Download Congruence Supplementary Packet

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GeometryFall2016
Name:Period:
Unit2–Congruence
Designedtobeusedtogetherwith
MathematicsVisionProjectPacket(Year2)Module5
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GeometryHomeworkGuideUnit2
Period_____Name______________________
Date
LearningTarget/Standard
Homework
EntryTask
EntryTaskTotal:
ImportantDates:
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Name_______________________________________________
FindingCongruentTriangles
1) Alltrianglesaremadeupofthreesidesandhavethreeinteriorangles.Usingaprotractor,measureallthree
anglesineachofthetrianglesbelowmeasuredtothenearestdegree:
a. ∆"#$ m∠" =
m∠# =
m∠$ =
A
C
B
b. ∆'() m∠' =
m∠( =
m∠) =
c. ∆*+, m∠* =
m∠+ =
m∠, =
P
Q
R
d. Makeaconjectureaboutthemeasuresoftheanglesfoundinatriangle.Whatdothemeasuresofthe
anglesintrianglesallseemtohaveincommon?(Wewillprovethisconjecturetobetruelater.)
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2) Giventhattherearesixpartstoeverytriangle(threesidesandthreeangles)itturnsoutthatwhenwecompare
anytwoofthemtoseeiftheyarethesametriangle“congruent”,weonlyneedtocompareanycombinationof
threeofthoseparts.Listallthewaysthatwecanchoosethreepartsofatriangletocompare:
A
S
S
A
A
S
3) Whenwechoosethreepartsofatriangleandcomparethemtotheircorrespondingpartsofadifferenttriangle,
sometimeswefindthetwotrianglesarecongruentandsometimeswefindtheyarenotcongruent.Inthis
activity,youwillchooseanythreepartsofatriangleandinvestigatewhetherthosethreepartscreateexactly
onetriangleorifthosethreepartscanmakemorethanonedifferenttriangle.Namethepartstakenandcircleif
onlyonetrianglecanbeformedorifmorethanonetrianglecanbemade:
3PartsTaken
NumberofPossibleTriangles
ExactlyOneTrianglecanbeformed
MorethanoneTrianglecanbeformed
ExactlyOneTrianglecanbeformed
MorethanoneTrianglecanbeformed
ExactlyOneTrianglecanbeformed
MorethanoneTrianglecanbeformed
ExactlyOneTrianglecanbeformed
MorethanoneTrianglecanbeformed
ExactlyOneTrianglecanbeformed
MorethanoneTrianglecanbeformed
ExactlyOneTrianglecanbeformed
MorethanoneTrianglecanbeformed
ExactlyOneTrianglecanbeformed
MorethanoneTrianglecanbeformed
4) Ifonlyonetrianglecanbeformedgiventhreepartsofatriangle,thenthatissufficientevidenceneededto
provetwotrianglesarecongruent.Ithethreepartsofonetrianglearecongruenttothreecorrespondingparts
ofanothertriangle,thetwotrianglesarecongruent.Showthetickmarksthatwouldindicatethetrianglesare
congruentinfivedifferentways:
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5) Indicateifthetwotrianglesarecongruent≅ornotcongruent≅giventhemarks.Iftheyarecongruent,justify
thisbyindicatingthethreecorrespondingpartsaslistedonthepreviouspage:
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CONGRUENCE, CONSTRUCTION AND PROOF- 7.5
7. 5 Congruent Triangles
to the Rescue
A Practice Understanding Task
Part1
ZacandSioneareexploringisoscelestriangles—trianglesinwhichtwosidesarecongruent:
Zac:Ithinkeveryisoscelestrianglehasalineofsymmetrythatpassesthroughthevertex
pointoftheanglemadeupbythetwocongruentsides,andthemidpointofthethirdside.
Sione:That’saprettybigclaim—tosayyouknowsomethingabouteveryisoscelestriangle.
Maybeyoujusthaven’tthoughtabouttheonesforwhichitisn’ttrue.
Zac:ButI’vefoldedlotsofisoscelestrianglesinhalf,anditalwaysseemstowork.
Sione:Lotsofisoscelestrianglesarenotallisoscelestriangles,soI’mstillnotsure.
1. WhatdoyouthinkaboutZac’sclaim?Doyouthinkeveryisoscelestrianglehasalineof
symmetry?Ifso,whatconvincesyouthisistrue?Ifnot,whatconcernsdoyouhaveabout
hisstatement?
2. WhatelsewouldZacneedtoknowaboutthecreaselinethroughinordertoknowthatitisa
lineofsymmetry?(Hint:Thinkaboutthedefinitionofalineofreflection.)
3. SionethinksZac’s“creaseline”(thelineformedbyfoldingtheisoscelestriangleinhalf)
createstwocongruenttrianglesinsidetheisoscelestriangle.Whichcriteria—ASA,SASor
SSS—couldheusetosupportthisclaim?Describethesidesand/oranglesyouthinkare
congruent,andexplainhowyouknowtheyarecongruent.
4. Ifthetwotrianglescreatedbyfoldinganisoscelestriangleinhalfarecongruent,whatdoes
thatimplyaboutthe“baseangles”ofanisoscelestriangle(thetwoanglesthatarenot
formedbythetwocongruentsides)?
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https://flic.kr/p/3GZScG
SECONDARY MATH I // MODULE 7
CC BY Anders Sandberg
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SECONDARY MATH I // MODULE 7
CONGRUENCE, CONSTRUCTION AND PROOF- 7.5
5. Ifthetwotrianglescreatedbyfoldinganisoscelestriangleinhalfarecongruent,whatdoes
thatimplyaboutthe“creaseline”?(Youmightbeabletomakeacoupleofclaimsaboutthis
line—oneclaimcomesfromfocusingonthelinewhereitmeetsthethird,non-congruent
sideofthetriangle;asecondclaimcomesfromfocusingonwherethelineintersectsthe
vertexangleformedbythetwocongruentsides.)
Part2
LikeZac,youhavedonesomeexperimentingwithlinesofsymmetry,aswellasrotational
symmetry.InthetasksSymmetriesofQuadrilateralsandQuadrilaterals—BeyondDefinitionyou
madesomeobservationsaboutsides,angles,anddiagonalsofvarioustypesofquadrilateralsbased
onyourexperimentsandknowledgeabouttransformations.Manyoftheseobservationscanbe
furtherjustifiedbasedonlookingforcongruenttrianglesandtheircorrespondingparts,justasZac
andSionedidintheirworkwithisoscelestriangles.
Pickoneofthefollowingquadrilateralstoexplore:
•
Arectangleisaquadrilateralthatcontainsfourrightangles.
•
Arhombusisaquadrilateralinwhichallsidesarecongruent.
•
Asquareisbotharectangleandarhombus,thatis,itcontainsfourrightanglesand
allsidesarecongruent
1. Drawanexampleofyourselectedquadrilateral,withitsdiagonals.Labeltheverticesofthe
quadrilateralA,B,C,andD,andlabelthepointofintersectionofthetwodiagonalsaspointN.
2. Basedon(1)yourdrawing,(2)thegivendefinitionofyourquadrilateral,and(3)information
aboutsidesandanglesthatyoucangatherbasedonlinesofreflectionandrotational
symmetry,listasmanypairsofcongruenttrianglesasyoucanfind.
3. Foreachpairofcongruenttrianglesyoulist,statethecriteriayouused—ASA,SASorSSS—to
determinethatthetwotrianglesarecongruent,andexplainhowyouknowthattheangles
and/orsidesrequiredbythecriteriaarecongruent(seethefollowingchart).
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Licensed under the Creative Commons Attribution CC BY 4.0
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SECONDARY MATH I // MODULE 7
CONGRUENCE, CONSTRUCTION AND PROOF- 7.5
Congruent
Triangles
CriteriaUsed
(ASA,SAS,SSS)
IfIsayΔRST≅ΔXYZ
basedonSSS
HowIknowthesidesand/orangles
requiredbythecriteriaarecongruent
thenIneedtoexplain:
• howIknowthat RS ≅ XY ,and
• howIknowthat ST ≅ YZ ,and
• howIknowthat TR ≅ ZX soIcanuseSSScriteriatosayΔRST≅ΔXYZ
4. Nowthatyouhaveidentifiedsomecongruenttrianglesinyourdiagram,canyouusethe
congruenttrianglestojustifysomethingelseaboutthequadrilateral,suchas:
• thediagonalsbisecteachother
•
thediagonalsarecongruent
•
thediagonalsareperpendiculartoeachother
•
thediagonalsbisecttheanglesofthequadrilateral
Pickoneofthebulletedstatementsyouthinkistrueaboutyourquadrilateralandtryto
writeanargumentthatwouldconvinceZacandSionethatthestatementistrue.
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SECONDARY MATH I // MODULE 7
7.5
CONGRUENCE, CONSTRUCTION AND PROOF- 7.5
Afterworkingwiththeseequationsandseeingthetransformationsonthecoordinategraphitisgood
timingtoconsidersimilarworkwithtables.
6.Matchthetableofvaluesbelowwiththeproperfunctionrule.
I
II
x
-1
0
1
2
f(x)
16
14
12
10
III
x
-1
0
1
2
A.! ! = −! ! − ! + !
B.! ! = −! ! − ! + !"
C.! ! = −! ! − ! + !
f(x)
14
12
10
8
IV
x
-1
0
1
2
f(x)
12
10
8
6
V
x
-1
0
1
2
D.! ! = −! ! + ! + !
E.! ! = −! ! + ! + !"
f(x)
10
8
6
4
x
-1
0
1
2
SET
Topic:UseTriangleCongruenceCriteriatojustifyconjectures.
Ineachproblembelowtherearesometruestatementslisted.Fromthesestatementsa
conjecture(aguess)aboutwhatmightbetruehasbeenmade.Usingthegivenstatementsand
conjecturestatementcreateanargumentthatjustifiestheconjecture.
7.Truestatements: Conjecture: ∠A ≅ ∠C PointMisthemidpointof!"
∠!"# ≅ ∠!"#
a.Istheconjecturecorrect?
!" ≅ !"
b.Argumenttoproveyouareright:
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f(x)
8
6
4
2
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SECONDARY MATH I // MODULE 7
7.5
CONGRUENCE, CONSTRUCTION AND PROOF- 7.5
8.Truestatements
∠ !"# ≅ ∠ !"#
!" ≅ !"
Conjecture:!"bisects∠ !"#
a.Istheconjecturecorrect?
b.Argumenttoproveyouareright:
Conjecture:∆ !"# ≅ ∆!"#
a.Istheconjecturecorrect?
b.Argumenttoproveyouareright:
9.Truestatements
∆ !"#isa180°
rotationof∆ !"#
GO
Topic:Constructionswithcompassandstraightedge.
10.Whydoweuseageometriccompasswhendoingconstructionsingeometry?
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ExtraPractice/Homework
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Kuta Software - Infinite Geometry
Name___________________________________
Parallel Lines and Transversals
Date________________ Period____
Identify each pair of angles as corresponding, alternate interior, alternate exterior, or consecutive interior.
1)
2)
y
y
x
x
3)
4)
y
y
x
x
5)
6)
y
y
x
x
7)
8)
y
x
x
y
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-1-
Worksheet by Kuta Software LLC
9)
14
10)
y
x
x
y
Find the measure of each angle indicated.
11)
12)
?
84°
?
110°
13)
14)
?
100°
111°
?
15)
16)
?
125°
?
17)
47°
18)
53°
?
113°
?
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Worksheet by Kuta Software LLC
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Solve for x.
19)
20)
75°
11 x − 2
21 x + 6
21)
22)
x + 139
60°
132°
8x − 4
23)
24)
−1 + 14 x
23 x − 5
12 x + 17
21 x + 5
Find the measure of the angle indicated in bold.
25)
26)
x + 96
20 x + 5
x + 96
24 x − 1
27)
28)
6x
x + 109
x + 89
5 x + 10
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Worksheet by Kuta Software LLC
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Kuta Software - Infinite Geometry
Name___________________________________
Angles in a Triangle
Date________________ Period____
Find the measure of each angle indicated.
1)
2)
?
65°
40°
57°
?
3)
4)
85°
20°
50°
?
130°
?
5)
6)
137°
?
102°
100°
?
35°
7)
8)
155°
60°
?
30°
?
20°
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-1-
Worksheet by Kuta Software LLC
9)
17
10)
?
39°
35°
40°
20°
65°
50°
60°
11)
?
12)
86°
?
27°
84°
?
35°
23°
36°
13)
14)
156°
?
155°
115°
35°
20°
85°
?
15)
16)
45°
45°
68°
68°
79°
100°
?
75°
60°
?
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Worksheet by Kuta Software LLC
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Solve for x.
17)
18)
55°
x + 74
8x + 2
54°
70°
19)
60°
20)
64°
27°
x + 51
97 + x
60°
80°
Find the measure of angle A.
21)
22)
x + 59
x + 37
x + 67
A
84°
x + 51
A
23)
24)
A
A
3x − 6
130°
x + 23
8x + 4
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80°
4 x + 17
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Worksheet by Kuta Software LLC
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Kuta Software - Infinite Geometry
Name___________________________________
ASA and AAS Congruence
Date________________ Period____
State if the two triangles are congruent. If they are, state how you know.
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
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Worksheet by Kuta Software LLC
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Kuta Software - Infinite Geometry
Name___________________________________
SSS and SAS Congruence
Date________________ Period____
State if the two triangles are congruent. If they are, state how you know.
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
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Worksheet by Kuta Software LLC
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Kuta Software - Infinite Geometry
Name___________________________________
Congruence and Triangles
Date________________ Period____
Complete each congruence statement by naming the corresponding angle or side.
1) ∆DEF ≅ ∆KJI
2) ∆BAC ≅ ∆LMN
E
K
I
M
A
C
B
F
D
N
J
∠A ≅
FD ≅ ?
3) ∆TUV ≅ ∆GFE
?
4) ∆WVU ≅ ∆GHI
U
V
W
I
G
F
G
U
E
T
∠W ≅
V
∠U ≅
L
H
?
?
5) ∆ZXY ≅ ∆ZXC
6) ∆DEF ≅ ∆DSR
Y
E
D
Z
X
F
R
S
C
∠Y
∠F ≅
≅?
?
Write a statement that indicates that the triangles in each pair are congruent.
8)
7)
T
J
I
D
R
H
I
K
S
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C
B
G
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Worksheet by Kuta Software LLC
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9)
10)
R
S
I
P
Q
T
R
D
11)
12)
E
C
D
U
C
X
W
V
D
S
T
E
Mark the angles and sides of each pair of triangles to indicate that they are congruent.
14) ∆GFE ≅ ∆LKM
13) ∆BDC ≅ ∆MLK
D
L
K
M
G
E
B
C
L
15) ∆MKL ≅ ∆STL
F
M
K
16) ∆HIJ ≅ ∆JTS
K
T
I
S
M
17) ∆CDB ≅ ∆CDL
S
H
T
L
J
18) ∆JIK ≅ ∆JCD
C
K
I
L
J
B
D
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D
C
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Worksheet by Kuta Software LLC
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Kuta Software - Infinite Geometry
Name___________________________________
Isosceles and Equilateral Triangles
Date________________ Period____
Find the value of x.
1)
2)
7
x
x
6
3)
4)
6
x
4
5)
x
6)
40°
x
x
7)
75°
8)
x
75°
x
54°
9)
10)
65°
x
28°
x
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Worksheet by Kuta Software LLC
11)
26
12)
x
120°
148°
x
13)
14)
8
−1 + x
12
15) m∠2 = x + 94
2 x − 12
16) m∠2 = 4 x − 2
2
2
68°
47°
17) m∠2 = 12 x + 4
18) m∠2 = 13 x + 3
2
118°
146°
2
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Worksheet by Kuta Software LLC
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