Geometry - Unit 3 Torgets & Info Nome:This Unit's theme - Pcrollel Lines and TronsversolsApproximotely October 17 - November 2Use this sheet as o guide throughout the chopter to see if you are getting the right informationin reaching eoch target listed.

By the end of Unit 3, you should knowhow to...

Identify ond use correct vocobulory:Corresponding angles, olternote inferiorongles, olternote exlerior ongles,consecufive interior ongles, verticol ongles,lineor poir, tronsversal, porollel,perpendiculor, slope, y-intercept

Use ongle relationships to find themeosures of ongles in o diogrom

State if lines orestotement with o

Torget foundin...

Find the slope of a line given a groph, twopoints, or the equotion of o line

Chapter

Write the equotion of o line given:o) two pointsb) o point on the line ond the slopec) o point on the line and the equotion

of o porallel or perpendicular line

porollel and justify yourpostulate or lheorem

Did Ireoch thetarqet?

Complete a two column proof by providingredsons that justify each given slotement

Chopter 3

Seclion 2,poges 89-95

DIAGRAMS &EXAMPLES!

Complete o blank two column proof usinggiven informotion ond o diogram.

Chopler 2Section 3,Þdqes 98-IO4

t** You will be ollowed to use a sheet with oll theorems/postulotes from the unit on thetest. You do not need to memorize lhe theorems. t**

Chopter 3Section 5,poqes 113-119ChopterSections

3

s&6

Chopter 3Section 4 pogesto6-ILz

Lesson 1: Lines and Angles

parallel lines: lines that are coplanar and do not intersect

skew lines: lines that are not coplanar

parallel planes: planes that do not intersect

Parallel Postulate

If there is a line and a point not on the line, then there is exactly one line through thepoint parallel to the given line,

Perpendicular Postulate

If there is a line and a point not on the line, then there is exactly one line through thepoint perpendicular to the given line.

transversal: a line that intersects two or more coplanar lines at different points

Im

corresponding anglesZI and 25Z2 and 2623 and Z7Z-4 and Z8

alternate interior anglesZ3 and 26Z4 and Z5

alternate exterior anglesll and Z822 and Z7

consecutive (same-side) interior angles23 and Z5Z4 and Z6

Name apair of corresponding angles.

Name a pair of alternate interior angles.

Name apair of consecutive interior angles.

Name a pair of alternate exterior angles.

Tell which kind of angles each of the following are.

./.1 and.l3

Z.I and 22

Zl and 26

Zl and Z8

23 and Zll22 and 16

22 and l725 and Zll

Postulate

If two parallel lines are cut by a transversal, then the corresponding angles are congruent.

Given: I ll m

Prove: Z2=.13

If two parallel lines are cut by a transversal, then the alternate interior angles are congruent.

Given: I ll m

Prove: ZI = 23

If two parallel lines are cut by a transversal, then the alternate exterior angles are congruent,

Given: I ll mProve: Z2 and 13 are supplementary

If two parallel lines are cut by a transversal, then the consecutive interior angles aresupplementary,

Given:

Prove: t Lm

tllmI II

If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other

I

m

Complete the following proof:

1. Given: o ll b

tllmProve: Zl = 23

Statements

Lesson I Practice: Lines and Angles

1. ollbtllm

2. Z7=22

3. 22= 23

4. Zl=23

a

b

2, Given: r ll sProve: Zl and 23 arc supplementary

Statements

I

Reasons

r' 6.ive¡

].

2.

J.

4.

rll s

22= Z3

Zl and Z2 are a linear pair

ZI and 22 are supplementary

mll + mZ2:780o

mZ2: mZ3

mZl + mZ3:180"

Zl and Z3 are supplementary

:tr+ ¡r.llcl li¡rc¡ 4rt c-ul

#n"n øcæsPaoltg tls 4n¿

Í1 p,a.llcl llrt+{¡v,rr o,t{cr^a*c

fävrsi{,vc

5.

6.

7,

8.

ert cul [YMfcnor tJs

üy Â-

6 vcn

Et paa ltr/ ltaes etr' c'{ bY G

*rar'r..rersrl ¡ fii,cn cltws¡ordn5 ¿ls a'¡r- ã

OeÊ'¡r,*t,n of lineour Pcrr

l^cte ptif: aft- SvPleuucnkl

пI¡rtll¡on oP s'tPpl"*rchry

Icli¡¡ttort o{ cil! t"ctt*

tnrrrrcnn.I,

L *n t¡,æn*t¡q,f!- =

rs

svbs$-lY#on

Dcf,nl llon o( suppl, ¡n4

Page 153, #7 -10, 12-20, 22-39

Solve for each variable.

Lesson 2: Using Parallel Theorems

X: 75

J.

2.

X:

z:7010

": llo v:

v:

5.

Sytî +76 =lto3Y -- toz

Y=sls1

¿(o Fx: \)

X= 7Ò

lz=goILK = tno

LY +lo = ç16

v: Go

+2Y) .bo

X:

( x+tV = zo)'-s3*+zf

/Ò

- 3u -ltY = -Llo3r +2Y = oô

- tOY = -l.f¡-ye/€

Hint: Draw a third parallel line

70

9.

X:

\+ 4t =fz

LI L

X:

ll.

8o

10.

c

-|

x: LIO

E

D

BE bisects IABD

x: 5(t

1a

fY - '{o27 - t4o

12.

v:

x: 10 V: bÒ

Given: AS II

Zl=Prove: 23 = 2.4

S{a-tstænfs

BT

22

t Fs ilF2-. tl : ¿1

g. t?- ! t3

tLl

1.

Given:

?tz

L ¡ 7t3

:"'

S

peasorrs

BC

->BC

Prove: /.1 and /3 are supplements

t. Gvr¡z. T1 pcrr.llc I lt^,, .r¿ eu* LY ^'

+nrr.r*rr.'l , 1&^ c,næsPm d'\ ¿ls qt¿ Ë

3' f4 paøtle-l llvtes ^',Ir- ct bY o-(tzna¡e¡s*l , 14r.x a;l{rcmn[c- ¡r/r*r-tìor

t-f3 a'CrL :

{. -Tnns,*i¿t ^

II DF

bisects ZABE

ft Dr

2

3.

bisecf5

!¿z"^^Å L3

LItZ

q. ntLwttS c llÞf. nu I --¡,A¿L(t' t"'rtl+14L3 --l&)i t I oÅ ¿l o* s"PPb'nu'fs

L*B€

"n ryplu'ult

l. 6¡¡rn

psasra s

Ð.{u,'lb", o( åiso.-/t

* I l;¡es 4¡c c'''l + 4'tl¡en Consauk,rc ¡alcn'of Ul

0.(taÅiort o( s u¡/cfie"ltcY

t.$n'lcon d( gsubshlfuhon1c(aì{¡on oi supplrnnt"/

3.

,1.

fO.-?

l, Given: gE ll CoZ2= 13

Prove: Zl = 14

Lesson 2 Practice: Using Parallel Theorems

A

1.

2.

J.

4.

5.

6.

Zl and Z2 are supplementary

23 and Z4 are a linear pair

Z3 and 24 are supplementary

22= Z3

Zl=24

2. Given: OC ll AU

l.

aL.

3.

4.

5.

6.

G rVørr

l+ t( fthes e',rì¿ cÜ+ [y ^' *r¡vr¡ærsc'l'

å¿¡^ c,ortrt¿u+\'e- rh*c dor tl q'll' *il\'n""lt¿.{ryriil"n "-e lincor pnlfLiyrecr â¡çs erê soPPlemcnhcY

6:\lgn

AK bisects IDAB

Prove: Zl = 22

Statements

+

Dc ll fro--Ð

t

hÉ åricctst+ a,tL = L5

L37 Ll

t+ lvo 'n'5les c'rs 3 ' fu

SúpPlettu*is oút- i

23.

¿{. tli tt

LDABl. Gve"r

2- D.f'xll,on o1

, Tî ( l¡nes

1+r.^" ^l{c.rno-te

q.ln¡r,¡fiìln-

fâet(

5¿5ects

a,re- cn* !Y ^f'^+e c/or L';

*r*rr,.c"*l ,-1,a.fs =

Solve for each variable:

3. x: -7o v: 3o 4, *: 1A ,:

ml7: I LO

mZ3: bO

mZ5: I LO

?o

mZ2:

mZ4:

m./.6:

lollol7l

7. *: 3Ç y: Lll

L)oc í5t

6.

v:

{x +S = 3x+llx=b3.( +l = lll

3y o t5r>

b5c>

x+14=tox-- 3Ç

(9x+12)"

8. x:

ôV + t3V-to -- lE6

llY = ll6Y =lo

îx+l'L=12Ò1x = lo8

fr, = lL

145'

-fo110'

- c

x: l0ç

I l. a:8O 6=

": l2l a:

10. x:z:

| #{= toY=1r

,l{3l v:

/6öll

Llf

13, x: l'l

3y =71

12. x: Gl

3r+E = so31,'-'lL

t4.x: lÖ ,: I-?x -- -70

¡alô

If two parallel lines are cut by a transversal, then the corresponding angles are congruent.

State the converse.

Lesson 3: Proving Lines are Parallel

'1"ß'lçAlso a Postulate* * *

Given the following information, what can you conclude?

Given: 22 = 13Prcve: I ll m

If two lines are cut by a transversal so that the alternate interior angles are congruent, then thelines are parallel,

Zl=22

If two lines are cut by a transversal so that the alternate exterior angles are congruent, then thelines are parallel.

Given: Zl = 13What can you prove?

If two lines are cut by a transversal so that the consecutive interior angles are supplementary,then the lines are parallel.

Given: 22 and 23 are supplementary

What can you prove?

Given:

Prove:

iilkkll I

i ll I

If two lines are parallel to the same line, then they are parallel to each other.

j

k

I

In a plane if two lines are perpendicular to the same line, then they are parallel to each other,

Given: m a pnap

What can you prove?

SUMMARY

Name 6 ways to prove lines are parallel,

r' g\^) crtt'¿qoo^//t5 L1 ^cs Y

2. g\^, Êlþra¡

mn

3' SLww

4. flt\r'/

s.É6. f+

4{{arvì^-L

uwrsr4t,._ f,^+,4ro- ¿ls qß_ sv|plenc.fury

z

Which lines, if any, can be proved parallel from the given information? (TEST QUESTION)

fmø

lM.s

L Zt = Ze sllü2. z5=tt0 tltf3. Z7 = ZIt ¡)o¡. tt

4. 212 = 214 ttl f5, z6 = ze sttË6. sll tandsll u tilv7. z2=zt2 Sttü8. mZl3+mll4:180o tltf9. slwandul-w Slly

û.rL l(

arN- L+o J\. .fq'n c l,¡c

il. Sa*+t¿ ltn.

10.

11.

12.

13.

14.

15.

t2 = Z4 ¡/n- ll

t2 = 23 q-tt ft3 = tt4 6[ltmZ5 + mZ6 +mZ8 : l80o S tl tt3 = Zr2 ¡/anc- [l27 and Zl7 arc supplementu.y Sllt

Given:

Prove:

Zl=2213=24

A-B ll -D

LIL3

St^*.ø ,rrts

2.

L2^L'(

LL 7 t3

3. Lt 7 t4

"((

4 ED

Pns,rs6 ¡w".

Vql¡ ^lT¡ans'4¡v'

lJs oJÊ"

4AIY-

¡-ltÆor L/5

f ix.s a'FL I I

l. Given: .lO tt fNZl=12Z3=24

Lesson 3 Practice: Proving Lines are Parallel

JKA

Prove: KO ll AN

Statements

t.

)

3,

4.

5.

6.

7.

Jo ll KN

Zl=23Zl=22

22= 13

23=24

Z2=24

Reasons

Ko ll AN

2. Given: Zl = 22

Prove: 23 = 24

5

6

7

frwrsiliv.-l+ crçß?onlrg '",,4r{-

-¡-fa'r- l¡ncs ø\Îe- ' l '

L ? tt t

Statements

LI 7LL

3. L3 ?¿1

+ràr^s *rrJ,

Reasons

G ¡.r.rn

î+ftrcn

z,

*lt rt

a-t+.c'^^iL ùt-lcc,'or /t q'la ?'/

freÌl*",

linc a,cc- ll .

ll l¡Ytes a'rL

a-t{ernaþ

.u* by

Clc+c rtor

a-

/ç

frarls.l rs^l,..lJa.,(g =

Page 160-163 , #7-10,12-29,32,34, 54-57

7{

-zx---11,

X: 8o

Slope:

Lesson 4:

citeL-- -rv^

Find the slope of the line passing through points (3, 5) and (-2,l).

Parallel and Perpendicular Lines and Slope (Atgebra Review)

V¿_yl __ lvX¿ -xr

Find the slope of the given line.

:î-l n-J3--L =L? I

Slope-Intercept Form:

Find the slope of the following lines:

l) y = 3x+2

tl =M x+b

4) )=-5

2))y=--x- |"5

_2rvLà I

s)@l,'14f rü

3) 3x-Zy=-$-Zf = -3x-L

V ã lx*S

Paraltet Lines: $auvrt S fg c

Perpendicurar Lines , SltoTCS a,fc- O??OS;h- rc./p Cæ¡-/S

Are the following lines parallel, perpendicular, or neither?

t) !=3x+2 y=3x-6 p^r^llrlm t 3 m=9

2)Iv=-x-5'2

$ ,L!=2fv|--O

3)

4) the line through (-2,6)and (8, l) ìA = Ithe line through (4,3) and G,Z) -¿'t

r,l.= 1=Ll- (,

!=2x+3m'- L

x=9

Find the equation of the given lines.

l) m : 2, through the point (-2,5)

m--urrh{r*l

il¿ihh.r

2) vertical line through (0, 9)

f = 2r,+1

?ufr,^*'Lr

3) passes through (-2,7) and (3, -3)

f,Y

€- -'¿lO -I

-tL

4) passes through (5 , 2) and is parallel to y - 2x -t I* Y t¡rL

-!z

5) passes through (- 1, 3) and is perpendicular to 2x * 3y = |

1--t 4 =-zl4= fr 2-s

y= f,x++

h¡:st?.

tr\2L

3 *8

3Y = -tx#l\='â**å,n ---þ

u = 2*-+B; . z!s'l +B2s lO+B-tÇ B

t^*2\=â*+g-3 -- *?t) +g3e -p+ß+ --ß

Lesson 4 Practice: Parallel and Perpendicular Lines and Slopes

L Find the slope of each of the following lines:

slope =

-ar^, L

-l

2

2b. slone: *----------T-

Find t e slop of th line through the following points:

a) (0. 4) and (2, -3) b)Lt_ -s . -7o- ' l

c) (-4,3) and (2, -l) t-71#-*=l-11Find the slope of the following lines:3

a) y:5x-l

d. slope: unrle+M¿J

slope =

c) v:3

slope: O

(5,2) and (1,2)? -L ôt-l - Ll

d) (3, l) and (3, -2)

¡--L3-3

b) 5x-2y:6 Ssiope: -L

d) lx-3y=5'2

slope:

2Ò

-Lþ

-ZV =./=

-sy =

t=

-lr+b$x.+ -3

- 4,* rEf* -l

Use the slopes of the following lines to determine if the following lines are parallel, perpendicular, orneither. EXPLAIN WHY.a) y:4x- 7

fr\.4b) 3x-2y: g

,^" = 4.c) x:3

rA=

v =!x+2"4rt^ --.rr

31v=-x --'22ü=2

y:.2ln =o

d) the line through (2, 5) and (-1, -1) ¿,r ,the line through (l, -3) and (3, -4)

tA=

,rrl-r{n J

Find the equation of the line following lines.

slope : 2, through the point (3, -5)

:;!?{u*-l=:;fr ,-î=-.--1

a)

N¿'4L'r

P"ÅM - 9"¡^'c- sW'

through the points (-1, 4) and (1,7) d) slope : 0 and the y-intercept : 5

Q*fúulff- sbses Jff;t",'

e) through the point (3, -2) and parallel to 4x - y : 6

t=/!el-

through the point (-1, 5) and perp.endicular to y: 3x-2

b) vertical line through (4, -1)

Puprr'Åic¿lnF

t*2+8'* le

y---!(++ I ^'t

F h)

ç=t+B| =s

+8\ +B+ß

Chapter 3 Test ReviewComplete the following proofs.

l. Given: x ll yqllr

+Prove: Z1cZ4 af,L

s"pptør'udt

Statements

xf(v r tl{rtl î¿3

L 3 "r^L L L( are-Svp/ev*onls

ML3 imt( = lfot L l= nL3fiLl *rq¿{ --t1O

Ll ,*rL ¿ I q/ß suPlc*uttb

Given: m ll n

Prove: Zl and 24 are supplementary

Statements

z.

tf.

f.lt.

1.2.

f . 6vtn2-. T+ tl lìrtes â.f,s ¿¿* b{ â {'rswul'

thc.,n a-t*ern^jr- et{t'rlor ts o'cg ã

3. ç( lt lt'^?: 4re cvl bY À *nt':*i*,i$e-n c¡ns¡"v+íJc fvrþrtoc /t a¡c s'\plc

¡{.}ç{tn¡#on '4 suP/eøøh'l5, 0c{mJ{iov't J t

"nlo¡/\m

l. nnlln

2. Ll "^.1

3. ¡-z ? t3

{.

t.(t.7.t.1.

L3!úLZ !u4

L L af,? s"ppl*rrnlt

t.L

3.

6 i v",,'..

l-Merr faffs are svfPlut-l^rY

Tf f t line¡ q¡t a* bY À' *X,.-. t l¿ añL

=

\,t.u.1.a.1.

*'l^ø¡r CocASf n"t'rf5 LtS.V

VcrJÌ cn( L/s a{É =Tn tsil,vt

^D"f¡n,'fton of

3:*1'H,.iD¿{¡n;4¡wt ët

â.o^.î

*artsrærs"t,

supplcuu^1*Y

Given: m ll n, Zl= 22

Prove: n ll p

l. vrtltn )

2. n4 llPs. /ì llp

Statements

Lt ? tz f.

2,

3.

Ç, vcn

T1 ,.|-l"c,q{g

Given: ll and Z5 are supplementary, 23= 15

Prove: n ll p

ftll.e,n

T+

¿cnr+c s¡'[errbr L,'3,

I trtes a,ft- Ttvllcl

I

2

Statements

L3 ! tS

^tIP

2 I tttcs Â,îr- para.llclt,lnc 1,,,r \ ) #^ thos'

paca-ttc\

m

n

p

Lf a. Ce Z,

{"I ¡rtcs

I

2

Q,vcn

T+ At+<¡v,iß- ex+en'ol ¿is

+'ù\!,v\ lMes a'rc- P*czllel

fltc4l¿

oúc

7. Given:

Prove:

Z3=24

ll=12Statements

l. L s "¿1?' Filq-

v. Lly LL

f . G,vttnz. E+ o-l{.¡,to/c- ¡þrbr Lts

{.hen lùct a'rt- ll3. T1 ll littes are c'[ LY 4-

{+.å ørr1at{,,t¡ tls e'ce X

m

Given: ll = 22

Prove: 13 = 24

Statements

ü 9tzz. r^ llrt

3. tg ! ¿rl

a.fl- .,w

+ønsvefY( t

G¡,'ut

t+ a- f{.¡nn'{e

"iârn ltnc5 ¿tT€-

s.T+'lâ"r.

¡ Jen'or L3 q.rc

u.

*lbY.tis 4rc

qlt

=,

'ln,,,suers^lr4-t

Given: Zl= 22,23= 24

t.

2.Ll iLZ I Lt?Lqk ll ,,n

3. LLI +¿l s"¡¡le*ra,rLr

4. tZ + L3 3uppl¿*trh

t u|?¿l(¿.

^llp

Extra PractÍce Proofs

Qensons

z. 4 ec¿n=Jpanl\ Utafc- É 41,\4r^ tntc5o,t- ino-llel

3 . æ ll lfrtcs. .,rÊ cnl \ o. . únrs,,rrs.l,,1.

I

Given: 25 = ZIO

Prove: .12 = ./.4

rfn*cctor L'3 al! Sypitu,nal^tyl.rheer pr¡Yl âf,c Sv?plerlz.ur.l^îlEÊ 2 ¿'3 o.¡z * t +l,r¡r^ {hcûc

b . Tt a"msg"^rt,tg y's arv t,

s{c'lc'vun-ls

ì. ¿ç ? tto2. 6 UcF

É.^son s

| ' Gr\'¿fi

tlt n cøvtlccø{lvc

L. TÈ cørÍl<5P6v\Ålv\J

aft- *t +d,t" lt'nc'

supplernuls oulc-

lÀcn llncs Àr¿

d.tt- ¡ca.llel

'ur Pa¡o-ttel

þ=

It

fr'ncjtt5

,f^a^

i\\ ft

comes çonÅiry

Itrtcs q.N- c*[-lJs

byra,{€,

Given: mlln

Prove: ZI and Z2 are supplementary.

Given: allb and clld

Prove: ZI = 22

Given:

Prove:

Ã3 tt erll=2213=24

Write aparcgraphproof

Given: BC ll DF

c

d

Prove: Z7 and 23 are supplements

BC bisects ZABE