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
