstitution Prop

Complete the proof using properties of equality,

Given: 5x - l8 = 3x * 2

A) Reflexive Prop B) Symmetric Prop ransitive Prop

Prove: x = 10

Statements

1.

2.

J.

4.

5¡-18 =3x*22x-18=2

2x=20

x=10

Given: 552- 3(92+12) = 44Prove: z=-I

Statements

1. Given

2. A)Addition

3. A) Addition

4, A) Addition

Reasons

552-3(92+12)= 44

552-27 z- 36 = -14

282- 36 = -64

282= -28

2.

J.

4.

5.

B) Subtraction C) Substitution

B) Division C) Substitution

B) Division C) Multiplication

Reasons

l. Given

2. A) Mult. B) Distributive C) Subst.

3. A) Subtraction B) Addition C) Subst.

4. A) Addition B) Mult. C) Transitive

5. A) Mult. B) Division C) Addition

Complete the proof using properties of equality.

Given: 3(4v - l)- 8u = 17Prove: v=5

Statements

l.3(4v-l)-8v=17

2.

J.

4.

lZv-3-8v=17

4v-3=17

4v =20

Given: AB = CDProve: AC = BD

Statements

2.

3.

4.

5

l. AB: CD

2. AB+BC:BC+CD

3. AC:AB+BC

4. BD: BC + CD

5. AC: BD

A B

1. A) Given B) Prove

2. A)Addition B) Symmetric

3. A) Seg + Post B) Reflexive

4. A) Seg + Post B) Transitive

5, A) Symmetric B) Reflexive

C D

C) Subst.

C) Subst.

C) Subst.

C) Subst,

C) Subst.

Given: mZl+ mZ2= 66"mZl+ ml2+ mZ3 = 99"mll = ml3mZI= mZ4

Prove: mZ4 = 33o

Statements

1. mlI+ ml2= 66o

z. mZl + ml2 + ml3 = 99o

3. 66" + ml3=99"

mZ3=33o

ml3= m.ll,mZl= mZ4

mZ3 = mZ4

mZ4=33o

Determine a valid conclusion using the Law of Detachment or Law of Syllogism.

Ex l)If proofs arepart of Geometry, then Geometry is fantastic,Proofs are fantastic.

Conclusion:

Ex2)If Eric drives the golf cart into the waferhazard, he will get thrown off the course.

If Eric drives the golf cart into the water hazard, then he will have to pay for the damages.

Conclusion:

Ex 3)If the two angles add up to 90o, then they are complementary,

m/.A+mlB=90o

Conclusion:

esson 3 Practice: Reasoning with Properties

Name the algebraic property that justifies each statement.

1. Ifx : y + 2 andy'r 2 : 12,thenx: 12,

2. Ifx+ 3 :7,thenx:4 S.¡b{trgùhn

3. xy: xy ß*Çt.X¡t"4. If 7x:42,then x:6.5. If XY -YZ: XM, then XY : XM + YZ

6. 3(x - a) : 3x - t2 0i+¡ hrtNe

7. lf mZA+mZB:90o andmZB:30o, thenmZl+ 30o:90o. SuLS1,'1u{,'ot^

8

9

. lf mZA : mlB,then mZB : mZA, Synrrn.*n'C

DNlsPn

Tr¿¿rstl+¿'

. If 3x t 2x:40, then 5x: 40. SrbSI,'{"{¡on

r0. ttä - ]:t*,rhenx-10:12x. ¡l\vlh¡tic-*ianl1

l2

. lf mZA: mZB and mZB : mZC, then mZA: mlC. T'þ¡sl+V .

. If 4x - 5:31, then 4x:36. A//¿tV^

pÅt$,ø

13. mZA: mzA Ê.4fcr|.t4. If AB + BC: AC and BC: 6, then AB + 6: AC. s"bgfti+lto^15. If 4x(x- t): 12,then4x2 -4x:12. Oishl LJionComplete each proof by naming the property that justifies each statement.

16. Given: 2(x - 3) : 8Prove: x:7Statements

t.2(x-3):82. 2x- 6: 8

3. 2x: 14

4. x:7

Reasons

1.

2.

3.

4.

6l'l"nD¡sla btl'tv¿

M¡4¡^Oì't¡slov't

t7. Given: 3x-4= 1x+62

Prove: x:4Statements

1. 3x-4 = 1x+62

2. 1x-4:62

3. lx= 102

4. x:4

Reasons

18. Given: wZRPQ= mZkPSProve: wZSPQ=2(ryZRPQ)

Statements

1. 6¡11""

2. 5¿Llr"-cl.to¡

3. A{rlt*on

1. wZRPQ=qZfuPS

2. wZ.SPQ= qZRPQ+ryZSPR

3. wISPQ= wZRPQ+ryZRPQ

4. wZSPQ=2(ryZRPQ)

ON¡s,pn I n'l,r,Plìc"l'u'n

19. Find the distance between the points (3,4) and (-2, -l).

20. Find the midpoint of the segment from (6, -1) to (-2,5).(t+,-$

Reasons

21. Solve for x and y.

X: T

subs{:ìIuilon

J/./¡al/iþon

3¿bs*ì4" l,b'

22. One angle is 18o less than twice its complement. Find the measure of the angle.

rt= ?C¡ß ¡l=zClo_,+)-t! J¡l=lbzh* ê=1o * = ¡gO-?¿1 -lt

2l+2d =

t6,!11 , zE

(l0y + 12)"

Make a valid conclusion (if possible) for each set of statements using the Law of Detachment or the Lawof Syllogism.

23. If Kirby climbs the tree, then he will fall.If Kirby falls, then he will break his foot.

conclusiont Í+ F:cby e-ln¡bs +1,- {rce t +h* h..

24. If Copper is a hound dog, then he will howl when he finds what he's been tracking.Copper howled when he found what he was tracking.

conclusiont tlo v*lil c,,otclrs¿-,

25. If two angles are adjacent, then they are not vertical angles.If two angles are not vertical angles, then they are a linear pair.

conclusion | 4 lwo or6(e, qf,€- -l¡.*tt, fh*. -lÃ"Y

26. If the basset hound parade is canceled, Mr, Leman will be devastated.The basset hound parade is canceled.

conclusiont ,yl, . Lerurn"^

Determine if each statement is true or false.

F 27. Perpendicular lines always intersect to form four right angles.

T 28. Deductive reasoning uses patterns and observations to make conjectures.

F 29. The inverse of "If you do not vote, then you cannot complain about the elected" is "lf you vote,then you can complain about the elected."

@'oIftheinverseofaconditionalstatementistrue,thentheconVeISeisalsotrue'

@ F 31. A biconditional statement is true if both the conditional statement and converse are true.

T @ tr. If two planes intersect, then they intersect at exactly one point.

F 33. Vertical angles are congruent.

T @ ,0. Vertical angles are never supplementary.

witl årc-þ hi, {"+'

vtll åc Jertså.+rJ.

T 35. A biconditional statement is false only if both the conditional statement and converse are false.

l,'n..r/e,rî.

Lesson 4: Proving Statements About Segments

Theorem - a statement that follows as a result of other true statements.*MUST BE PROVEN

Two - Column Proof:

GIVEN:PROVE:

Tyre

Statements

1. PQ=XY

2, PQ= XY

3. XY= PQ

4. fr=-Pl

Symmetric Property of Congruence: If 7B = CD, then Ø =78

X

Property of Congruence: Fo AB = AB

e Property of Congruence: If = EF, then

Reasons

Y

Paragraph Proof: (of Symmetric Property of Congruence)

You are given that-PQ=-XY. By the definition of congruent segments, PQ= XY. By the symmetric property of equality, XY= PQ. Therefore, by the defînitionof congruent segments, it follows that -XY=-PO

AB= Tp

Ex 1) GIVEN: LK== 5¿K - 5, JK=JLPROVE: LK= JL

Statements

l. LK:52. JK: 5

LK- JK

R=-JK

Reasons

Ex 2) GIVEN: Q is the midpoint of PRPRovE : PO=! Pnand OR -! pn-2-2

l. Q is the midpoint of Æ2. PQ-QR

Statements

PQ+QR- PR

PQ+ PQ= PR

2.PQ- pp

Po-!pn?2

oR-!pne2

Reasons

Ex 3) GIVEN:PROVE:

Statements

VW=WX,WX=

l. W=Xy,VW=WX,WX=yZ

2. W=TZ3. UV=XYantVW-YZ

4. UV+W=XY+YZ

5. UV+W=UWXY+YZ= XZ

6. UW= XZ

7. UW=-XZ

Z

Complete the following proofs.

J. Given: B is the midpoint of AC

BC=DC

Prove: AB = DC

Statements

Lesson 4 Practice: Proving Statements about Segments

1. B is the midpoint of AC

2. AB=BC

3. BC=DC

4. AB=DC

2. Given; AB = AE

BC=ED

Prove: AC = AD

Statements

A

Reasons

L

2.

aJ.

4.

@ùvcrr

P¿4¡n¡|'orn

ê iv¿ltfo^rsl l¡,,,.

A

AB=AEBC=EDAB:AEBC: ED

AB+BC:AE+EDAB+BC:ACAE + ED: ADAC: ADAC=AD

of /a iJpornt

3. 'Write a paragraph proof on a separate sheet of paper.

Given: nZPMN = nZRBCProve: wZ.ABR+ wZPMN = nZABC

1' G vr'".,,

ttu2. D¿{r"i ffo^ o+ =

¡),{'lthonSêgnt..rr* AJÅt{¡^ þ"+uln+'

5rbá+''þ+,'^p¿{¡ntlior, c+ 3

Tell which property is illustrated by each of the following.a. If 3x + 7 :4},then 3x:33. tb{ß-¡-{dø^b. If 6x: 30, then x: 5. DtUtsiorn

c.

d.

e.

f.

If 4(2x+ 1) : 20,then8x + 4:20. O;striL" {i,¡.If 3x + 2y : 24and y - * a 3, then 3x + 2(x+ 3)= 24. S¿bstilo{,o^

If AB + BC: AC, then AB : AC - BC. 5.¡b*c"¡.{donIf mZA + mlB: 90o and mZB: 40o, thenmZ|+ 40o : 90o. 5¿bs*ì-[u*lon

o

h.

i.

If 45 : 3x'r I2,then 3x + 12: 45. Syrnnrtt.þtc

If a: b and b = c, then a: c. fce¡r,6j,k,/eIf 4x + 2x + l0 = 16, then 6x + l0: 16. S"\gllþhonmZX: mZX. R.,Çt.xfu.J

termine if each statement is true or false.

F a. If a statement is true, its negation is false.

@ r b. The converse of p + q is q -ì p.

T @ c. If a conditional statement is true, its converse is also true.

r O ¿. Theinverseofp+qispì-q.

@ n e. The converse of a conditional statement and the inverse of the conditional statement mean

@err@e

the same thing.

Inductive reasoning uses patterns and observations to make conclusions.

Inductive reasoning is what we use in proofs.

Find the midpoint of the segment from (-4, 5) to (6, -9).

-t'l + u t+-'lTITFind the distance between the points (-4, 5) and (6, 9).

(t )-z)

Reflexive:

Symmetric:

Transitive:

Lesson 5: Proving Statements about Angles

For any angle A, ZA = ZA.

If ZA= lB,then ZB = ZA.

lf ZA= ZB and lB = ZC,then lA= ZC.

Given:

Prove:

ll is a right angleZ2 is a right angle

Z7=22

Statements

Z1 is a right angleZ2 is a right angle

mZl:90"mZ2 = 90o

3. mZl: mZ2

4. Zl=22

Reasons

l. Given Where does statement 2 come from?

V/e know that all right angles measure90o because that is how we defïne andidentifr a right angle.

?k?k?kNow that\rye have proven this theorem true, we can use it as a REASON inother proofs.***

Where does statement 3 come from?

We need to show that angle I andangle 2 have equal measures so thatwe can say they are congruent andprove statement 4.

V/hy isn't it substitution? Substitutiononly replaces one item at a time, forexample 2x + 3x can be replaced by5x. But the transitive property statesthat two items equal the same thing,for example mzl : 90o and mZ2: 90o, so mll : mZ2

Given:

Prove:

Zl and 22 are supplementsZ3 and 24 are supplementsZ1=2322= 14

Statements

ll and 22 are supplements23 and 24 are supplementsZl=23

mZl+mZ2=180"mZ3+mZ4=180"

mZl+mZ2=mZ3+mZ4

m/.l = mZ3

mZZ: mZ4

22= 24

Theorem: If two angles are congruent, then their complements are congruent.

If two angles are congruent, then their supplements are congruent.

Reasons

J.

5.

Postulate: If two angles form a linear pair, then they are supplementary.

Given:

Prove:

Zl and Z2 are vertical angles

Zl=22

Statements

l. Zl and 22 are vertical angles

2. Zl and Z3 are a linear pairZ2 and Z3 are a linear pair

3, Zl and 23 are supplementary22 and 23 arc supplementary

23=23

Zl=22

Given:

Prove:

Z3 and 14 arc vertical angles

23=24

Justify each of the following statements with a definition, postulate, or theorem.

Z2= 13 A

If BelD€, then ZABC is a right angle.

lf mZ2 = 40o then mZ4 = l40o

Given:

Prove:

mll + mZ2:90"22 and 13 are complementary angles

Zl=23

mZl + .12:90o

Zl and Z2 arecomplementary angles

12 and 23 arecomplementary angles

Z2 = .12

Z.l = Z3

1)Given:

Prove:

Zl is a right angleZ2 is a right angle

Zl=22

2)

Given:

Prove:

ll and 22 are supplements23 and Z4 are supplementsZI=23Z2= 24

3)Given:

Prove:

Zl and 22 are vertical angles

Zl = .12

Complete the following proofs:

1. Given: ll is aright angle

22 is a right angle

Prove: 23 = Z4

Statements

Lesson 5 Practice: Proving Statements About Angles

1. ZI=23

2. Zl is a right angle22 is a right angle

3. Zl=/24. 22= 23

5. 22= 24

6. Z3=24

2. Given: ZI = 2.4

Prove: 12 = Z3

y'¿c*ra.l a^1glcs a,r( e'n¡',o*l

G v.-J.

4.

5.

6.

4tl

frarrusflPr-

fiSh* o.45tcs a,{s cong ru€l\+

Z.l and Z2 are a linear pair23 and 24 are a linear pair

Zl and 22 are supplementary23 and Z4 are supplementary

2.7 = 2.4

Z2= 23

Statements

Ltnc.r fuc

a.re- sy/eøurh"y

14t.,:r

crr1î)v¡

3. Given: 22= 23

Prove: Zl = 14

l. Lz !¿32. LI ? LL3. Lt 9t3¿1. Ls ?ç'l5 ¿ tÏt4

Statements

f.

2.3.q.

5.

G,Yeøy'ec1t'cn-l annS les

Tt¿r¡sìttteVer*',fa( a*5les

T-f¿rrç;lrV<

5' z I = tt /*+îr-ul a,^Jlcs ^æ *"rm^l15 and 26 arc ì4 oI Lß^YIf Z7 and ZB are complementary, thenmZ.

+If AC bisects ZDAB,then Z7 = 2.8.

D.{v,,'.fù^'t't't t'4t' ålkccfDE + EC: DC Í*.gr*** e,t/;lm ft'þ kt

10. If 25 isa right angle, thenmZS:90o. D.hÅV, ,( fìþkl *C.

I t. If Z7 = Z2 and Z2 = 24, then Z7 = 24. TTa,nSthVe

EC

12, If BE bisects Ñ, then E is the midpoint of OC.

D.{,v,r'l¡^ r( 6¡"'-fIf ml7 + mZ\:90o andmlS = 50o,thenmZT + 50o:90o. S¿bsli{"h,øl

13.

14. 22= z2 {.{t.XN"15. If DE + EC : DC, then DE : DC - EC. S , L{cac}finn

16. mz7 +mzB:mZDAB 4"Sl- Nl,*o", PotJuL{-

17. If 3(x - 4) + 10 = 16, then 3x - 12 + l0 : 16. 0,'s{ri LrltVc

18. rf 7x - 2 : 26,then 7x : 28. Atl[ 1+o"

lg. If 3x + 4x+ 12:20, then 7x+ t2:20. SobSlî4¿Ilont

20. rf x+ 4

21. If 3x + 5y :24andy : xt 2,then 3x+ 5(x + 2) : 24. SubE4il',rlio^

* 1 : 5, then 2x + 8 + 1 = 30. yyttlltglic^{do'^

22. If AB : CD, then CD: AB. Syfrn.r,e-{n'c-

23. If 6x: 42,thenx:7.

24. 24=3(x-5)+2x, then3(x-5) r2x:24. 9l r'*.¿lft"

True or False.

zs.@ p

26. T

27.T@28. T

Dv,'sø'"

The contrapositive of p* q is -q*-p.The negation of p is -p* -9.

Deductive reasoning uses patterns and observations to make conjectures.

Vertical angles are never complementary.

For each of the following determine a valid conclusion, if possible, using either the Law of Detachment orthe Law of Syllogism.

29. If two angles are supplementary, then their sum is 180o.mZl+mZ2=180"

Conclusion: ¡1/¿ Vnlü

30. If two points are collinear, then they are also coplanar.If two points are coplanar, then they must lie in the same plane.

Conclusion t TF Jv,, Vot¡fs 4rq col[îneo,^î, flr1 +h./ fli'ùsJ

li¿ )n tÄt sþ*tl< 1lo.,*e-'

31. If a biconditional statement is false, then either the conditional statement or its converse is false.The biconditional statement is false.

conclusion' 7-h en f+r<- c!"ott'Í¡ùr,vl qt unt)efSe ills l- b'

øac.Lsøn

32. If two angles are vertical angles, then they are congruent,If two angles congruent, then they must have the same measure.

Conclusion t ç+ f-rt- angles a. cÊ yerli<,n/ o,vt1le5 ,Irr^- +1^t 6wte t:nea'svfe-'

33. If the measure if an angle is more than 90o, then it is obtuse.mZA = 60o.

Conclusion:

34. V/rite the iÊthen form, converse, inverse, and contrapositive of the statement:

All right angles are congruent.

rf-then: T+ ø a*g(z È c- rf¡hl a,,rgle ) +h^ i+ is .cø'tt r';-^{ *a,tl cJghf a',ryþs ,

converse: Jr4 4r^ .45fe øv+gles r 'l+'^ J+ isInverse: L

'.h*^ il- '\ "ol co'.Jîv?Af

contrapo ' o^ a''tJlc

ft q,ll nlhf ^nict. ,

4 ^,t\ ar.glc ,'s ,ú cþ^ôpe*\; +-. ltt r'þht o"vylcs ' +âe'n'

¡+ ìs not -; r-''ðht a".q,le-'

Aryl. 4 i s not oLlw.._ '

(*f". .

4+tA^^ J-l,r*¡ ,utusf

Chapter 2 Test Review

State the negation of each statement.

a. vertical angles are congruent. l/r¡Iùc:';l a,r15les q,ce ^t %Nr.,n+'

b. Monday will not be my birthday. /n"^4 v¡ ú L î^y L;r*l^Jy,

Write the following statements in "If...then" form.

a. Geometry students learn to think. f+ G.^^t-hy s+v"L^l"

b. Congruent angles have the same measures. É+ a^lgles ace *tyunr*,¡ fJ*.^ +4"yÅÀ,c- '++v- 5úe. rrrzqsules.

For the statement "If angles form a linear pair, then the angles are supplementary" write the

a' converse: t( aa6bs q.re s,:pler'rtrnlocy ) #sn *h.Y 0" c\ "åW

b' Inverse: ç1 6a6s d" nol, &* q. l,tnt r Pltç, lhen ty't.-ø6lns d/æ- Ao* 5'7yle*rc.n1^fV.

c' contrapostive: T,ç otgles q. rg not suyb*r"n{uy ¡ fiø'n /' not

T

ôoGo

F

F

F

F

F

F

F

F

{oC,nn 4. livreor F4}C.4. If a conditional statement is true, its converse if false.

5. If a conditional statement is true, its contrapositive is true.

6. If the inverse of a conditional statement is false, then the converse is also false.

7 . If a statement is true, its negation is false.

8. If two angles are congruent, their supplements are congruent.

9. If two angles form a linear pair, they are supplementary.

10. All right angles are congruent.

11. If two lines form a right angle, then they are perpendicular.

12. Vertical angles are congruent.

/ccrìr 1+\,ø lO*rl{

Tell which property, postulate, or definition is illustrated in each of the following:

13. If 3x - 4:14, then 3x: 18,

14. zA= zA {*01*vu15. If a+ 3b: l6 and b: at 4, then a+ 3(a+ 4):16.

16.

17.

18.

19.

20.

21.

rf 2x+ 5 -7x:25,then-5x + 5 :2s. áqffi*3* ,Ls+ilù{io"tIf 24 :3x * 6, then 3x + 6 :24.

If 4(x + 2):40, then 4x + 8 :40.

If ZA = ZB, then mZA: m.lB.

If x: y and y:12, then x : 12.

If *-l:2.thenx-l:lo.5

AColìIøv',

22. If 6x : 48, then x : 8.

S,¡wwac*n'c.

Subs{ìhho"

ptrfti Lullve

23. Given: 37 :2(3x - 4) + 3x

Statements

D.fini{rta o+ b'Iw'^)Trolslþe

I

2

J

4

5

6

/n"lh7lte-þo^

D,'vl slovt

37 :2(3x - 4) + 3x

37:6x-8+3x

37:9x-845:9x

5:xx:5

Prove: x: 5

Reasons

1

2

J

4

5

6

6NenDisfr¡'br{'v"Suislilutlo.".þl'li[r"n

24. Given: nD r ocBCIDCZl=22

Prove: .13 = 24

l. anrrcBC I- DC

Statements

ZADC is a right angle

ZBCD is a right angle

nZADC:90onZBCD: 90o

mZ3 + mZl: mZADC

mZ4+mZ2=nZBCD

mZ3 + mZI :90o

ml4 + mZ2: 90"

23 and Z-l are complementary

24 and Z2 are complementary

Zl=1223=24

r' Q ueø

2. D.4rni{rb^ o+ P"+dic"k

7.

8.

D.{rn i{ran o1

4. A.6t. hloli'l¡o't Øt'l"l*4'

5., bsJ', J.,, {¿on

nlgh* o'r'ylc

O.{+ni{rrr, o( e}'vt1t',',n1tu1

o1é

G Nen

T+

Trar¡sllv"

z4fur^

a*gbs++.-l'fa/L u$6vev\+ '

25. Given: AE = BE

Prove: AC = BD

EC=ED

AE=BEEC=ED

AE: BE

EC: ED

AE+EC:BE+ED

AE+EC:ACBE+ED:BD

AC: BD

AC=BD

Statements

l. GVo'"

2. 0.{¡n i{'o"t o€ tu\¡r-"e,.^+

fublÌ{4'",3e0,i^-,r\+ N¿I +¡* P' tþl^,L-.

5.¡bc'f¡'fr#^Du-$n, {,u,n n+