Lines and angles

Chapter 8

LINES AND ANGLES

Pn Sazazila Abdullah

CONTENTS

Lines and angles
Angles related to intersecting lines

Angles related to parallel lines and transversals

Why study this chapter ?

- Lines and angles constitute the basic knowledge in
geometry and construction .

Lines and angles are used in various fields . In engineering ,
lines and angles provide the visual beauty and stability to the

structure of a building .
In visual art , lines and angles produce stunning patterns .

Word link

Intersecting lines - garis bersilang
Transversals - garis rentas lintang
Parallel lines - garis selari
Perpendicular lines - garis serenjang

CongruTeintcley t-ekxetkongruenan
Perpendaidcudliatriobnisector - pembahagi dua sama serenjang

Angle bisector - pembahagi dua sama sudut
Adjacent angle - sudut bersebelahan
Vertically opposite angle - sudut bertentang bucu
Angle of elevation - sudut dongak
Conjugate angle - sudut konjugat

Word link

Interior angles - sudut pedalaman
Complementary angles - sudut pelengkap
Supplementary angles - sudut penggenap
Reflex angle - sudut refleks

AlternatTeitalengtleexst- sudut selang-seli
Correspaodnddiintigonangles - sudut sepadan

Angle of depression - sudut tunduk
Line segment - tembereng garis

Lines and angles

An angle is denoted using the symbol ‘  ‘

and capital letters at the vertex and at the ends of the two
arms of the angle . For instance ,

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Lines and angles

An angle is denoted using the symbol ‘  ‘

and capital letters at the vertex and at the ends of the two
arms of the angle . For instance ,

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How do you estimate and measure the length of a line
segment and the size of an angle ?

The size of an angle can be estimated by recognising the right
angle first . For instance ,

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How do you estimate and measure the length of a line
segment and the size of an angle ?

From what you have learnt , the size of an angle can be

measured more accurately using a protractor .

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addition ABC  135

What are the properties of the angle on a straight line , a
reflex angle and the angle of one whole turn ?

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The angle on a straight line is 180 . The sum of angles on a
straight line is 180 .

What are the properties of the angle on a straight line , a
reflex angle and the angle of one whole turn ?

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A reflex angle is an angle with a size more than 180 and less
than 360 .

What are the properties of the angle on a straight line , a
reflex angle and the angle of one whole turn ?

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addition

The angle of one whole turn is 360 . The sum of angles at a
point is 360 .

What are the properties of complementary angles ,
supplementary angles and conjugate angles ?

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The sum of the two angles is always 90 . These angles are

known as complementary angles .

What are the properties of complementary angles ,
supplementary angles and conjugate angles ?

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The sum of the two angles is always 180 . These angles are

known as supplementary angles .

What are the properties of complementary angles ,
supplementary angles and conjugate angles ?

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The sum of the two angles is always 360 . These angles are

known as conjugate angles .

How do you solve problems?

Example 1
In the diagram , p and 54 are complemetary angles . It is given
that p and q are supplementary angles and the angles
conjugate to r is 260 . Calculate the value of p , q, r, and s .

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addition

How do you solve problems?

Solution

p + 54 = 90 .... complementary angles

p = 90 - 54 = 36

p + q = 180 .... suplementary angles
36 + qTi=tle18t0ext
q = 18a0d-d3it6io=n=144

r + 260 = 360 ... conjugate angles

r = 360 - 260 = 100

s + 100 + 144 +36 + 54 = 360
s = 360 - 334 = 26

Angles related to intersecting Lines

What are vertically opposite angles and adjacent angles at
intersecting lines ?

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Vertically opposite angle at intersecting lines is 180.

Angles related to intersecting Lines

What are vertically opposite angles and adjacent angles at
intersecting lines ?

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The sum of adjacent angles at intersecting lines is 180 .

a  c a  d  180 c  b  180
b  d d  c  180 b  a  180

Angles related to intersecting Lines

What are vertically opposite angles and adjacent angles at
intersecting lines ?

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If two intersecting lines are perpendicular to each other ,
then all angles at the intersecting lines have the same
size of 90 .

How do you solve problems?
Example 2
In the diagram , POR and TOQ are straight lines . Find the
value of x and y .

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addition

How do you solve problems?

Solution

POT  72

x + 60 = 72
x = 72 - 60 = 12

TOTRaidtldei7ttie2oxnt 180
TOR  180  72  108

y + y = 108
2y = 108
y = 54

Angles related to Parallel lines and transversals

What are parallel lines and transversals?

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addition

Parallel lines can be denoted using arrow heads → as
shown in the photograph .

Angles related to Parallel lines and transversals

What are parallel lines and transversals?

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A straight line that intersects two or more straight lines is
known as transversal . For instance , straight line AB
shows in the photograph is a transversal .

Example 3
In the diagram , determine if the line labelled as PQ is a
transversal or not

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Solution :
Yes

What are corresponding angles , alternate angles , and
interior angles?

When a transversal intersects two or
more parallel lines,

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What are corresponding angles , alternate angles , and
interior angles?

When a transversal intersects two or
more parallel lines,

How do you determine whether two straight lines are

Example 4 parallel?

In the diagram , determine whether straight line PQ and

straight line RS are parallel.

i)

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addition

Solution :
PQ and RS are parallel

How do you determine whether two straight lines are

Example 4 parallel?

In the diagram , determine whether straight line PQ and

straight line RS are parallel.

ii)

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addition

Solution :

PQ and RS are not parallel because 42 128  170  180

How do you determine the values of angles related to
parallel lines and transversals?

Example 5
In the diagram , PQ , RS and TU are parallel lines . Find the
values of a , b ,c .

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addition

Solution :
a = 60 .... alternate angles

How do you determine the values of angles related to
parallel lines and transversals?

Example 5
In the diagram , PQ , RS and TU are parallel lines . Find the
values of a , b ,c .

Title text
addition

Solution :
b = 118 .... corresponding angles

How do you determine the values of angles related to
parallel lines and transversals?

Example 5

In the diagram , PQ , RS and TU are parallel lines . Find the

values of a , b ,c .

Title text
Solutioan:ddition

b+c = 180(interior angles)

118 + c = 180

c = 180-118 = 62

Example 6
In the diagram , PQR and UTS are straight lines . Find the
values of x and y .

SolutionT: itle text
135 + x =ad1d8i0ti(oinnterior angles)

x = 180 - 135 = 45
y= 180 - 45 - 42 = 93

What are angles of elevation and depression?

When a person sees an
object which is at a
higher level , the angle

betweTeintlethteelxinte of sight
and thaedhdoitrizoonntal line is

known as the angle of
elevation .

What are angles of elevation and depression?

When a person sees an
object which is at a
lower level , the angle

betweTeintlethteelxinte of sight
and thaedhdoitrizoonntal line is

known as the angle of
depression .

Example 7

The diagram shows the position of

Devi and Umi at two buildings on a

horizonTtaitl lgeroteuxndt . Draw and label
a) anglaedadtiotiodnenote the angle of

elevation of Umi from Devi

b) angle b to denote the angle of

depression of Devi from Umi .

Example 7 Solution :

The diagram shows the position of

Devi and Umi at two buildings on a

horizonTtaitl lgeroteuxndt . Draw and label
a) anglaedadtiotiodnenote the angle of

elevation of Umi from Devi

Example 6 Solution :

The diagram shows the position of

Devi and Umi at two buildings on a

horizonTtaitl lgeroteuxndt . Draw and label
b) anglaedbdtiotiodnenote the angle of

depression of Devi from Umi .