Mathematics Grade8

1

Alineisaninfinitenumberofpointsbetweentwoendpoints.Wheretwolinesmeetorcross,theyformanangle.Anangleisanamountofrotation.Itismeasuredindegrees.

TypesofanglesNameofangle Example SizeofangleAcuteangle

Between0°and90°

Rightangle Equalto90°

Obtuseangle Between90°and180°

Straightline Equalto180°

Reflexangle Between180°and360°

Revolution/anglesaroundapoint

Equalto360°

Anglelanguage:Labellingangles:𝐵or𝐴𝐵𝐶Also:

Geometry(Part1)

Linesandangles

A B

C

vertex

arm

arm

angle

B

Werefertothereflexangleas‘reflex𝑩,′

Mathematics Grade8

2

TerminologyIntersect

ABandCDintersect(crossorcut)atE

Bisect ABbisect(cutsinhalf)CD

ComplementaryanglesAnglesthataddupto90°SupplementaryanglesAnglesthataddupto180°

E.g.thecomplementof48°is42°E.g.thesupplementof130°is50°

Adjacentangles

Anglesthathaveacommonvertexandacommonarm→ 𝑝and𝑞areadjacentangles.

Perpendicularlines

Linesthatmeetorcrossat90°.𝐴𝐵 ⊥ 𝐶𝐷

A

BC

ED

A

BC

D

𝑝𝑞

Adjacentangelsonastraightlineaddsupto180°

𝑚 𝑛∴ 𝑚 + 𝑛 = 180°

C

A

B D

Thislittleblockindicatestousthatthelinesareperpendicular.

Symbolfor‘perpendicular’

Mathematics Grade8

3Exercise1:(a)Inthediagrambelowname: (1) 5acuteangles (2) 2rightangles (3) 10pairsofadjacentangles (4) 3obtuseangles (b)Inthediagrambelow,classifytheangleslabelleda–j.Thefirstoneisdoneforyouasanexample: a:Acute b:

A

B

1

E

1

2

2

D

34

1 12 23C 4

a

b

c

d

e

fgh

i

j

Mathematics Grade8

4c: d: e: f: g: h: i: j: (c)Considertheanglesmarked𝑥and𝑦.Statewhethertheyareadjacentornot: (d)Completethetablebyfillinginthemissinginformation:

Measureofangle Complement Supplement37° 90° − 37° = 59° 180° − 37° = 143°20° 77° 101° 90° 96° 𝑥 𝑦

𝑥𝑦

𝑥

𝑦

𝑥 𝑦 𝑥 𝑦

𝑥𝑦

𝑦

𝑥

Mathematics Grade8

5REMEMBER:Adjacentanglesonastraightlinearesupplementary.

Iftheyareadjacentanglesonastraightline,thentheyaddupto180°.Example:Determine,withreason,thevalueof𝑥:

Statement Reason𝑥 = 180° − 120° Adj∠′sonastrline

Ingeometrywealwaysneedtoprovidereasonsfor‘why’westatesomething.

Exercise2:Calculatethesizeofthevariables(𝑎, 𝑏, 𝑐𝑎𝑛𝑑𝑑).Giveareasonforyouranswer. Statement Reason(a)

(b)

(c)

(d)

Verticallyoppositeangles:Whentwostraightlinesintersecttheanglesoppositeeachotherarecalledverticallyoppositeangles.

𝑥120°

Weusetheseabbreviationstomakeourlivesalittlebiteasier!*thereisacompletesummaryonpage

50°𝑎

𝑏10° 60°

2𝑐120°

𝑑𝑑 + 20°

𝑥 𝑥𝑦

𝑦

Verticallyoppositeanglesareequaltoeachother.

Mathematics Grade8

6Example:Determine,withreason,thevalueof𝑥:

Statement Reason𝑥 = 110° Vertopp∠’s

TransversalsIfalinecutsortouchesanotherline,itiscalledatransversal.Transversalscreatesthreeimportanttypesofangles,namely:1.Correspondingangles2.Co-interiorangles3.Alternatingangles1.Correspondinganglesareinthesamepositionaseachother.TheymakeaFshape:2.Co-interioranglesarebetweenthelinesandonthesamesideofthetransversal.Theyare“insidetogether”.TheymakeaCorUshape.

𝑥

110°

e.g.𝐴𝐵isatransversalbecauseitcuts𝐶𝐷and𝐸𝐹, 𝐶𝐷and𝐸𝐹arealsotransversalsof𝐴𝐵.

A

CD

E

F

B

Mathematics Grade8

73.Alternateanglesarebetweenthelinesandonalternate(opposite)sidesofthetransversal.TheymakeaZorNshape.RememberthewordFUNwheneveryouseeatransversal!Exercise3:Usethediagrambelowtofind:(a) 10pairsofcorrespondingangles (b)8pairsofverticallyoppositeangles (c)4pairsofco-interiorangles (d)8pairsofalternateangles (e)6pairsofadjacentsupplementaryangles

AB

C

D

11

1

1

22

2

2

33

3

3

44

4

4

Mathematics Grade8

8

80°𝑥𝑦

Exercise4:Findthevalueofeachvariable,inalphabeticalorder(wherethereismorethanonevariable),providingreasonsforyourstatements: Statement Reason(a)

(b)

(c)

(d)

(e)

UsethefollowingreasonstohelpyoucompleteEx4and5• Adj∠′sonastrLine• Adjcomp∠′s• Vertopp∠′s•∠′satapt

𝑥145°

40°

40°𝑥𝑦

50°60°𝑥𝑦𝑧

95°

𝑥

10°

Mathematics Grade8

9Exercise5:Usethediagramtowritedownanequation,withareason,inordertocalculatethevalueof𝑥: Statement Reason(a)

(b)

(c)

ParallellinesParallellinesarelinesthatstaythesamedistanceapart,nomatterhowlongthelinesare(theyarelinesthatnevermeet).Iflinesareparallelthen: Reasons:

• Thecorrespondinganglesareequal corr∠′s;…//…• Thealternateanglesareequal alt∠′s;…//…• Theco-interioranglesaresupplementary co-int∠′s;…//…

Toprovelinesareparallel:Provethecorrespondinganglesareequal corr∠′s=Provethealternateanglesareequal alt∠′s=Provetheco-interioranglesaresupplementary co-int∠′s=Let’sseeinExercise6howtheseparallellinescanhelpusdeterminethevalueofunknownangles…

𝑥 + 20°70°

2𝑥 − 50°𝑥 + 20°

3𝑥 − 10°140°

Arrowsareusedtoindicatethatlinesareparallel.

NB:Youhavetomentiontheparallellinesused!

Mathematics Grade8

10Exercise6:(a)Determinethesizesoftheanglesmarkedwithvariables,inalphabeticalorder.Givereasonsfor youranswers.(Thefirstoneisdoneforyouasanexample)

Statement Reason(1)

𝑥 = 108°𝑦 = 180 − 108°𝑦 = 72°

Corr∠′s;AB//CDAdj∠′sonastrline

(2)

(3)

(4)

(5)

108°𝑥𝑦

A

B

C

D

F

E

H

G

88°

𝑥

𝑦

51°

100°𝑥

𝑦𝑧

I J

KL

M

N

O

P

𝑥

62°

Q

R

S

V

X

T

U

W

71°

𝑥𝑦

Mathematics Grade8

11(b)Ineachcase,statewhether𝐴𝐵isparallelto𝐶𝐷.Providereasonsforyourstatements. (1) (2)

(3) (4)

SummaryofstatementsandreasonsStatement ReasonAnglesonastraightlineaddsupto180° Adj∠′sonastrlineComplementaryanglesaddsupto90° Adjcomp∠′sVerticallyoppositeanglesareequal Vertopp∠′sAnglesaroundapointaddsupto360° ∠′sataptCorrespondinganglesofparallellinesareequal Corr∠′s;…//…Co-interioranglesbetweenparallellinesaddupto180°

Co-int∠′;…//…

Alternatinganglesofparallellinesareequal Alt∠′s;…//… *Pleasenotethatnoneofthediagramsinthisworkbookaredrawnaccordingtoscale.

A C

B D

69° 69°

A

B

C

D

69° 69°

AB

CD

69°

69°

AC

D

69° 111°

B

Mathematics Grade8

12

MEMO

Exercise1:(a.1) 𝐴K; 𝐴M; 𝐸N; 𝐷K; 𝐷M; 𝐵N(anyfive)(a.2) 𝐸𝐶𝐵and𝐸𝐶𝐷(a.3) 𝐴Kand𝐴N;𝐴Nand𝐴M;𝐴Mand𝐴O;𝐵Kand𝐵N;𝐸𝐶𝐵and𝐸𝐶𝐷;𝐷Kand𝐷N;𝐷Nand𝐷M;𝐷Mand𝐷O 𝐴Kand𝐴O;𝐷Kand𝐷O;𝐸Kand𝐸N(a.4) 𝐴N;𝐴O;𝐸K;𝐷N;𝐷O;𝐵K(anythree)(b) b:Obtuse c:Reflex d:Obtuse e:Obtuse f:Right g:Acute h:Acute i:Reflex j:Obtuse(c.1) Adjacent(c.2) Notadjacent(doesnotshareacommonpoint)(c.3) Notadjacent(doesnotshareacommonarm)(c.4) Adjacent(c.5) Adjacent(c.6) Notadjacent(doesnotshareacommonpoint)(d)

Measureofangle Complement Supplement20° 70° 160°77° 13° 103°101° Nocomplement 79°90° 0° 90°96° Nocomplement 84°𝑥 90° − 𝑥 180° − 𝑥𝑦 90° − 𝑦 180° − 𝑦

Mathematics Grade8

13Exercise2:

Statement Reason(a)

𝑎 = 180° − 150°∴ 𝑎 = 130°

Adj∠′sonastrline

(b)

𝑏 = 180° − 10° − 60°∴ 𝑏 = 110°

Adj∠′sonastrline

(c)

2𝑐 = 180° − 120°2𝑐 = 60°𝑐 = PQ°

N

∴ 𝑐 = 30°

Adj∠′sonastrline

(d)

𝑑 + 20° + 𝑑 = 180°2𝑑 = 180° − 20°2𝑑 = 160°𝑑 = KPQ°

N

∴ 𝑑 = 80°

Adj∠′sonastrline

Exercise3:(a) 𝐴Kand𝐵K;𝐴Nand𝐵N;𝐴Mand𝐵M;𝐴Oand𝐵O;𝐴Kand𝐷K;𝐴Nand𝐷N;𝐴Mand𝐷M;𝐴Oand𝐷O 𝐵Kand𝐶K;𝐵Nand𝐶N;𝐵Mand𝐶M;𝐵Oand𝐶O;𝐶Kand𝐷K;𝐶Nand𝐷N;𝐶Mand𝐷M;𝐶Oand𝐷O (anytenpairs)(b) 𝐴Kand𝐴M;𝐴Nand𝐴O;𝐵Kand𝐵M;𝐵Nand𝐵O;𝐶Kand𝐶M;𝐶Nand𝐶O;𝐷Kand𝐷M;𝐷Nand𝐷O(c) 𝐴Mand𝐷N;𝐴Oand𝐷K;𝐴Nand𝐵K;𝐵Oand𝐶K;𝐵Mand𝐶N;𝐶Kand𝐷N;𝐶Oand𝐷M (anyfour)(d) 𝐴Nand𝐵O;𝐴Oand𝐷N;𝐴Mand𝐷K;𝐵Kand𝐴M;𝐵Oand𝐶N;𝐵Mand𝐶K;𝐶Kand𝐷M;𝐶Oand𝐷N(e) Anytwoanglesthatareonastraightlineandsharethesamepoint.Exercise4:

Statement Reason(a)

𝑥 = 95° Vertopp∠′s

50°𝑎

𝑏10° 60°

2𝑐120°

𝑑𝑑 + 20°

95°

𝑥

Mathematics Grade8

14(b)

𝑥 = 180° − 145°∴ 𝑥 = 35°

Adj∠′sonastrline

(c)

𝑥 = 90° − 40°∴ 𝑥 = 50°𝑦 = 90°

Adjcomp∠′s

(d)

𝑥 + 50° + 60° = 180°𝑥 = 180° − 50° − 60°∴ 𝑥 = 70°𝑦 = 50°𝑧 = 60°

Adj∠′sonastrlineVertopp∠′sVertopp∠′s

(e) 𝑥 = 90°𝑦 = 90°

Adj∠′sonastrlineVertopp∠′s

Exercise5:

Statement Reason(a)

70° = 𝑥 + 20°∴ 𝑥 = 50°

Vertopp∠′s

(b)

𝑥 + 20° = 2𝑥 − 50°20° + 50° = 𝑥70° = 𝑥∴ 𝑥 = 70°

Vertopp∠′s

(c)

2𝑥 − 10° + 140° = 180°2𝑥 + 130° = 180°2𝑥 = 50°𝑥 = 25°

Adj∠′sonastrline

𝑥145°

40°

40°𝑥𝑦

50°60°𝑥𝑦𝑧

10°80°

𝑥𝑦

𝑥 + 20°70°

2𝑥 − 50°𝑥 + 20°

2𝑥 − 10°140°

Mathematics Grade8

15Exercise6: Statement Reason(2)

𝑥 = 88°𝑦 = 88°

Vertopp∠′sCorr∠′s;EF//GH

(3)

𝑥 + 51° = 180°∴ 𝑥 = 129°𝑦 = 100°𝑧 = 180° − 100°∴ 𝑧 = 80°

Co-int∠′s;IJ//KLCorr∠′s;IJ//KLAdj∠′sonastrline

(4)

𝑥 = 62° Alt∠′s;MN//OP

(5)

𝑥 = 71°𝑦 + 71° = 180°∴ 𝑦 = 109°

Alt∠′s;UV//WXCo-int∠′s;QR//ST

(b.1) AB//DCbecausecorrespondinganglesareequal.(b.2) ABwillnotbeparalleltoDCbecausetheco-interioranglesarenotsupplementary.(b.3) AB//DCbecausethealternatinganglesareequal.(b.4) AB//DCbecausetheco-interiorangleswillbesupplementary.

51°

100°𝑥

𝑦𝑧

I J

KL

M

N

O

P

𝑥

62°

Q

R

S

V

X

T

U

W

71°

𝑥 𝑦

E

H

G

88°

𝑥

𝑦

F