Cambridge IGCSE Mathematics Core and Extended

Cambridge IGCSE TM DA1501
Mathematics Cambridge IGCSE TM





Core and Extended COMPREHENSIVE REVISION GUIDE

ASTEP-BY-STEP APPROACH for Effective Mastery Mathematics

TM
Cambridge IGCSE Mathematics Core and Extended is written specifically for
students who will be sitting for the Cambridge IGCSE examination. This book provides full
coverage of the Cambridge IGCSE Mathematics syllabus 0580. The book has been written Core and Extended
with the focus on a complete revision that will put students at ease in their preparation for the
examination. This is made possible by the winning combination of comprehensive notes,
helpful tips, clear examples, carefully formulated tests and practices, exam-style COMPREHENSIVE REVISION GUIDE
questions, vivid illustrations and creative layout designs. The exam-oriented approach COMPREHENSIVE REVISION GUIDE
adopted by the writers of the books will ensure that excellent results are within the reach
of every student. TM A STEP - BY - STEP APPROACH for Effective Mastery

Special Features:
• Learning Outcomes • Test Yourself
• Flashback • Concept Map
• Maths Online • Mastery Practice
• Tips • Online Quick Quiz Cambridge IGCSE Mathematics Core and Extended
• Worked Example • Online Exam Questions
• Calculator Corner • Specimen Paper


Also available:












ACE YOUR
MATHEMATICS
Workbook
Exam
Exam


www.dickenspublishing.co.uk



DA1501 Dickens
ISBN 9781781872710
Dickens Publishing Ltd
Suite G7-G8, Davina House, 137-149 Goswell Road,
London, EC1V 7ET, United Kingdom. Based on the
Latest Syllabus
E-mail: [email protected]





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Cambridge IGCSE TM
Mathematics









Core and Extended



COMPREHENSIVE REVISION GUIDE





A STEP - BY - STEP APPROACH for Effective Mastery















© Dickens Publishing Ltd 2023
All rights reserved. No part of this book may be
reproduced, stored in a retrieval system,
or transmitted in any form or by any means,
electronic, mechanical, photocopying, recording
or otherwise, without the prior permission of
Dickens Publishing Ltd.




ISBN: 978-1-78187-271-0
First published 2023

Printed in Malaysia

ANSWERS
ANSWERS
Dickens Publishing Ltd
Suite G7-G8, Davina House, 137-149 Goswell Road,
London, EC1V 7ET, United Kingdom. Based on the
E-mail: [email protected]
Latest Syllabus


i



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Special Features in


This Book






Learning Outcomes States the learning objectives of the chapter.


Flashback Helps students to recall the basic concepts for
the chapter.

Lists the suitable websites related to the
Maths Online
chapter.


Highlights important points for students to take
Tips note of.



Consists of sample questions with complete
Worked Example
and comprehensive solutions.

Shows detailed steps in the use of scientific
calculator to solve problems under relevant
subtopics.


Provides students with the necessary practices
Test Yourself that apply the various concepts learnt in each
subtopic.

Consists of brief and concise notes that summarise
Concept Map
the concepts learnt in each chapter.


Consists of subjective questions covering all the
Mastery Practice
learning outcomes of each chapter.
Consists of past-year exam questions as further
practice which familiarise students with exam-style
questioning patterns.

Provides speedy QR-accessible exercises which
are just a scan away using your mobile phone.


Specimen P
Specimen Paperaper Provides practices of exam-oriented questions.
Specimen P
r
ape
Answers Allow students to monitor their progress.




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Contents







UNIT 1

Chapter 1 Whole Numbers 1
Chapter 2 Number Facts and Sequences 13
Chapter 3 Real Numbers 28
Chapter 4 Fractions 51
Chapter 5 Decimals 77
Chapter 6 Percentages 91
Chapter 7 Integers 102
Chapter 8 Algebraic Expressions I 115
Chapter 9 Indices 121
Chapter 10 Lines and Angles 133
Chapter 11 Polygons I 147
Chapter 12 Circles I 162

Chapter 13 Geometrical Constructions 178
Chapter 14 Statistics I 192

UNIT 2

Chapter 15 Perimeter and Area 209
Chapter 16 Solid Geometry I 220
Chapter 17 Directed Numbers 227

Chapter 18 Squares, Square Roots, Cubes and Cube Roots 240
Chapter 19 Ratio, Rate and Proportion 256
Chapter 20 Basic Measurements 279
Chapter 21 Polygons II 294
Chapter 22 Circles II 305
Chapter 23 Standard Form 326
Chapter 24 Bounds 332


UNIT 3

Chapter 25 Solid Geometry II 344
Chapter 26 Sequences 357
Chapter 27 Algebraic Expressions II 370
Chapter 28 Linear Equations 380




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Chapter 29 Pythagoras’ Theorem 394
Chapter 30 Trigonometry I 402
Chapter 31 Similarity and Congruence 417
Chapter 32 Solid Geometry III 430
Chapter 33 Coordinate Geometry 448


UNIT 4
Chapter 34 Algebraic Expressions III 473

Chapter 35 Algebraic Formulae 489
Chapter 36 Linear Inequalities 497
Chapter 37 Simultaneous Equations 509
Chapter 38 Quadratic Expressions and Equations 518
Chapter 39 Sets 538


UNIT 5
Chapter 40 Scale Drawings 558

Chapter 41 Bearing 565
Chapter 42 Trigonometry II 578
Chapter 43 Statistics II 603
Chapter 44 Personal and Small Business Finance 622


UNIT 6
Chapter 45 Graphical Representation of Inequalities 631

Chapter 46 Graphs in Practical Situations 643
Chapter 47 Function Notation 667
Chapter 48 Statistics III 690
Chapter 49 Probability 720
Chapter 50 Graphs of Functions 746
Chapter 51 Direct and Inverse Proportion in Algebraic Terms 769
Chapter 52 Transformations 779

Chapter 53 Vectors in Two Dimensions 811

SPECIMEN PAPERS

Specimen Paper 1 Core 841
Specimen Paper 2 Extended 845
Specimen Paper 3 Core 849
Specimen Paper 4 Extended 855



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11 Polygons I






P
olygons I












At the end of this chapter, you
Flashback should be able to

1. Name these shapes. recognise and name polygons.
(a) (b) draw lines of symmetry of shapes.


find the order of rotational symmetry.

state the geometric properties of
different types of triangles and
(c)
quadrilaterals and name them.

determine the sum of the angles of
a triangle and a quadrilateral.
2. State one property of square. solve problems involving triangles and
quadrilaterals.
3. State one difference between a square
and a rectangle.











Maths Online


https://thirdspacelearning.com/gcse-maths/
geometry-and-measure/regular-polygon/

http://www.math.com/tables/geometry/
polygons.htm

https://www.mathsisfun.com/geometry/
symmetry-rotational.html











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11.1 Polygons


A. Recognising polygons

A polygon is a closed plane figure bounded by three or more straight lines as its sides.

In a polygon, TIPS
(a) vertices are points where two sides meet,
(b) diagonals are lines joining two non- adjacent vertices.

For example, This is not a polygon
because it is not a closed
A
In quadrilateral ABCD, points A, B, C and shape.
D are vertices. Lines AB, BC, CD and AD
D B
are sides. Lines AC and BD are diagonals.
This is not a polygon
C
because one of its sides is
a curve, not a straight line.
B. Naming polygons

Polygons are named according to the number of sides they have.

Number of Example Number of Example
Polygon Polygon
sides of shape sides of shape



Triangle 3 Heptagon 7



Quadrilateral 4 Octagon 8




Pentagon 5 Nonagon 9




Hexagon 6 Decagon 10



C. Properties of a polygon

For a polygon,
(a) number of vertices = number of sides,
n × (n – 3)
(b) number of diagonals, Dn = ,
2
where n is the number of sides of the polygon.
(This formula is valid for certain polygons only.)




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Example 1
TIPS
(a) Name the polygon on the right.
(b) Find • A triangle does not have
(i) the number of vertices, any diagonal.
(ii) the number of diagonals • We usually use capital
of the polygon. letters to label the vertices
of a polygon.
Solution
(a) Octagon The polygon has 8 sides. 1
8 2
(b) (i) Number of vertices = Number of sides
= 8 7 3
n × (n – 3)
(ii) Number of diagonals = 6 4
2 5
8 × (8 – 3)
=
2
8 × 5
=
2
= 20

Test Yourself 11.1


1. Name each of the following polygons. 2. Determine the number of sides, vertices and
(a) (b) diagonals in each of the following polygons.
(a) (b)



(c) (d)
(c) (d)






11.2 Symmetry

A. Determining and drawing line(s) of symmetry

A shape is said to have symmetry if it has two parts that overlap each other perfectly when folded along
a line. This line is known as the line of symmetry or the axis of symmetry.
For example,
Lines of
symmetry



Line of
symmetry
(a) (b)

Diagram (a) has one line of symmetry whereas Diagram (b) has two lines of symmetry.




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Example 2
The diagram is drawn on a grid. Determine the
number of line(s) of symmetry and draw the
line(s) of symmetry of the diagram.

Solution



2 lines of symmetry






B. Completing a symmetrical figure

Example 3
Complete the following shape M
if line MN is the line of
symmetry.





N
Solution
Step  Identify the vertices of the given shape that do not lie
on the line.
Step 2 For each of these vertices, mark the point that has the
same perpendicular distance from the line.
Step 3 Join all the points marked as well as the points lying
on the line to complete the shape.

M M M







N N N
 2 3
C. Rotational symmetry of plane figures


A diagram has rotational symmetry if it still looks the same when rotated. The order of rotational
symmetry is the number of times a diagram looks the same when it is rotated through 360°. There is
a centre of rotation about which the rotational symmetry occurs.
180° 180°

A B C D A B
D C B A D C



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When we rotate the diagram through 180°, it fits exactly its original shape. When we rotate through
another 180°, it fits too. We say such diagram has rotational symmetry of order 2.
The following diagrams have rotational symmetry of other orders:

TIPS
If a figure fits its original
shape once only after a full
rotation, we say it has no
order 3 order 4 order 5
rotational symmetry.
Example 4
For each of the following diagram, state
(i) the number of lines of symmetry,
(ii) the order of rotational symmetry.
(a) (b)





Solution
(a) (i) Number of lines of symmetry = 0
(ii) Order of rotational symmetry = 2




(b) (i) Number of lines of symmetry = 6

(ii) Order of rotational symmetry = 6

Test Yourself 11.2

1. For each of the following diagrams, (c) A (d) A D
determine the number of line(s) of symmetry

and draw the line(s) of symmetry. C D
(a) (b)
B C B


3. For each of the following diagrams, state
(c) (d) (i) the number of lines of symmetry,
(ii) the order of rotational symmetry.
(a) (b)



2. Copy and complete each of the following
diagrams if AB and CD are the lines of
symmetry. (c) (d)
(a) (b) A

A B

B

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11.3 Triangles

A. Identifying triangles

P Ve rtex
Vertex
A triangle is a polygon having three sides. A triangle can be represented by
the symbol ‘∆’. For example, a triangle with vertices labelled as P, Q and R is Side Angle
Side
Angle
represented as ∆PQR.
Q R
B. Line(s) of symmetry of triangles
Example 5

Determine the number of line(s) of symmetry and draw the line(s)
of symmetry for each of the following triangles.

(a) (b)


Solution
(a) (b)




1 line of symmetry 3 lines of symmetry
C. Drawing triangles

A triangle can be drawn using a protractor and a ruler. While drawing a triangle, we usually start with
a given side as the base of the triangle, then proceed with the other given sides or angles.
Example 6
Draw triangle PQR with PQ = 6 cm, ∠QPR = 60° and
∠PQR = 40°.
Solution

Step 
Draw line PQ of length 6 cm using a ruler. P 6 cm Q
Step 2
Draw an angle of 60° at point P using a
60°
protractor. P 6 cm Q

Step 3
Draw an angle of 40° at point Q using a
60° 40°
protractor. P 6 cm Q
Step 4 R
Extend the lines drawn in steps 2 and
3 until they meet at one point. Label the
60° 40°
intersection point as R.
P 6 cm Q

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D. Stating geometric properties and types of triangles

The following shows different types of triangles and their respective geometric properties. These triangles
are categorised according to the length of sides or the size of angles.
(I) According to the length of sides:
(a) Equilateral triangle


• All the sides have the same length.
• All the angles are 60°.
• It has three lines of symmetry.

(b) Isosceles triangle
• Two of the sides have the same
length.
• The base angles are equal.
• It has one line of symmetry.

(c) Scalene triangle
• All the sides have different lengths.
• All the angles are of different sizes.
• It has no lines of symmetry.



(II) According to the size of angles:
(a) Acute-angled triangle
• All the angles are acute angles.

TIPS
Many triangles could be of
(b) Obtuse-angled triangle category (I) as well as of
category (II). For example,
• One of the angles is an obtuse
angle. an equilateral triangle is also
an acute-angled triangle. A
(c) Right-angled triangle right-angled triangle could

• One of the angles is a right angle also be an isosceles triangle.
(90°). A scalene triangle could also
be an acute-angled triangle
or an obtuse-angled triangle.

E. Sum of angles of a triangle

The sum of the angles of a triangle is 180°. These angles are known as interior angles.
For example,

y
x + y + z = 180°
z
x




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Example 7
Find the value of x in the triangle
on the right. 80°
41° x
Solution
x + 41° + 80° = 180° The sum of the angles of a triangle is 180°.
x = 180° – 41° – 80°
= 59°
F. Exterior angle of a triangle

An exterior angle of a triangle is the angle formed when one side
of a triangle is extended. Exterior angle

The exterior angle is equal to the sum of the two opposite interior angles.
For example,
a
c = a + b
b c

Example 8
Find the values of x and y. 80°
y
x
Solution
In the isosceles triangle, the base angles are equal.
x + x + 80° = 180°
2x + 80° = 180°
2x = 180° – 80°
2x = 100°
x = 50°
y = 80° + x Exterior angle = Sum of the
y = 80° + 50° opposite interior angles.
= 130°
G. Problem-solving

Example 9
P T
In the diagram, PR and ST are straight x 67°
lines. Find the value of x. Q
40°
Solution S
20°
In ∆ QRT, R
/TQR + 67° + 20° = 180°
/TQR = 180° – 67° – 20°
= 93°

/PQS = /TQR = 93° Vertically opposite angles
In ∆ PQS,
x + 40° + 93° = 180°
x = 180° – 40° – 93°
= 47°

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Test Yourself 11.3

1. For each of the following triangles, determine
whether it has line(s) of symmetry. Draw the (c) (d)

line(s) of symmetry for the triangles that 48° 28°
42°
have lines of symmetry. 56°
(a) (b)
5. Find the value of x in each of the following
triangles.

(a) 80° (b) 76°
(c) (d) x
45° x
28°

(c) x 30° (d) x
2. Draw each of the following triangles using 22° 124°
a protractor and a ruler.
x
(a) An equilateral triangle with sides 6 cm
each.
(b) An isosceles triangle with base 8 cm 6. Find the unknown angles marked in the
and base angles 58° each. following triangles.
(c) ∆KLM with KL = 6.4 cm, /LKM = 40° (a) (b)
and /KLM = 105°.
70°
(d) ∆PQR with PQ = 7.5 cm, PR = 8.4 cm
a
and /QPR = 50°. x
30°
(e) ∆XYZ with YZ = 5.6 cm, XY = 7.2 cm
and /XYZ = 60°.
(f) ∆ABC with AB = 8 cm, /BAC = 46° (c) (d)
and /ACB = 80°. (Hint: Determine the t
third angle in the triangle first.) 128° y
42°
3. Determine whether each of the following
triangles is an equilateral triangle, an
7. In each of the following triangles, find the
isosceles triangle or a scalene triangle.
value of x.
(a) (b)
(a) (b)
65°
70°
40° 60° x 72° x
(c) (d)

50° 130° (c) x (d) 38°
25° 80°
30° 28°
10°
4. Determine whether each of the following x
triangles is an acute-angled triangle, an (e) (f)
obtuse-angled triangle or a right-angled 46° 62°
triangle. 34° x 45°
(a) (b) x
50°
(g) 114° (h) x 67°
44°
47°
x 46°
32° 20°



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11.4 Quadrilaterals

A. Identifying quadrilaterals
Vertex
A quadrilateral is a polygon having four sides.
Side
Angle
B. Line(s) of symmetry of quadrilaterals


Example 10
Determine the number of line(s) of symmetry and draw the line(s)
of symmetry for each of the following quadrilaterals.

(a) (b)


Solution

(a) (b)



4 lines of symmetry 2 lines of symmetry
C. Drawing quadrilateral

Example 11
Draw a quadrilateral PQRS with PQ = 8 cm, QR = 6.4 cm,
RS = 4 cm, /PQR = 90° and /QRS = 75°.

Solution
Step 
Draw line PQ of length 8 cm using a ruler. P Q
8 cm
Step 2
Draw an angle of 90° at point Q using a
protractor. P 8 cm Q
Step 3 R
Mark point R on the arm of the angle 90°
using a ruler so that QR = 6.4 cm. 6.4 cm
P Q
8 cm
Step 4
Draw an angle of 75° at point R using a R
protractor. 75°
6.4 cm

P Q
8 cm



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Step 5
4 cm R
Mark point S on the arm of the angle 75° S
75°
so that RS = 4 cm.
6.4 cm
Step 6 P 8 cm Q
Join point S to point P.
D. Stating geometric properties and types of quadrilaterals

The following shows different types of quadrilaterals and their respective geometric properties.
(a) Rectangle

• Opposite sides have the same length and are parallel.
• All the angles are 90°.
• Diagonals are equal and bisect each other.
• It has two lines of symmetry.
(b) Square
TIPS
• All the sides have the same length.
• Opposite sides are parallel. • A square is also a rhombus.
• All the angles are 90°. This means that a square
• Diagonals are equal and bisect each other has all the properties of a
rhombus.
at 90°. • A rhombus is also a
• It has four lines of symmetry.
(c) Rhombus parallelogram. This
means that a rhombus
• All the sides have the same length. has all the properties of a
• Opposite sides are parallel. parallelogram.
• Opposite angles are equal.
• Diagonals bisect each other at 90°.
• It has two lines of symmetry.
(d) Parallelogram
• Opposite sides have the same length and are parallel.
• Opposite angles are equal.
• Diagonals bisect each other.
• It has no lines of symmetry.
(e) Trapezium
TIPS
• Only one pair of opposite sides are parallel.
• In general, it has no lines of symmetry. • Some trapeziums may
have a line of symmetry.
For example,
E. Sum of angles of a quadrilateral

The sum of the angles of a quadrilateral is 360°.
For example,
q
p + q + r + s = 360°
p r
s




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Example 12
Find the value of x in the quadrilateral.
x 108°
68°
71°
Solution
The sum of the angles of a
x + 108° + 68° + 71° = 360°
quadrilateral is 360°.
x = 360° – 108° – 68° – 71°
= 113°
F. Problem-solving
Example 13 M

In the diagram, JKLN is a quadrilateral and 70° N
L
KLM is a straight line. Find the value of y. 120°

y
K J
Solution
Sum of adjacent angles
70° + /KLN = 180° on a straight line is 180º.
/KLN = 180° – 70°
= 110°
Sum of angles of a
y + 90° + 120° + 110° = 360°
quadrilateral is 360º.
y = 360° – 90° – 120° – 110°
= 40°
Test Yourself 11.4

1. For each of the following quadrilaterals, (e) Quadrilateral KLMN with KL = 7.2 cm,
determine whether it has line(s) of symmetry. LM = 5.6 cm, /LKN = 60°, /KLM = 70°
Draw the line(s) of symmetry for the and /LMN = 100°.
quadrilaterals that have lines of symmetry. (f) Quadrilateral PQRS with PQ = 8 cm,
(a) (b) PS = 7 cm, QR = 5.4 cm, /QPS = 70°
and /PQR = 120°.

3. (a) Name the quadrilaterals that have
(c) (d)
equal interior angles.
(b) Name the quadrilaterals that have four
sides of equal length.
(c) Quadrilateral A has equal opposite
2. Draw each of the following quadrilaterals
using a protractor and a ruler. angles and its opposite sides are
(a) A square with sides 4.2 cm each. parallel and of equal length. State the
(b) A rectangle with adjacent sides 5.6 cm type of quadrilateral A.
and 4.5 cm respectively. (d) Which quadrilateral has only one pair
(c) Parallelogram ABCD with AB = 6.8 cm, of sides that are parallel?
BC = 4.6 cm, /ABC = 70° and (e) Name two common geometric properties
/BCD = 110°. of a rectangle and a parallelogram.
(d) Trapezium EFGH with two
(f) List the quadrilaterals that have their
parallel sides, EF = 6 cm and
diagonals which are equal and bisect
GH = 4 cm respectively, EH = 5.2 cm
each other.
and /FEH = /EHG = 90°.
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(g) Which quadrilaterals have its diagonals 5. Find the value of x in each of the following
that bisect each other at a right angle? diagrams.
(h) Quadrilateral B has two lines of (a) (b)
x 40°
symmetry. Determine the type of 28° x
quadrilateral B. 106°
4. Find the value of x in each of the following (c) 18° (d)
x 24° x
quadrilaterals.
(a) (b) x 37° 80° 136°
124°
x 114° 110°

145° 104°
43° (e) (f)
125° x
88°
x
43° 50°
(c) 50° (d)
x 109°
x 124°

100° 78° 50° 77°
Concept Map


Polygons I
A closed plane shape made up of three or more straight lines.



Pentagon Hexagon Heptagon Octagon







Triangle Symmetry Quadrilateral
• A closed plane shape made up A shape that has symmetry has • A closed plane shape made up
of three sides. parts that match each other of four sides.
• Sum of all angles is 180°. perfectly when folded along a line. • Sum of all angles is 360°.

Equilateral triangle Rectangle
Isosceles triangle
60°
Square
60°60°
Acute-angled Scalene triangle Rhombus
triangle

Parallelogram

Obtuse-angled
Right-angled triangle Trapezium
triangle







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Mastery Practice 11

1. The diagram is drawn on 6. In the diagram, PST Q
a grid of equal squares. is a straight line. x R
State the number of Find the value of x. 76°
line(s) of symmetry of
80° 130°
the diagram. T
P S

2. 7. In the diagram, PQS P
R R
and TQR are straight x 32°
Q
lines. Find the value 100°
of x. S
P Q T
A
8. In the diagram, ABD
S is an equilateral
D
triangle. Find the
The incomplete shaded shape is drawn on a
value of x. 50°
grid of equal squares. Complete the shape
x
if PQ and RS are the lines of symmetry of
the shape. B C
3. 9. In the diagram, PQR P
and SQT are straight
32°
lines. State the value
of x. 50°
S T
Q x

R
The diagram is drawn on a grid of equal
squares. Draw all the lines of symmetry of 10. In the diagram, ABC A
the diagram. and FED are straight 80° B C
lines. State the value x
4. R
P of x.
x Q 78° 65° 95°
F E D
46°
S
T 11. In the diagram, SRQ P
In the diagram, PQS and TQR are straight is a straight line. Find 80° 25°
lines. State the value of x. the value of x. x
R Q
5. K
S
L P
118° 12. In the diagram, QRS
x is a straight line. Find x
N M
the value of x.
In the diagram, triangle KNM is a right-
110°
angled triangle and KLM is a straight line.
Q R S
State the value of x.


160 Cambridge IGCSE Mathematics




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13. In the diagram, PQT is Q 20. In the diagram, URS P Q S
an equilateral triangle R S and QRT are straight x 160° R 65°
and PTU is a straight x lines. State the value T
line. Find the value of x. 70° of x. 40°
P T U
U
14. In the diagram, PQR is S Q
a straight line. Find the 85° 30° 21. In the diagram, PRST
25° x
value of x. R is a rhombus. QRT
x Q
P and PS are straight R S
lines. Find the value
Q of x. 30°
15. In the diagram, TSP R
is a straight line. Find 110°
P T
the value of x. x
T
S T
P 22. In the diagram, PQRS
is a square. PRU x
16. P Q
and SQT are straight y
Q lines. Find the value
of x + y.
R
P S R 50°
U
S
23. In the diagram,
PQR is an isosceles
The diagram is drawn on a grid of equal R
triangle. Find the
squares. Mark point S on the grid so that x Q
value of x.
PQRS is a parallelogram. P 254°
P
17. In the diagram, TSR S
is a straight line. Find 95° 24. In the diagram, x R
the value of x. 115° x Q PTQ is a straight 36° 110°
T line and TQRS is a
S 100°
parallelogram. Find
R P y
the value of x + y. T
18. In the diagram, find Q
x
the value of x + y. S
y T
25. In the diagram, PRS 45°
70° R
65° and QRT are straight 65°
lines. Find the value 55°
x Q
19. In the diagram, P Q of x.
x P
PQRS is a
rectangle. PR and 142°
QS are diagonals of S R
the rectangle. Find
the value of x.










Chapter 11 Polygons I 161



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