MPM 1D – Unit 7: Geometric Relationships – Lesson 1 Date: ______________ Learning goal: how to apply angle relationships to find missing angles in triangles and quadrilaterals. Angle Relationship in Triangles and Quadrilaterals REVIEW: GRADE 8 THEOREMS Opposite Angle Theorem (OAT) - opposite angles are ____________ Complementary Angles (CA) - two angles add up to ______________ Supplementary Angles (SA) - two angles add up to _________ Angle Sum Triangle Theorem (ASTT) - three angles in a triangle add up to _________ Isosceles Triangle Theorem (ITT) - angles opposite the equal sides are _________ Equilateral Triangle Theorem (ETT) - all three sides equal and all three angles are________ Exterior Angle Theorem (EAT) - exterior angle of a triangle is equal to the sum of the interior and non-adjacent angles Angle Sum Quadrilateral Theorem (ASQT) - four angles in a quadrilateral add up to _________ Example 1: State the measure of the supplementary angle for each of the following angles. a) 40° b) 170° c) 93° Example 2: State the measure of the complementary angle for each of the following angles. a) 14° b) 64° c) 1°
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
MPM1D–Unit7:GeometricRelationships–Lesson1 Date:______________Learninggoal:howtoapplyanglerelationshipstofindmissinganglesintrianglesandquadrilaterals.
AngleRelationshipinTrianglesandQuadrilaterals
REVIEW:GRADE8THEOREMSOppositeAngleTheorem(OAT)-oppositeanglesare____________
ComplementaryAngles(CA)-twoanglesaddupto______________
SupplementaryAngles(SA)-twoanglesaddupto_________
AngleSumTriangleTheorem(ASTT)-threeanglesinatriangleaddupto_________
IsoscelesTriangleTheorem(ITT)-anglesoppositetheequalsidesare_________
EquilateralTriangleTheorem(ETT)-allthreesidesequalandallthreeanglesare________
ExteriorAngleTheorem(EAT)-exteriorangleofatriangleisequaltothesumoftheinteriorandnon-adjacentangles
AngleSumQuadrilateralTheorem(ASQT)-fouranglesinaquadrilateraladdupto
_________
Example1:Statethemeasureofthesupplementaryangleforeachofthefollowingangles.
a)40° b)170° c)93°Example2:Statethemeasureofthecomplementaryangleforeachofthefollowingangles.
a)14° b)64° c)1°
Example3:Determinethemeasureofeachindicatedangle.Stateyourreasoning.
a) b) c)Example4:Determinethevalueofx,andthenthemeasureofeachangle.Stateyourreasoning.
a) b)
67° 44° z
57°
c
b
a
2x
3xx
36
b
95
23x+7
25x-7
EXTERIORANGLESProof:
ExteriorAnglesofTrianglesandQuadrilaterals
Theexterioranglesofatriangleaddto.
Theexterioranglesofaquadrilateraladdto.
Example5:Determinethesizeofeachindicatedangle.Stateyourreasoning. a) b) d
105
95
95
4y
30
yc x
Assignment7.1:AngleRelationshipsinTrianglesandQuadrilaterals1. Determinethemeasureofeachindicatedangle,stateyourreasoning. a) b) c) d) e) f)2. Findthevalueofx,thenfindthemeasureoftheunknownangle. a) b)
3.Determinethevalueofthevariables,stateyourreasoning. a) b)
7.1Answers:
1.a)z=65º,r=65º,t=78º b)a=122º,b=53º c)x=66º,y=114º,z=129ºd)a=83º,b=98º,c=104º,d=75º,e=105º e)x=55º,y=70º f)x=580,y=420
2.a)x=20,<A=170,<B=110,<C=80b)x=94,<AEB=112,y=34,<BEC=68
3.a)x=62,y=35,z=145 b)x=152,y=18
82°
33°
z r
t
37°
127°
100°
85° b
a
51° 117°
x
y
z
97°
82°
76°
d
c b
e a
//\\
55° x
y
z
40°
y 28° x
105°x
y 72°
28°
A
BD
C
AB
C
42° x
80° y
80°
MPM1D–Unit7:GeometricRelationships–Lesson2 Date:______________Learninggoal:howtoapplyanglerelationshipsinparallellinestofindmissingangles.
AngleRelationshipinParallelLines
Term Definition Diagram
ParallelLines
Transversal
CorrespondingAngles
AlternateAngles
InteriorAngles
Ifthetransversalcutstwoparallellinesthentheparallellinetheorem(PLT)states:
Alternateangles___________________.
WecallthisthePLTZ-Pattern.
Correspondingangles_________________.
WecallthisthePLTF-Pattern.
Interiorangles___________________.
WecallthisthePLTC-Pattern.
Example1:Determinetheanglemeasureindicatedbyeachsmallletter.a)b)
c)
A
B
C
D
E
F
75o
wxy
z
49o
98o
mn p
64o
c
a
Statement Reason
Statement Reason
Statement Reason
G
H
Assignment7.2:AngleRelationshipsinParallelLinesDeterminetheanglemeasureindicatedbyeachsmallletteralgebraically.Donotuseaprotractor.Stateyourreasoning.1) 2)3) 4) 5.Writeanequationandsolvefortheunknown.Statethetheoremusedtomaketheequation.
a)
b)
c)
d)
7.2Answers:1.y=75o,x=105o2.r=112o,s=112o,q=68o3.a=44o,b=30o,c=106o4.x=62o,y=87o,z=93o
5.a)b=110,b)c=20,c)x=6,d)h=32
75°
y x
q
r
112°
s
30°
44°
c
b
a
118° x y
87°
z
(2c+111)°
(7c+11)°
(6x-14)°
(3x+4)°
MPM1D–Unit7:GeometricRelationships–Lesson3 Date:______________Learninggoals:howtoapplyinterioranglerelationshipstosolveunknownanglesinpolygons.
InteriorAngleRelationshipinPolygons
Investigatehowtodeterminethesumoftheinterioranglesofapolygonbycompletingthefollowingtable.
Polygon NumberofSides SketchofPolygon Numberof
TrianglesSumofInterior
Angles
Triangle 3
1 180°(180°x1)
Quadrilateral 4
2 360°(180°x2)
Pentagon 5
3 540°(180°x3)
Hexagon 6
Heptagon 7
Octagon 8
n-gon n
INTERIORANGLESOFAPOLYGONExample1:Findthesumoftheinterioranglesofeachpolygon.
a) b) c)
Canyoudeterminethemeasureofeachinteriorangleineachquestion?Ifso,determinethemeasureofeachinteriorangle.
Example2:Findeachinteriorangleofthefollowingregularpolygons.
a) b)
c)
RegularPolygon
Aregularpolygonisapolygonwithallandall.
Eachangleinaregularpolygon=
Example3:Findthesumoftheinterioranglesofapolygonwith18sides.Example4:Findthemeasureofeachinteriorangleofaregularpolygonwith20sides.
Example5:Findthenumberofsidesofapolygonwithasumof2340°interiorangles.
Assignment7.3:InteriorAngleRelationshipsinPolygons1.Findthesumoftheinterioranglesinapolygonwith: a)8sides b)12sides2.Howmanysidesdoesapolygonhaveifthesumoftheinterioranglesis: a)1440° b)2520° 3.Findthemeasureofeachinteriorangleinaregularpolygonwith: a)5sides b)6sides4.Determinethemeasureofx.Justifyyouranswersusinggeometricproperties.
a)
5.Determinethevalueofx,andthendeterminethemeasureofeachinteriorangle.Justifyyouranswersusinggeometricproperties.a) b) 6.Determinethevaluesofeachoftheunknownsinthediagrambelow.Justifyyouranswersusinggeometricproperties.7.3Answers:1.a)1080°b)1800° 2.a)10b)16 3.a)108°b)120°4a)36°5.a)102b)x=72à144,108,108,90,906.a=95,b=85,c=60,d=150,e=150,f=50,g=130,h=85,p=125
85°
130°
120°
150°130°
g 140o
e d c p
b a
f
xx
x
xx
MPM1D–Unit7:GeometricRelationships–Lesson4 Date:______________Learninggoals:howtoapplyexterioranglerelationshipstosolveunknownanglesinpolygons.
ExteriorAngleRelationshipinPolygons
Fromlessononeweknowthesumoftheexterioranglesofaquadrilateralis360°.Letslookattheexterioranglesofapolygon.INVESTITATIONInvestigatingapentagon.
1. Drawapentagonandlabelthevertices.Estimatethemeasureeachofthefiveinteriorangles.
2. Extendonesideateachvertexofyourpentagontocreateanexteriorangle.Nameandmeasurethefiveexteriorangles.Findthesumoftheseexteriorangles.Comparethissumtothosefoundbyyourclassmates.
3. Makeahypothesisaboutthesumoftheexterioranglesofanypentagon.
ExteriorAnglesofaPolygon
Forapolygonwithnsides,thesumoftheinterioranglesis.
Thesumoftheexterioranglesofapolygonis.
Example1:Consideraregular20-sidedpolygon,whatisthemeasureof….. a)Thesumoftheinteriorangles b)Thesumoftheexteriorangles c)Eachinteroirangle d)EachexteriorangleExample2:Sixexterioranglesofa7-sidedpolygoneachmeasure50°.Findthemeasureofthemissing
angle.Example3:Eachexteriorangleofaregularpolygonis12˚.Determinethenumberofsides.
Assignment7.4:ExteriorAngleRelationshipsinPolygons1.Findthemeasureofeachexteriorangleinaregularpolygonwith: a)7sides b)9sides 2.Eachexteriorangleofaregularpolygonis36o.Determinethenumberofsides.3.Astopsignintheshapeofaregularoctagonisrestingonabrickwall,asshownintheaccompanyingdiagram.Whatisthemeasureofanglex?4.Ifthesumofalltheinterioranglesofaregularpolygonistwicethesumofalltheexteriorangles,findthenumberofsidesintheregularpolygon.5.Determinethevalueofeachexteriorangleinthediagrambelow.6.Determinethevalueofx,andthendeterminethemeasureofeachexteriorangle.Justifyyouranswersusinggeometricproperties.7.4Answers:1.a)51.43°b)40° 2.103.45o
4.65.x=100o,120o,140o
6.x=40à80,70,55,65
Related Documents
