SEMESTER FINAL REVIEW: Part L: chapter L - 3.4Test expectations:L/3 - L/4 of the test points will be allotted to the sections 5.I-5.4This portion of the test will consist of vocabulary, short answer/calculation problems, andapplication problems. You will have a short section on which you may use geogebra tocompute your answers.

The Rest of the test:There will be a multiple choice and true/false sectionThere will be a short answer section.There will be a conceptual understanding section.There will be an application section.There will a section of proofs.

VOCABUTARY:

Chapter 1- Know how to identify and draw examples of highlighted terms,- Know how to define the starred terms

þ Choprer Vocobulory¡ acut€, right, obtuse, o congruent angles (p. 29)

straight angles 1p. 29) . congruent segments. adjacent angles (p. 34) (p.22). angle bisector (p. 37) . construction (p.43). collinear points, coplanar r isometric drawing (p. 5)

(P. 12) o linear pair (p. 36). complementary angles . measure of an angle(p.34) (p.28)

. net (p. 4)

. orthographic drawing(p.6)

. perpendicular bisector(p.'t4)

r perpendicular lines (p. ¿H)r point, line, plane (p. 1l)r postulate, axiom (p.'13)

. ray, oppos¡te rays (p, 12)

. segment (p. 12)

. segment bisector (p. 22)

. space (p. t2)

. supplementary angles(p.3¿)

. venex of an angle (p. 27)r vertical angles 1p. 34)

Chapter 2- Know the meaning of the highlighted terms, as you will be asked to find them in a

question.- Know how to provide examples of the starred term.

2.biconditional (p.98)Chopter Vocobulory

. conclusion (p. 89)

. conditiona¡ 1p. 89)

. conjecture (p. 83)o contrapositive (p. 91). converse (p. 91)o counterexample (p.84)

. deductive reasoning (p. t06)

. equivalent statements (p. 91)

. hypothesis (p. 89)¡ inductive reasoning (p.82). inverse 1p. 91). Law of Detachment (p. 106)o Law of Syllogism (p. 108)

o negation 1p.91). paragraph proof (p. 122). proof (p. 1 15). theorem (p. 120). truth value (p. 90). two-column proof (p. 115)

Chapter 3- Know how to identiff the highlighted terms in a diagram.- Know how to define the starred terms

?"1l:rH,Y::i:ïJ:ï. alternate ¡nterior angles (p. 1a2). auxiliary line (p. 172). corresponding angles (p. 1a2). exterior angle of a polygon (p. 173)

. flow proof (p. 158)

. parallel ¡¡¡ss (p. 140)o parallel planes (p. 1a0)o point-slope form (p. 190)r remote interior angles (p. 173)

. same-side interior angles (p. 1a2)

. skew lines (p. 140)

. slope (p, 189)o slope-intercept form (p. 190). transversal (p. lal)

. vertex angle of an isosceles tr¡angle(p.2s0)

o inscribed in (p.303). median of a triangle (p.309). midsegment of a tr¡angle (p. 285)o orthocenter of a triangle (p. 311)o point of concurrency (p. 301)

Chapter 4You will not be asked any specific questions related to definitions of the vocabulary below,though you should know the relationships that are highlighted (See chapter 4 review formore information)

J Chopter Vocobuloryr . base angles of an isosceles triangle

(p.2s0). base of an isosceles tr¡angle (p. 250). congruent polygons (p. 219)

. corollary (p. 252)

. hypotenuse (p. 258)

. legs of an isosceles triangle (p. 250)

. legs of a right triangle (p. 258)

Chapter 5- Know what the highlighted terms mean, as they will be used within questions.- Know how to define the starred terms and how to find them in a triangle

) Choprer Vocobulory¡ altitude of a triangle (p. 310). centroid of a triangle (p. 309)o circumcenter of a tr¡angle (p. 301)o circumscribed about (p.301). concurrent (p. 301)

. distance from a point to a line(p.2e4)

o equidistant (p. 292). incenter of a triangle (p, 303)o indirect proof (p. 317)r indirect reasoning (p. ¡tz)

Chapter 1 Review:

Section 1.2: Points. Lines and PlanesSkill 1: Name points,lines, and planes.Points are named with just one uppercase letter that identifies the point.Example: PointA

Lines are named using the line symbol and two points on the lineOR Lines are named using the lowercase italicized letter thatlabels the line. Example: Þf or Line L

Planes are named using 3 non-collinear points on the plane fpoints that do not form a line)OR Planes are named using the lowercase italicized letter that labels the plan. Example:Plane ACM or Plane y

Skill 2: Name segments and rays.

Segments are named using the segment symbol and two points on the segment.Ex:,4N

Rays are named using the Ray symbol and the end point of the ray and another point that ison the ray.

Ex: A+ +AC or AB

I

cI

Example:Example: Use the figure below for Exercises 1-8. Note that ffipierces the plane at t/. It is not coplanar

ø,î;y "i#ff",i1.ìff"-n in rhe ngure ññ,ñ''lo w:ce-r;; \ z.What is the intersection of ffi ¿rr¿ ffiz ?oi.'{ ¡l!ì¿¡14'liøtì * 3. Name three collinear points. A , N anô \

X'4. What are two other ways to name plane V? Pfôffi ptï\0

p\ene MNX5. Are points R, N, M, andXcoplanurh' \ -

6. Name two rays shown in the fieureffi , ññ

7. Name the pair of opposite rays with endpoint i/. ñì . ---às,d f$X

o

Skill 3: Find intersections between lines and planesLines intersect at a point,

Planes intersect at a line.

Skill4: Understand the difference between collinear and coplanar.Collinear means the points lie on the same line.

Coplanar can refer to both points and lines. Coplanar points are points in the same plane.Coplanar lines are lines in the same plane,

Section 1.3: Measuring Segments

Skill 1: Measure segment lengths using the ruler postulate.

Rule Postulate: Every point on a line can be paired with a real number. This makes a one-to-one correspondence between the points on the line and the real number.

Skill 2: Find the measure of segments using the segment addition postulate

Example: Use the figure at the right for Exercises 2l-29Name the intersection of each pair of planes or lines.

1. Planes ABP and BCD 'fè2. hdu,'¿ hd Qcrra{ (3. Planes ADR and, DCQ "DR'

4. Planes BCD and BCQ ßï€5. or an¿ Qn

?o\ ^li P

BA

oc

RQ

Example: Use the ruler postulate to find the length of each segment.

A B cÐ1. AB 2. ncR6 = \ -4êrl\ = l-4*rl ' 3 þC '. \4 -' ll = 5 -4-3-2-1 0 t 2 3 4 5 6

@

EXAMPLE : Use the segment addition postulate to find the length of each segment.

1. lf PQ = 20 and QR = 22, then PR =2.lf PR = 25 and PQ = L2, then QR =

PR= ?o+ &f¿ -Ð 25 = \2+QR -:>-(L - (t-

â{? = 13

PQR?ß =Prù*QG-- ?,¡ç'+x-Z= 4Z

SegmentAddition Postulate: lf three points, A, B, and C are collinear, and B is in between Aand C, then AB + BC = AC

Skill 3: Compare segment lengths and determine if they are congruent.

Segments are congruent if the have the same lengths.

Skill4: Use information regarding segments being congruent to find segmentlengths.

To find the missing values of congruent segments, given algebraic expressions to representthose segments, just set the expressions equal to each other, as congruent segments areequal.

\L(. 7Yx :I ó+to = 7Ö

Skill S:Use information regarding segment lengths.A midpoint is the halftuay point on a segment. It creates two equal halves

On a number line, the coordinates of and l4/are -7,, 5,X!

-7e -!,

z=l{rl=5

respectively. Find the lengths of the two segments. Then tell whether theyare congruent.

l-+ -+

and, ZW-\\ \"-

23. XY-, E'

;,I Y

.------:€ |

\YTy¿r.,1 I

5l25. YZ

\*> r' andxwl@'s \-a'- 5l+ -\L24. zxand rry-', Éãõï ì

\\-;1\; ts4tv\fiffK-

ÐÐ; î)& +

B

ÐE c

_f-\ .-'ftc ''t(the diagram

âb(3.bat

AD - T¡C3ò " 2l\D -

Algebra llse

+ a

3

n!¿r

Y&

40. A

of 1,.

'Z)"*?x 8

If ,,lD : 12 and AC : 4y - 36, find the value

eD= üt4 =lÞ

'Il'ren find ,4C and DC

40.IftrD:x*4+4t

DB :3x -H.l6 ,ÀC. =T)¡t t

When given an expression for both halves of the segment: Set them equal to each other

SU. = ü+("=V

Example:T is the midpoint of SU.

a. If .ST = 2x + 3 and SU=26, find the measure of TU.

TU= lã sr=\? = 2xj? > i{ a->\c<-d þt F¿)

-*Ë '=zY/; I li^¿ voLu¿ -{ x'X=f )

b. If .ST = x + 3 and TII = 4x - 6, find the measure of SU

2ç

l7 'f q

f[- =1e1 _* #1? 4x-6-Y +t6 >4[ -6>

z

* --1+ ã= 6s^Y,l?

When given the total value of the segment, and the expression for one half. Set the total totwice the given half,

Section 1.4: Measuring AnglesSkill 1: Correctly name angles.-lf the vertex of an angle is not shared with any other angle, you may name an angle by theletter of the vertex.-lf the vertex is shared, use three points on the angle to identify an angle. The center pointmust be the vertex, and the other two points must be on each side of the angle.-You may also name angles by numbers if indicated.

l KExample: z2 can also be named LKML or LLMK

M

Example: What are two other names for zI?¿ l|AL o{" L L f^T I K

1 2L

M

Skill 2: Measure angles using a protractor and identify angles.Example:Acuteangle: 0o<x<90oRightangle: x=90oObtuse angle: 90o< x < l-B0oStraight angle: x = l"B0o

When using a protractor to measure an angle,1,. Set the center hole of the base of the protractor on the vertex of the angle.2. Align the base of the protractor with the base of the angle.3. Measure the angle starting at zero of the base. Remember - you can also choose

which number to use based on whether the angle is acute or obtuse.

2L

@

-{\* ,. dìô \rt¡t Ñrr* up I

* r'ina

L. ZMLN

6=Yyy\tfrØC = Btø).3=45'

yb/ÞøL>b(e)t4=b16

¿_U0"ç'. 4s +g4 : 1q' t/

of each angle and identif,i it as acute, right, obtuse or straight

o 3. zvrp

A B

Skill 3: Use information about angles being congruent to find missing angle values.Ifangles are congruent, just set them equal to each other.

Skill 4: Use the angle addition postulate (and algebra skills) to find missing anglemeasures.Angle addition postulate:lf point B is in the interior of zAOC,lhenmzAOB * mzBOC : mLAOC

Example oc

L\?x

"11

i56F,=us

w, ¿f.(i9 = ¿(tq.r) ¡.2o : t31"

yrnlÊ-ôE = L(t4.5)* 4 = 4b"

(5x + 4)'(8x - 3)"

5

= ZþØL + /_ gùgb*^ z-r fx +4tlx+l-l

19{1 b

¡ /ÊQs 4 rvr lTû5 : tgo6X+Lo + 7x+4 \30

rðx + 24 \4CI

(6x + 20)" (2x + 4)"AIaT R

L.fim¿anD: Tg,whatarewLABC and wLDBC? \qÔo

LRQr i. " 9gq'ght_ú1".

What are mLRQS and wLTQS?

¿ frg'Þ?q

Tq-t

o

ol,a-e,lc'.13{ +4b' : llÓ" r,/

Section 1.5: Exnlorins Anqle PairsSkill 1: Identify and be able to draw all types of angle pairs: supplementary angles,complementary angles, adjacent angles, linear pai¡ and vertical angles.

Example: Draw an example of each angle pair mentioned above, J Øz Sb ce- fs r vrc\uek-qd. r¡a-þ-o-S .

¿ta.-- vc",hL-o ¿5

Name an angle or angles in the diagram described by each of the following.

a. Complementary to IBOC', /-lÓgb. Forms a linear pair with IDOE i L Dob oR ¿€o*c. Is a vertical angle to ID}E? ) ¿ B 6k

D

E c

Ba

A

Supplementaryangles &''/E/

¿\ + cL= \?6'Complementaryangles e-&

¿1+ Lz = 1.ôoAdjacent angles

Or--0.À\teo',* /g¿t bLz

I ¿

Linear pairT^

¿l +LZ =\ßO"Vertical angles ¿l ù¿L{u¿

¿1 L ¿¿\t

3 Iz

Supplementaryangles

zL and z2are supplementary¿l;+ LZ = 1B0o

Complementaryangles

z'1, and z2 are complementaryzl * z2 -900

Linear pair zI and z2 form a linear pairzl+ zZ - 1B0o

Vertical angles zI and z2 are vertical angles.LL-L2

Skill 2: Find missing angle measurements using algebra and understanding of anglearrs.

EXAMPTE

4 6

e

Supplementaryangles

¿l and ¿2 are supplementary.LL = 43o . Find the measure of¿2.

Show your work:

4?_+,¿ e.= \aö-4¿- -4?

w\LZ= ffio

Complementaryangles

zL and zZ are complementary.LL= 6x*7 andL2 : 4x * 13. Find the value ofeach angle.

ll - 41' LL'- 4\6

6x++¡{¡1*ì3 =ïÕ"[ov+zo = 1o

-?-rJ -1-t¿

JÀfÓY=?olrox--n

¿L--4{q) +t3:

4lo/1=Ql" Ltz

Show your work:L\ + ¿7=q6o Ll=

6(t\+t:4qo

Linear pair zl and z2 form a linear pair.LL =7x- 46and z2 - 3x * 6,Find the measure of each angle.

Show your work:Lt+tz -- \t6"+t(-4b ¡ãx+6=\3olOv -46 =r1ó' +4ó {¿{o --¡õ8"

Vertical angles LL and zZ are vertical angles.LI= xIl6and z2:4x+5.Find the measure of each anglett 2 .T+ tb = 11

^,lb.TA'l( ìñh=4(+).t=Sr6.ilq'

5xx> /ttx36q

Show your work:Ll z Lt-\* tu-{ -5

\\

4x+S-\ -s

EXAMPLE:Find the measures of zq and zBEC

A--

t

I

Skill 3: Use information about an angle bisector to find missing angle measurements.ExampleAngle bisectors create two congruent angles.

, Algebra In the diagram, d bísects LFGI.a. Solve forrand fnd wLFGH,b. Find mLHGLc. Find nLFGL \ .rr. \ .\ (3x - 3)'

Ð _?Iîáfç .',t Ðiíìgl,; " q iÍÏt;l -'

o\\ +\ VO" b6"

F H

(4x - 14)I

Skill4: Evaluate diagrams to assess the validity of information and to find missinginformation.

Section 1. 7: Midpoint and Distance in the Coordinate Plane.

Skill 1: Find the midpoint given two coordinates.Midpoint formula = (ry,ry)

Example: Find the midpoint of the two coordinates:

L. A(6,7), B(4,3) Z, A(-t, 5), B(2, -3)

þ# ,2#)=F,r)

9(,+'=)=U'\)

Skill 2: Find an endpoint given one of the endpoints and midpoint.Set up the midpoint formula using the given information:

(*^,y^)= (ry,'+)where(X- ,!*)arethe coordinates of the midpoint.5o... x^=ry and.y^:ry. Once you put in the values of the midpoint coordinate forx^ ønd Vm, andthe values of the given point îor xzandlz [or xr and yr), you can solve for themissing coordinate values.

Example: The coordinates of point Y are (-10, 5) given. The midpoint of Xf is (3,

-5). Find the coordinates of point XXv,,.-3=Xr+-lD 9vn.-$=3,+9

Z26=lr+-tô -lO=Ur-tS

+lo +to -S -E\6=Yr

Skill 3: Use the distance formula to find the distance between points.

Distance formula = d (x, - xr) +(vz - v)

Section 1.8: Perimeter, Circumference and Area.Skill 1; Find the perimeter of rectangles, triangles and irregular shapes.

The perimeter is the distance around a figure.To find the perimeter of a figure, just add up the values of its side lengths.

I ?= 4+4+7+7+2"þ?.+2 + 3 = 4.2 * ?.2 + 4= ß+ìg + 4. Z6(.}n'Ê51 -4 r3

4

Skill 2: Find perimeter of above shapes in the coordinate plane by counting units andusing the distance formula.

42

22

-15 - g,

Example: Find the distance between each pair of points. If necessary, tound to the nearest tenth.

L. A(6,7), B(-t,1) 2. C(5, -5), D(5, 3)

À = \ (-r-6)¿+ (1-1)¿ = ¡{=1 la (s-s)z -+ (3--5)¿ =$@ =¡

@

Rå "Þ\ î

TPl t (4-1¡r-* {-z-z¡, = {fr4-z .içç;[ , (7;,5

?=4+ãtS=lz(t-t)"+ Lz-z1z(a-t L+ ? --)

L. R(|,2),,S(1, -2), T(4, -2)=W ={G=4={?ã =t(T-3

Exam Find the

Skill3: Find the area of rectangles, triangles and other irregular shapes that can bebroken down into triangles and rectangles.

Break down irregular shapes into triangles and rectangles.Then use the area formulas to find the area of each triangle or rectangle.

Area of a rectangle = base x height = bhArea of a triangle = 1x base x height = !!

Then add up all of the areas of the triangles and rectangles to find the total area.

Example: Find the area of the shape.A, = 4 -11= AA Î-ta

Az-- g a=þ7ff=lbf+-12_

A=48+\b=

4

4ft

12 ft6 4+f

0 tt-42-B

Skill 4: Find the circumference of a circle. Remember, c= rrd = ZnrThe circumference is the distance around the outside of a circle.It is given by the formula c = nd : 2rrr, where d is the diameter of a circle and r is theradius.

Skill 5: Find the area of a circle.The area of a circle is given by the formula A= nrz where r is the radius.

Example: Find the area of a circle. Leave your answer in terms of rr.

 =T[r'A = ï [rz"¡)'ft = 156.2øfr 51vr,orc vntlS

Examp le: find the circumference of the circle. Use an estimate of z to find your answerRound to the nearest hundredth.

0=2nrC = 2{lb)rte = ãZfr Ê, iô0. bå uvrnls

\o

Section 2.4Skill r: Students should know how deductive reasoning differs from inductive reasoning.(lnductive reasoning is based on patterns and observations, where as inductive reasoningis based on premises such that ¡f the premises are true, the conclusion is guaranteed to betrue)

Skill z: Students should know the structure of the law of detachment and the chain rule.Lawof detachment: lf Athen B. Chain Rule: lf Athen B

A is true lf B then CTherefore, B is true Therefore, if A is true, then c is true

V..-

a. lt normally takes you 20 minutes to walk home from school. By walking faster one day,you make it in r5 minutes. The following day, you make it in rz minuteslou conclude that the next day, you can make it to school in ro minutes.

b. Because the brand A product costs $r.5o and the brand B product costsconclude that the brand A product is 5oo/o more expensive.

Þed\À¿h\' : -Þ'¿ tOvr û(u¿iøa ß W\O,A -6'Y1 &p'n4 '\i\Aêtfh¿vìrtohrcq

c. The united States Census Bereau collect data on the earnings of American citizens.Using data for the three years from zoor to 2oo3, the bureau concluded that thenational average median income for a four-person family was s43t 527.

-Ðrrdr,r.rrhVe, . 1*- ø,, øla¿t"^ i= bûWÅ ân d\ 3'rv1ô\\ Sna\\ lorw

reasonrng rs u to reac youror conc usronetecr CW

rue.*ib".}r¿L'u.cvh,"vt¿ cY:âe,ilY\vYô i3 Vtue),c\C6n1 t^¿ìøn *ut*+ t*tuê+ ô {1,.{- h4^Fo t\0L áoer¡ rrrtee- lÀ i, ftäuc¡,V¡uf WrVpür*+<' t4$we v1,^õ

t$ youoo,

te stu- - 66,

reasoning. (r pt each)

150-bo=-Çô

fÞlct

Wason r y ,#= 5o,/o

State the conclusions and whether the law of detachment or the chain rule can be used to drawthe conclusion. lf neither law can be used, write invalid. (r pt each)

a. lf an angle measures more than 9oo, then it is not acute.mZABC = r.2oo'lhrt$.t, uLÙW þ M? o*ruF<-

L

-\døto¿hrnorvf P-rQr. _

c5t

b. All45" angles are congruent. 7zeãza 5 *" conc\ustÕñ f6ssrT:r\e ì<-c. lf lzis acute, Then 23 is obtuse. P-+4.

lf 23 is obtuse. then Z¿ is acute. -qJ_ß_

, E¡"1'3rß tu'u+e, î+n ¿4 \5 d.crft] *

2\ Tu c¡",* \Zul¿

(Ð

Skill r: Students should know the properties of equality and congruence

Name the property of equality or congruence that justifies going from the firststatement to the second statement.

AB=AB LA= LAlî AB = õ, then CO = ¡A.lî LA = LB,lhen LB = LA,lf AB = CD and CD = nF,then ¿B = EF.lî LA = LB and LB = Lc,then LA = LC,$ LB = LAand LB = Lc,then LA = LC,

Transitive Property

Key Concept Properlies of Congruence

Reflexive PropertySymmetric ProperÇ

a.lf Z.Q = ZÇ and lE= lC b.

ID=ZE d= 4

t* øn t¿âþ=t6 3. AB=CD

CD= AB

TrÂ"ruihk(nQ

d,rviSr'anr'.,r h4vnvt oz+.n'c fLç ,

Section 2.6Skill r: Students should know how to use theorems to defend their reasoning in solving formissing angles.

Skill z: Students should know how to use theorems, postulates, properties and definitionto defend statements in two column proofs.

a b. what is the value of x7SHORÍ RESPOTSE T\ryo lines intersect tolotm Zl. ¿2,13,and ZL.Ttre measurc of13 is ¡hree times the meaCu¡e oî ll andmll = m/,2. Find all fourangle measures.Explain your reasoning.

?r óìv o- Á,U n)Stø'^J uY

'{ \.+?\ = \164x = \âóY-4SLl = 4s*

. -òL2= 4\

1l

\

?y"-b 44

¿3 = gàs\=t3s"¿¿+ = \25'"

G

a.

GIVEN: 23 and Z2 are complementary.mZl * ml2:90o

PROVE: Zl= 13Statements

l. 13 and 12 are complementary.

2. mll * ml2:90o3.m23 * nZZ:90"4. mll * ml2 : ntZ3 * ml25.mZl: mZ3

6.Zl=13

l. ZS is a right angle.nIRTS: 40o, mZRTU: 50o

2.,/gtt).: mZRTS + mtRTIJ3. mlSTLt : 40 r 6ò4. mlSTI.J: IO5. ISTU is a right angle.

6. {}s L ãf-d

l. âtf¡en--a-Z. ¡¡rü<n--v-3. -ô9î. f c6n'P\rvnevr\9

a. j4,vutþìF<- W)îE-3S. $ t \' ¿,+l'* 9*Y "C 2'

6. -0\gt. f cóY\.(vw¿ wlv '

2

3

Reasons

s

b.Use the given iriformation and the diagram to complete the proof.

GIVEN: ZS is a right angle.m/.RTS : 40o, nIRTU : 50o

PROVE: ZS = I.STU

Statements Reasons

T

U

1.+\rcw

2. -ha1\¿- ¿r¡[i;lr;rn Paç-J3. Substitution Property of Equality

4. Simplify.

s. Ð<Ã. ñ VÀ|\w+ ¿ .J

6. RightAngles Congruence Theorem

G

Flll ln the blanks to complete the two-column proof,10. Glven: IHKJ is a straight angle.

R bisects ¿HKJ.Prove: ZIKJ is a right angle.Proof:

Statements Reasons+rl(' L1 ú. 1. Given

2. nIHKJ = 18Oo 2. b. ô¿,t . f sîør¡¡r+ L3. Ls v 3. Given

4.llKJ= tlKH 4. Def. of Z bisector

5.m/.lKJ=mllKH 5. Def. of = .á6.0 6. Z Add. Post.

7.ZmZIKI=180 7. e. Subst. (Steps 2tç ' ø |

S.mllKJ=9O' L Div. Prop. of =

9. ZIKJ is a right angle e. f. Ðe.î. æ oF É¡thl- {

H

cd. Try:

Use the gÍven plan to write a two-column proofof the Tlansitive Property of Congruence.

Given:¡4,8- =Ø,6 =ÑPmve: en = nf

BE Dc

F

Write a two-column proof.

GTVEN: ,qE = CEAB and CD bisect each other.

PROVE: gg = ED

Statements

A,

DA

Bc

Reasons

0't''z) $tve n

¡r¡ ge-r sf vv¿-ttr

Plan: Use the definition of congruentsegments to write the given congruence statements as statements of equalÍty. Thenusã the Tfansitive Property of Equality to show that .,{B = EF. So .ãB--= EFUy ttredefi nition of congruent segments.

ùFvæ,) ¡i[j 8- ã VtÇzc* ?AÊwlrl\e{

å)}resãBcE ã gr>

4) ¿E é€øÀ+) 6vz eÞ

ò øøtt.. tv't-i an 6 t'*t')- (,.e'

D ò r**"ittþ fØ5 (a'¿)

Hl--Al

6ofe

Chaoter 3:Section 3.1:Skill 1: Students should be able to identify parallel, skew and perpendicular lines in adiagram.

Key Concept Porollelond Skew

DefinitionParallel lines are coplanarlines that do not intersect.The syurbol ll means "isparallel to."

Skew lines arenoncoplanar; they arenot parallel and do notintersect.

€<-¡1-B and CG are skew

SymbolsîÈ ll'f,Èffittft

Diagram

D

E

c

G

F

Use anows to showe<+eeAE llEFand AD ll 8c.

Parallel planes are planesthat do not intersect.

plane;IBCD ll plane EFGII

Example: Think of each segment in the diagram as part of a line. Complete thestatement with pørallel, skew, or perpendicular.

X Y

t.2.3.

WZMF* çrtrc\\øt{4 14WZ ú.OWSY asAWX

are

are

ttcolors

frt

rt

Section 3.2Skill: Students should know the angle relationships when two parallel lines are crossed bya transversal and how to solve problems related to these relationships. Notes below:

If lntft lln€s tÌì and I are P.4R.lLLELfhefol¡owlngholdstrue:

IuII

3 4A,LIf, RNATE INTERIOR An$er arc 3 ;

¿.J=¿.6¿-{=¿-sî 6

7

CORRf,SPONDING Angles ¡re Êír¿.r=¿-s¿-3=.¿-?¿2=¿,6

Ìrì ¿.{=¿-t

I7 I

'.J rl .,

ALTERNATE EXTERIOR Angfcs S;.¿-t=¿-8¿.7=¿-2

CONSECLITI1T Interlor Angfes aresupplementar,r:

ø23 + mZ5 ={E0o l{nâ{ + mZ6 =l8oo

Example: Find the values of x-andy.

r?6o{3 + 3tr

{6x + 8}*

?y +3g

3l = \T6o bl" r+ At+- ex+.

= lbbI

ib(e + fox+b = l8Ò144 -+ bX = (2,Ò

* \¿tq -\,{qCrx. = 36rËãl{i-d

33

Section 3.3 :

Skill: Students should know how to lines el usi the converse theoremsThen....¿l- Ll øq¿ 7.- Lb a*¿þn¿a'qLçI =¿í

-o l\n

Corresponding Angle Postulate Converse

lf two lines and a transversal intersects toform corresponding angles that arecongruent, then the two lines areparallel

rf

IF(1 2

4 3

m 5 67

'1!¿n

Consecutive (Same-Side)lnterior Angles Theorem Converse

lf two lines and a transversal intersects toform consecutive angles that aresupplementary, then the two lines areParallel

rf(

1 B

1þ

4 3

m 5 67

Then....' ¿4 + lS=\8Þ

?t'l\.+¿6 =-tlo

n.e\\ fir\

Alternate lnterior Angle TheoremConverse

lf two lines and a transversal intersect toform alternate interior angles that arecongruent, then the two lines are parallel

rf.....(

1

4 3

m5 6I 7

+herr-.iF ¿3:Zç

z 1; /-ßtu^ì,Q \\vn¿2

Alternate Exterior Angle TheoremConverself two lines and a transversal intersect toform alternate exterior angles that arecongruent, then the two lines are parallel

rf (1

4 3

m 5 68 7

Then::'..i e ¿tÊ ¿+' o4..

¿.LE ¿8

fft.n".

-t t\^

o

Example: . Find the value of x that makes mll n.-ì ?Y +tS+n5:\8O

3y+Qo= 1$Ò

3x * 15)" n

Examples:Carpentry A carpenter plans to install molding on thesides and the top of a doonray. The carpenter cuts theends of the top piece and one end of each of the sidepleces at 45o angles as shown. Will the side pleces ofmolding be parallel? Explain.

!< s ,Wc.a,uy't- \tÞs J +Þ tr'¿ sa'vt"e-'vaù^ü 1'4- tnrwr ¿g o rc--k>"

What about these sides?

\{0 I b¿co^^x- 6¿, *4o ?qo

Given: NP l- NO; NP .f PQ

ZPQS and ZQSR are supplementary.

Prove: fid ttffi\lì.tP \ ñó/ffitd

. ¿--: .. É-->\Nc'll R) reos ø 4ôsq

ôre, 8.4tg\rt^,^f!4) Pd r ft-g

t?\ *--+ç)No t\ Ks

QD=5r+?o

n to use

ns'rcrSe-Q êY<-- pavaJle lL

5

?<¡ =ÉY'1-L= \

v :30Section 3.4Skill: Students should know howthe parallel line and

{5r f 20}"

perpendicular line theorems to prove lines parallel or perpendicular

topTA

?es€6ê

ÉLoot,vl

o

0

À,

P

u Jrwn R

Lrnns I tDt sêvt{-tzr¡¿w-'¡¿"4 *ra gaFall eQ

l) 5'ta1

e) 9an^" sr de ¡:*þr ,o-r conrr ø !-< -i* o^-,,.^r

400400

"4 u*@t{ .f" f.- 6.¡-c- L(,--r- .-.- ({

Rev\ao '. V¡sl 2Section 3.8Skill 1: Students should be able to tell if two lines will be parallel, perpendicular, orneither, by evaluating their slopes. If lines are parallel they will have the same slope.If lines are perpendicular, they will have slopes that are opposite reciprocals of each other

Are the lines arallel endicular

Skill 2: Students should know how to find a line parallel to a given line, given theequation of the given line, and the point the parallel line crosses through.Step L: Find the slope of the parallel line from the given equation of a line (it is the same asthe slope of the given line)

Slope=m= ?,1-'.|(xz-xt)Step 2: Use the slope found in step 1-, and lhe given point to write an equation of theparallel line (Use point-slope form, y-yL= m(x-xr) )Step 3: Solve for y by distributing and simpliSring. (y=mx +b)

,r pçr I

f[,=3 ]42"3çX( \\\\ú-'r¿1ø{É

bhne S arg pnÅuu

-3x -}}

LrvttS ùr¿,

4.¡l\,r z/q M¡'t

Lrh¿s &rú.

y=(-g| + 53y=15x* J

3y" o,.t- n

Mz't -SM, =-5Lrt^¿S

Example: Write an equation of the line passing through the point (-2,L) that is parallelto the line with the equation 10x + 4y=

\-8. Check your solution by graphing the

x-2 lÂ'-e.5iP+ (-2, t)

equations.øc/rr,,JJ,t-Q tl ='195 -9.àa Stoçc af 'Qt v<-, à

31 4

* ua ç+ slrr¿ñrvn:

U-t --2.9(xr4+ Y, *2.6 X -l¡,1S = -2.5x*

e

*

Skill 3. Students should know how to find a line perpendicular to a given line, giventhe equation of the given line, and a the point the perpendicular line crosses through.Step L: Find the slope of the perpendicular line by finding the slope of the given equationof a line and taking its opposite reciprocal.Step 2: Use the slope found in step 1, and the given point to write an equation of theparallel line (Use point-slope form, y-yr = m[x-xr) )

3: Solve for distrib and sim

rÉ

Parallel and perpendicular lines of horizontal and vertical lines.

Horizontal lines are modeled the equation y = k, where k is where the line intersectsthe y-axis.

x

Vertical lines are modeled by the eqintersects the x- axis.

on x = k, where k is where the liner-\

:,

xParallel lines:

{¡Lines that are parallel to a horizontaly=n

\rtrx>N\ Lines that are parallel to a vertical line are any line modeled by an equation of the form.q

<-

X=n

14 Parallel lines:4- Lines that are perpendicual to a horizontal line are any line modeled by an equation of the

formx=n

,\iiI

ii line are #y line modeled by an equati?n of the form

{z(. *--À

Lines that are perpendicual to a vertical line are any line modeled by an equatioform y-n ì*

Y?N

Example: Write an equation of the line passing through the point [2,3) that is perpendicularto the equation of the line y - 4 = -.2(x +3). Check your solution by graphing the equations.

^d Stopa df J \írr¿I J=-z*-ø+4à U'€y-z ltwa-I-$i*-t''^i\s-zbl = \â gorh\: (2,i)

' o so stoç'J á ¡ tì;.e rs å

ú¿ É. stç 6,r:l'n; U- 3 = 7z(X-z)v-3+1 =Yz.x-[+å

= \/2X

o

n of the

For l/our flotebookTHEOREM

Chapter 3/ Chapter 4: Stuff about triangles:Skill 1: use the triangle sum theorem. The sum of the angles is a triangle is 1800.

THEoREH tt.t Trlangh sum Theonm'l'lre sum <lf ¡ltc nleasurcs of tlm i¡ttcriorangtcs of a trian¡¡tc is 180'.

Am¿A + m¿E + m¿C: tgO,

SKItt 2: The exterior angle theorem: The exterior angle of a triangle is equal tosum of its remote interiors.

n)

?

TuEoREir 4.2 Exterlor Angle Theonem'fl¡e mcasure t¡[an exter¡or anglc of a trinn¡¡leis cqual to thc sum <¡f tl¡e nteasurcs of thetrvo nonatljacent interior an¡¡lcs.

Ic

m¿l * m¿^+ mlB

=?5The base angles and sides in an isosceles triangle are equal.

The angles and the sides in an equilateral triangle are all equal.

450

0

950 f

l=6ôol=25o

{=

¡!lì30 - (92+ toZ) = 25o \ îô - (40 +Bo)=øo

1

¿2

For l/our l{otebookTHEOREM

700 2

38"

IK

Ll ¿?'0+38 = lô36/2 = fþ-\0î =426

L

2Y- d = fio+f-Y +lç ts -4bx =150mgf * 1i+:15e.

(all of the the angles are all 600)

@

ANctE RELATIOilSHIPS Flnd the r¡æae¡re of the numbe¡ed anglo21. ¿l 45ÔÔ Z¿, ¿2 3t3ÔÔ

29. ¿3 r 5ô' 21. ¿4 ?. l?öo

\ zt ¿s =4ô' n 26 = gço2V ¿l :gO-40 =Sôô t\ tz= Eô- 9ô s \3ôo2ò ¿S = gO - So '40ô L6) L6 = \?(¡ - I z6

Chauter 4:

Skill 1: Classify triangles by sides and angles : Students should be able to classifytriangles based on given side lengths and angles.

Clasilying Triangler by SiderScalene Trlangle tsoscel€s Trlângle Equllareral Trlangle

No congrucnt lder At ha'l 2 congruenl rldcr 3 contruenl sldcJ

2.3>L3?5ôö+ \3ó) =},6"

,*\ ¿t+ 313ô'

W a'e G(>"

Classllylng Trlangles by AnglesÂcute RightTrlângle Trlangle

Obtu¡cTrlangle

EquiangularTrlangle

Example: Classify triangle by sides and/or angles W c y

l¿rr¡r.¡ alll. Anglemeeture3:.30',60",90' (tW+nn6r¡ Ia $dctengrhr: 2cm,2cm,Zcm (c1rai h¡ø¿aA/ *írÅ.)& Anglemcrlurel:60o,60o, 60" ( ttr4^l¿l oqri bJftà¿)t srderengrhs: uT,llf-r ( i s¿,cc1&Ð)$ Sldelengths¡5fr, Tfr,gfi LS ca.)*n¿-\

¡ æute ôngles 1 rlght ândc 1 obtute tngle 3 congruent anglcs

G Anglcmo¡rurc3:zf,l?.î,S Co btr,

KFY (ON(FPT For lour âlotebaak

@

år<- 5

Skill 2: Students should be able to classify triangles by sides and as a right triangle bycalculating side lengths and slopes.STEP 1; You can classifii a triangle in the coordinate plane by its sides, by finding the

length of each side using the distance formula. d - (xr-xt)z+(yz-!t)z

STEP 2: You can classify a triangle in the coordinate plane as a right triangle by finding theslopes of the lines and seeing if any two slopes are opposite reciprocals, thus implyingperpendicular lines and therefore, a right triangle.

Flxd

nhôS4¡ôcÊZlftríi{ðrw^,h MPo " -t Vuz

O ( ô,0)

Æ T,?' &$,3¡

&0, *¡

so APoo E À nìÉ tnovc6te,QOJ ?O

!

¡

{-E-=W-rq'\æ-6 7

p(-r,z) ô(6,3

t+(i{ i s Scø,V.me.

QO =

Example:

Mra= ffi =+Mqo = å = t/z

Pô "r¡ftaç¡¡¿p

P6={Ttrtr6}={5o-

Claeslfy ercA by lts sldec, Thendetermlne lf the trlangle lc a rlghtlrlangla

Triangle ABC has the vertices A(0,0), B(3,3) and C(-3,3). Classify it by its sides. Thendetermine if it's a right triangle.

Ð aft6c ìsSo

AB = \(a-oY+(3-¿,), = \,E-laB -3)t+ (g -¡T =

3 (e-q' =-0

AABC ìs Cvt lsosce-l¿¿ trrìor"¿\e

= + =6ffi3c¡s4â- hlv a, =-1.lo

PLe = å#Møc ¡ .-1:å

ßC.Aô =

.f %-\lã/

Mtg3l l{*c= -1Lrir¿ AB J L,ne ftC

t'À o-3 -0 '3

@

TRIANGLE PROOFS:You will be asked to complete proofs using SSS,SAS, ASA' AAS, and HL

You will be asked to complete proofs using HL.

You will have a maximum of 3 proofs you must write on your own.All other proofs will require you to either fill in blanks on the statement or reason side.

Things to remember:

Proving Sides Congruent: Proving Angles Congruent:-Segment bisector

-Perpendicular bisectors

-Midpoint

-Reflexive property

- Parallel lines (most likely result in a paircorresponding or alternate interior angles)

-Angle bisector (creates 2 equal angles,

-Vertical Angles

@

l¿

Chapter 5:

Skill 1: Students should know what a midsegment is and be able to find it in thecoordinate plane. A midsegment is a segment that connects the midpoints of the sides ofa triangle. To find the midpoints, use the midpoint formul

^, (ry,'+)Example:Coordinate Geometry The coordinates of the vertices of a triangle are K(2,3), L(-2, -1), and M(5, l).Findthemidsegments. $tdgorñt ar ffi: ({-L,ç) =þ.Srz)

urnd"pa,frI of ñl¡-r (ry,5!) =(å,o) *(¡.s,ø\wtartpoñt o( Eíc, t-+: ,+\ =(o, r)

Skill 2: Students should be able to identify the unique features of the midsegment(that it is half the length and parallel to the third side of the triangle) and use them tosolve problem.

ZtêY) = 4Y r?þ

rn

L

Find the value of x

5x

6x=*4( 4* tzo-41ß

6 y = Z(ro)sx =zo

íøx

a R

L\= 2öE3

L

Srh.¿tÎ ¿LNM = 53 find ¿LRQ and zSQM sO l\ NL

¿n50=s3.30 ¿ s6ìw\ = eo - s fi,

53'

Skill 3: Students should know the meaning of the terms perpendicular bisector, anglebisectot median and altitude are and be able to use those terms to define what acircumcenter, incenter, centroid and orthocenter is . They should know the definingfeatures of these points of concurrency.

s3= 34" Mo\/rhÀ,

LßQ=68'

o

Skill 4: Students should know how to find a centroid in the coordinate plane, byfinding the midpoints of each side and finding lines that extend from vertex tomidpoint.

Skill 5: Students should know how to find an orthocenter graphically as explained insection 5.4 notes.

Skill 6: Students should be able to use the defining features of the points ofconcurrency to solve basic calculation problems.

Skill T: Students should be able to use geogebra to model real life problemsinvolving the circumcenter and incenter. Problems will be similar to the airportproblem and the De Fun is Here Fun Center problem.

Point of ConcurrencyGive the name the point of concurrency for each of the following.

1. Angle Bisectors of a Triangle i ne¿,n û+,r

2. Medians of a Triangle c¿nbor$

3. Altitudes of a Triangle OrfVu e¿n{¿"r

4. Perpendicular Bisectors of a Triangle O,tYc,¡,tøt cei\4+4

Complete each of the following statements.

5. The incenter of a triangle is equidistant from thetriangle.

sù4,S of the

6. The circumcenter of a triangle is equidistant from the $Cfh'r¿¡; ofthe triangle.

7. The centroid is z lZ of the distance from each vertex to the midpointof the opposite side

B. To inscribe a circle about a triangle, you use the 1\ ¿eyllar

9. To circumscribe a circle about a triangle, you use the *t taw\cl,r1++,t

@

Acute  Obtuse  Rieht ÂCircumcenter rnsid¿- ..r4si d. an {¡u Vq pofvt,,, rcIncenter tnst&. 1¡5,än- thsñe¿Centroid ln srðI¡ t' Sld*. tù-Sìd<-Orthocenter Meùc- outsrij¡. ãÌ1 \ranrcx "f Þ.

10. Complete the following chart. Write if the point of concurrency isinside., outside, or on the triangle.

In the diagram, the perpendicular bisectors (shown with dashedsegments) of A^ABC meet at point G--the circumcenter. and are showndashed. Find the indicated measure. B

j. j.. AG = 25 1.2.8D = 2o

13. CF = 7q 1,4. AB = 40

15

E20

1,5. CE = 15 I6. AC= 4n

17. nZ-ADG = loo18. IF BG = (2x- 15J, find x.

2Y-tS = zstr5 +r S

2x = 4t¡Y=Zo

,= La

A C24F

.Ð

In the diagram, the perpendicular bisectors (shown with dashedsegments) of LNINP meet at point O-thecircumcenter. Find the indicated measure.

19. MO = Lb.L 20. PR = ?("

21. MN = 4A 22. Sp = LL

23. mZMQO = loo24. If OP = 2x, find x.

zY=26.1\ = \7.4

tv.4

Point T is the incenter of LPQR.

25. lf Point T is the incenter, then Point Tis the point of concurrency of

the l¿ hlil'ho

26. ST = 15

27. lf TU = (2x- 1), find x.

Tt4- 2X-l = lS2x=16\=?

M 200

22

s

W

R

lv

26

P

fr

U

x

oP

(D

26.8

x= t

See previous previous page for triangle PQR28. ff mZPRT = 24e, then mZQRT = 2ry

29. ff mZRPQ = 62e, then mZRPT = ølo

Point G rs the centroid of A ABC, AD = B, AG = 1-O, BE = 10, AC = 16 andCD = 18. Find the length of each segment. B

30. If Point G is the centroid, then Point Tis the point of concurrency of

Ðthe l4ølr I

31. DB= B 32. EA= Ç) A

33. CG = $[rr)= lt 34. BA = lbIt¿

35, GE= 5 36. cD= f {ra)=a

37. BC= 2llo) :2 Ò 38. AF =

Point^S is the centroid of LRTW, RS -- 4, VW = 6, and TV= 9. Find the length

CF

{{ro)'3

of each segment.

39. RV= 640.SU= L41. RU = 6ç42. RW = lz43. TS= âþ) = t"

-T\l-Q

U0

fr V 6 W

\l

44. SV 3

Point G is the centroid of A ABC. Use the given information to find thevalue of the variable.

45. FG=x+SandGA=6x-4 B

^ g * F b * AF

rg ls + óF Gh so.... T,6_ å(q n>

t(6x-4)7x -L-xó

E F

D c

5 ZY.

Y+ zxþs-f r¿

lôX

46.lfCG=3y+TandCE-6y

v T

:ru = ¿ (cø)

3,g *? r' å /¿v)Sgno '49]-t

@

Is segment AB a midsegment. perpendicular bisector, angle bisector.median, altitude. or none ol these?47) 48)

B

I bìs¿c/+þr M tdtqvn¿nf

s0)

A

A

4e)

B

V Vtvclo<

s1)

Name the point of concurrency, pB

B

wPis tn¡- rhc¿vrlar

AA

È1l"i{wd<.

s2) AA

flt¡n¿ f 4\rr24 Þ{¿¿ Ìat,tB B

C

cA

? i: 'trv" c¿Valw\cenle-,A

\3