1A_Ch6(1)
1A_Ch6(2)
6.1 Basic Geometric Knowledge
A Points, Lines and Planes
B Angles
C Parallel and
Perpendicular Lines
Index
1A_Ch6(3)
6.2 Plane Figures
• Introduction to
Plane Figures
A Circles
B Triangles
C Polygons
Index
1A_Ch6(4)
6.3 Three-dimensional Figures
A Introduction
B Sketch the Two-dimensional
(2-D) Representation of
Simple Solids
Index
1A_Ch6(5)
6.4 Polyhedra
A Introduction to
Polyhedra
B Making Models of
Polyhedra
Index
6.1 Basic Geometric Knowledge 1A_Ch6(6)
A) Points, Lines and Planes
1. Refer to the right figure. A
i. A is a point.
ii. BE is a line.
B E
iii. CD is a line segment, C D
C and D are called
the end points of that line segment.
iv. Figure ACD represents a plane.
Index
6.1 Basic Geometric Knowledge 1A_Ch6(7)
Example
A) Points, Lines and Planes
2. Relations among Points, Lines and Planes
i. The straight line in the common part of two
planes is called the line of intersection.
ii. The two lines meet each other at a point,
that point is called the point of intersection.
line
point of
intersection
line
Index 6.1 Index
6.1 Basic Geometric Knowledge 1A_Ch6(8)
M
(a) Name all the line segments and P
planes in the given figure. O
(b) Which point is the point of
intersection of MQ and PN ? N
Q
(a) Line segments : MN, MO, NO, OP, OQ, PQ, MQ, NP
Planes : MNO, OPQ
(b) Point of intersection of MQ and PN : O
Key Concept 6.1.1
Index
6.1 Basic Geometric Knowledge 1A_Ch6(9)
Example
B) Types of Angles
Angles can be classified according to their ‘sizes’ as follows:
Acute Right Obtuse Straight Reflex Round
angle angle angle angle angle angle
• Note : In a figure, a right angle is usually indicated by the symbol
‘ ’ but not an arc ‘ ’.
Index 6.1 Index
6.1 Basic Geometric Knowledge 1A_Ch6(10)
What kind of angle is each of the following A
angles in the given figure?
180 °
(a) ∠AOB (b) ∠BOD C O
140°
(a) ∠AOB = 180° D B
∴ ∠AOB is a straight angle.
(b) ∠BOD = ∠BOC – ∠COD
= 140° – 90°
= 50°
∴ ∠BOD is an acute angle.
Index
6.1 Basic Geometric Knowledge 1A_Ch6(11)
What kind of angle is each of the
following angles in the given figure?
(a) ∠AFE (b) ∠AHD (c) ∠EFB
(a) ∠AFE = 120° (b) ∠AHD = 90°
∴ ∠AFE is an obtuse angle. ∴ ∠AHD is a right angle.
(c) ∠EFB = ∠EFA + ∠AFB
Fulfill Exercise Objective
= 120° + 60°
Classify an angle.
= 180°
∴ ∠EFB is a straight angle. Key Concept 6.1.2
Index
6.1 Basic Geometric Knowledge 1A_Ch6(12)
Example
C) Parallel and Perpendicular Lines
i. RS and TU are a pair of parallel R T
lines. We can write RS // TU.
ii. AB and TU are a pair of B
A
perpendicular lines. We can
S U
write AB ⊥ TU.
iii. Parallel and perpendicular lines can be constructed by
a ruler and a set square.
Index 6.1 Index
6.1 Basic Geometric Knowledge 1A_Ch6(13)
A
G
Name all the parallel lines and F
perpendicular lines in the given B
figure.
E
D
C
Parallel lines : AG // DC // FE, GF // DE
Perpendicular lines : AB ⊥ BC, AG ⊥ GF, GF ⊥ FE, FE
⊥ ED, ED ⊥ DC
Key Concept 6.1.3
Index
6.2 Plane Figures 1A_Ch6(14)
Introduction to Plane Figures
1. A geometric figure formed by points, lines and planes
lying in the same plane is called a plane figure.
E.g.
Circle Triangle Polygon
Index 6.2 Index
6.2 Plane Figures 1A_Ch6(15)
P
A) Circles
1. O is the centre.
A O B
2. OP is the radius.
3. AOB is the diameter. Q
4. The curve AQBPA which forms the entire circle is
the circumference.
5. The curve AP is part of the circumference, called an
arc of the circle.
6. Circles and arcs can be constructed by a pair of
compasses.
Index
6.2 Plane Figures 1A_Ch6(16)
Example
A) Circles arc
radius
Note :
i. ‘Circumference’, ‘radius’ and ‘diameter’ can represent
lengths as well.
ii. Diameter = 2 × radius
Index 6.2 Index
6.2 Plane Figures 1A_Ch6(17)
It is known that O is the centre of each of the following circles,
find the values of the unknowns.
(a) (b)
4.5 m
12 cm
O
O
ym
(a) Diameter = 12 cm (b) Radius = 4.5 m
∴ x = 12 ÷ 2 ∴ y = 4.5 × 2
Key Concept 6.2.2
= 6 = 9
Index
6.2 Plane Figures 1A_Ch6(18)
Example
A
B) Triangles
C
B
1. In the above triangle,
i. the line segments AB, BC and CA are called the
sides of △ABC,
ii. points A, B, C are called the vertices (singular :
vertex) of △ABC.
2. The sum of the three angles of a triangle is 180°.
Index
6.2 Plane Figures 1A_Ch6(19)
B) Triangles
3. Classification of triangles :
Acute-angled Right-angled Obtuse-angled
triangle triangle triangle
Index
6.2 Plane Figures 1A_Ch6(20)
Example
B) Triangles
3. Classification of triangles :
Scalene Isosceles Equilateral
triangle triangle triangle
Index
6.2 Plane Figures 1A_Ch6(21)
Example
B) Triangles
4. Triangles can be constructed by a protractor and a pair
of compasses etc. according to given conditions:
i. Given three sides of a triangle.
ii. Given two sides and the included angle of a
triangle.
Index 6.2 Index
6.2 Plane Figures 1A_Ch6(22)
Find the unknown angle a in the figure.
a + 120° + 40° = 180°
a + 160° = 180°
a = 180° – 160°
= 20°
Index
6.2 Plane Figures 1A_Ch6(23)
Find the unknowns x and y in
△ABC as shown.
In △ABD, In △ABC,
x + 62° + 90° = 180° 28° + 62° + 48° + y = 180°
x + 152° = 180° 138° + y = 180°
x = 180° – 152° y = 180° – 138°
= 28° = 42°
Fulfill Exercise Objective
Key Concept 6.2.3
Find an unknown angle in a triangle.
Index
6.2 Plane Figures 1A_Ch6(24)
A B C D
For the above triangles A, B, C and D, identify
(a) scalene obtuse-angled triangle?
(b) isosceles acute-angled triangle?
(a) C
(b) D Key Concept 6.2.4
Index
6.2 Plane Figures 1A_Ch6(25)
Construct △ABC, where AB = 4 cm, BC = 3 cm and
AC = 3.5 cm.
Steps :
1. Use a ruler to draw a line segment AB of 4
length cm.
2. With centre at A and radius 3.5 cm, use a pair
of compasses to draw an arc.
3. With centre at B and radius 3 cm, use a pair of
compasses to draw another arc.
4. The two arcs drawn should meet at C.
5. Join AC, then BC. △ABC is drawn.
Fulfill Exercise Objective
Construct a triangle.
Index
6.2 Plane Figures 1A_Ch6(26)
Construct △PQR, where PQ = 3 cm, ∠RPQ = 50° and
RP = 4 cm.
Steps :
1. Use a ruler to draw a line segment PQ
of length 3 cm.
2. Use a protractor to draw ∠TPQ that
measures 50°.
3. Use a ruler to mark a point R on PT
produced such that RP = 4 cm.
4. Join QR, then △PQR is drawn.
Fulfill Exercise Objective
Key Concept 6.2.6
Construct a triangle.
Index
6.2 Plane Figures 1A_Ch6(27)
C) Polygons
1. A plane figure formed by 3 or more line segments is
called a polygon.
2. A polygon is usually named by the number of its sides
or n-sided polygon (n is whole number).
Index
6.2 Plane Figures 1A_Ch6(28)
C) Polygons
vertex
side
3. The line segments that form a polygon are called sides
of the polygon.
4. The point where two adjacent sides meet is called a
vertex of the polygon.
5. The line segment joining two non-adjacent vertices is
called a diagonal.
Index
6.2 Plane Figures 1A_Ch6(29)
Example
C) Polygons
Classification of polygons :
Equilateral polygon
Regular polygon
Equiangular polygon
Index 6.2 Index
6.2 Plane Figures 1A_Ch6(30)
For each of the following polygons, state whether it is
(a) an equilateral polygon; (b) an equiangular polygon;
(c) a regular polygon.
A B C
(a) A, C
(b) A, B Key Concept 6.2.8
(c) A
Index
6.3 Three-dimensional Figures 1A_Ch6(31)
A) Introduction
1. A solid is an object that occupies space.
2. The surfaces of a solid are called faces.
3. The line segment on a solid that is formed by any two
intersecting faces is called an edge.
4. A point that is formed by 3 or more intersecting faces
face
on a solid is called a vertex.
vertex
edge
Index 6.3 Index
6.3 Three-dimensional Figures 1A_Ch6(32)
Example
B) Sketch the Two-dimensional (2-D) Representation of
Simple Solids
1. We can use solid and dotted lines to
draw rough 2-D figures of solids on a
plane.
2. We can also use isometric drawings to draw more
accurate 2-D figures of solids on a plane.
Isometric Isometric
dotted paper grid paper
Index
6.3 Three-dimensional Figures 1A_Ch6(33)
Example
B) Sketch the Two-dimensional (2-D) Representation of
Simple Solids
The face obtained by cutting a solid along a certain plane
is called a cross-section of the solid. If we cut the solid at
different positions, we may obtain different cross-sections.
Different
Cross-sections
Note : If we obtain the same cross-sections by cutting a solid along
certain direction, then the cross-sections are called uniform
cross-sections.
Index 6.3 Index
6.3 Three-dimensional Figures 1A_Ch6(34)
4 cm
Use an isometric dotted paper to draw
the 2-D representation of the box.
8 cm
8 cm
2 cm
Index
6.3 Three-dimensional Figures 1A_Ch6(35)
Use an isometric grid paper to 6 cm 4 cm
draw the 2-D representation of the
given solid. 4 cm
6 cm
6 cm
2 cm
Key Concept 6.3.2
Index
6.3 Three-dimensional Figures 1A_Ch6(36)
Which of the following faces represents the
cross-section of the given solid when it is
cut vertically along the blue line?
A B C
The cross-section is B.
Index
6.3 Three-dimensional Figures 1A_Ch6(37)
Draw the cross-section of the given
solid when it is cut horizontally along
the yellow line.
The cross-section is :
Fulfill Exercise Objective Key Concept 6.3.3
Draw the cross-section of a simple solid.
Index
6.4 Polyhedra 1A_Ch6(38)
Example
A) Introduction to Polyhedra
If all the faces of a solid are polygons, then that solid is
called a polyhedron.
Note : The polyhedra can be named by their numbers of faces.
Index 6.4 Index
6.4 Polyhedra 1A_Ch6(39)
Determine which of the following solids is not a polyhedron.
A B C D
B
Key Concept 6.4.1
Index
6.4 Polyhedra 1A_Ch6(40)
Example
B) Making Models of Polyhedra
We can use a net to make a model of polyhedron.
For example, the net in Fig.(a) can be folded up to make a
model of the polyhedron in Fig.(b).
(a) (b)
Index 6.4 Index
6.4 Polyhedra 1A_Ch6(41)
The diagram on the right is a polyhedron.
Which of the following net do you think can
make that polyhedron?
A B C
Key Concept 6.4.2
A
Index