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