1. The document is a revision guide for mathematics chapters 9-12 covering topics like angles, parallel and perpendicular lines, polygons, area, perimeter, and geometric solids. 2. It provides definitions, diagrams, and methods for determining properties of different shapes as well as calculating measures like area, perimeter, and volume. 3. Formulas and step-by-step processes are given for finding missing values of angles, lengths of sides, areas of triangles, parallelograms, trapezoids, and volumes of cubes and cuboids.

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Mathematics (Chapter 9)
9.1 Angles
To recognise angles
The unit ofmeasurement forangles isthe degree(°).
To denote & label angles
In diagram 1, shown an angle is formed when two lines meet at a point. This point is called vertex.
The angle in the following diagram labeled as:
i. 𝐵̂ or 𝐴𝐵̂ 𝐶 or 𝐶𝐵̂ 𝐴
ii. ∠B or ∠ ABC or ∠ CBA
To compare and classify angles
Name of angle Value of angle Diagram of angle
Acute angle 0° < 90°
Right angle =90°
Obtuse angle 90° < 180°
Straight angle = 180°
Reflex angle 180° < 360°
Whole angle =360°
Determine the angle on a straight line equals 180°.
1. The sum ofangles onastraightlineis180˚.
2. In diagram 2, a + b + c = 180°.
3. The value of any angle on a straight line can be found by
subtracting the angle given from 180°.
Determine the angle on a complete turn equals 360°.
1. The sum ofangles onastraightlineis360˚.
2. In diagram 3, a + b + c = 360°.
3. The value of any angle on a whole turn can be found
by subtracting the angle given from 360°.
9.2 Parallel and perpendicular lines
To determine parallel lines
Table 1

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Parallel lines are: a) two straight lines do not intersect or meet a point.
b) distance between the lines are always the same.
In diagram 4, shown arrows are parallel.
AB is parallel to CD (AB//CD)
PQ is parallel to RS (PQ//RS)
To determine perpendicular lines
In diagram 5, shown a perpendicular lines.
Perpendicular lines are two straight lines intersect at a right angle.
AB isperpendicular to CD (AB CD)
To state the angle formed by perpendicular line
In diagram 6, shown two straight lines are intersecting.
Line AB intersects with line CD at point O.
The angles formed are ∠ AOC, ∠ BOC, ∠DOB, and ∠ DOA
∠ AOC = 90°, ∠ BOC= 90°, ∠ DOB= 90°, and ∠ DOA= 90°.
9.3 Intersecting lines
To identify intersecting lines
Intersecting lines are two straight lines meet a point.
The point is called point of intersection.
In diagram 7, AB intersects CD at point O.
Differences of parallel, perpendicular and intersecting lines
Lines intersect distance meet a point right angle
parallel lines - Always same - -
perpendicular lines yes - yes yes
intersecting lines - - yes yes
9.4 Verticallyopposite, adjacent, complementary and supplementary angles
To determine properties of vertically opposite angles
Table 2

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Vertically opposite angles are formed by two intersecting lines.
a and c are vertically opposite angles (a=c)
b and d are vertically opposite angles (b=d)
To determine properties of adjacent angles
In diagram 9, shown a adjacent angles.
Adjacent angles are two angles which is side by
side with a common vertex and a common side.
In the following diagram, ∠ABC and ∠CBD are side by side.
Point B is the common vertex. BC is the common side.
To determine properties of complementary angles
In diagram 10, shown a complementary angles.
Complementary angles are two angles whose sum is 90°.
a+b=90°.
To determine properties of supplementary angles
In diagram 10, shown a supplementary angles.
Supplementary angles are
two angles whose sum is 180°.
a+b=180°.
----------------------------------------------- Mathematics (Chapter 9) ------------------------------------------
Mathematics (Chapter 10)
10.1 Polygons
To recognise polygons
1. A polygon is a closedshape formed by line
segments.
2. A polygon is classified according to its number of
sides.
3. In figure 1, shown example of polygon and their
sides.

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To determine the number of side, vertex and diagonals according to example polygon
1. A vertex is a point where two sides meet.
2. A diagonal is a line joining vertices which are not next to each other.
3. In figure 2, shown the example of polygon and their sides, angles, vertices and diagonals.
10.2 Symmetry
To determine and drawing the line(s) of symmetry of shape
1. A shape has symmetry if one half of the shape can fit
exactly over the other half.
2. A shape can have one or more linens of symmetry.
3. In figure 3, shown a heart which has 1 line of symmetry.
4. In figure 4, shown a circle which has infinite of lines of
symmetry.
10.3 Triangles
To determine and drawing the line(s) of symmetryof triangle
1. In figure 5, shown an equilateral triangle which has 3
lines of symmetry.
2. In figure 6, shown an isosceles triangle which has 1 line
of symmetry.
3. A triangle is a polygon with 3 straight sides and 3 vertices.
Geometric properties and name of triangles
1. A triangle is a three-sided polygon.
2. A triangle is classified based on:
a) The length of its sides (Figure 7)
b) The size of its angles (Figure 8&9)
To solve problems involving triangles
1. The sum of the angles of a triangle is 180°.

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2. This is an example of following example 1.
10.4 Quadrilaterals
To determine and drawing the line(s) of symmetryof quadrilateral
1. A quadrilateral is a polygon with 4 straight sides and 4 vertices.
2. In figure 10, shown example of quadrilateral and their symmetry.
Quadrilaterals Symmetry of quadrilaterals (diagram) Number of line(s) of symmetry
Square 4
Rectangle 2
Parallelogram 0
Rhombus 2
Trapezium 1
Trapezium 0
Kite 1
Geometric properties and name of quadrilateral
In figure 11, shown example of quadrilateral and their properties.
Quadrilaterals Properties
Square
Rectangle
Parallelogram
Rhombus
Trapezium
Kite
Figure 10
Figure 11

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To solve problems involving quadrilaterals
1. The sum of the angles of a quadrilateral is 360°.
2. This is an example of following example 2.
-------------------------------------------- Mathematics (Chapter 10) -------------------------------------------
Mathematics (Chapter 11)
11.1 Perimeter and area
To identify the perimeter of a region
1. Perimeter is total distance around the outer edge of a figure.
2. The perimeter of figure 12 is the total length of all the sides.
3. The perimeter of figure 12 is (8+10+2+5+5+6) cm.
To find the perimeter of a region enclosed by straight lines
The perimeter of a region enclosed by straight lines can be found by adding the length of all the
sides of the region.
11.2 Area of rectangles
To estimate the area of a shape
1. Area is the amount of the surface covered.
2. A unit square is a square of length 1 unit.
3. The area of a unit square = 1 unit X 1 unit
= 1 unit2
4. Estimation of the area of any shape can be done by using unit square.
5. The standard units for measuring area are square centimeter (𝐜𝐦2
) and square meter ( 𝐦2
).
To find the area of a rectangle
Area of a rectangle (cm2
) = lengthX breadth

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11.3 Area of triangles, parallelograms and trapeziums
To identify the height and bases of triangles, parallelograms and trapeziums
1. The height of a shape is the perpendicular distance from the base to the highest point.
2. Height and base must meet at right angle (90°).
3. In figure 13, shown the height and bases of triangles, parallelograms and trapeziums.
To find the areas of triangles, parallelograms and trapeziums
1. Area of triangles = ½Xbase(b)Xheight(h) or base(b)Xheight(h)÷2
(because it is the half of square or rectangular)
2. Area of parallelograms = area of rectangle (because like a rectangular)
3. Area of parallelograms = base(b)Xheight(h)
4. Area of trapeziums = sum of parallel sides (sum of two bases)÷2Xheight(h)
or ½X sum of parallel sides (sum of two bases)Xheight
5. Area of trapeziums = separate or identify it into two triangle and one rectangular to find the area
of each shape and add up the three areas.
To find the height or base of triangles, parallelograms and trapeziums
1. Height of triangle =
2Xarea of triangle
base
2. Base of triangle =
2Xarea of triangle
height
3. Height of parallelogram =
area of parallelogram
base
4. Base of parallelogram =
area of parallelogram
height
5. Height of trapezium =
2Xarea of trapezium
base
6. (Sum of two parallel sides) Base of trapezium =
2Xarea of trapezium
height
To find the areas of figures are made up of triangles, parallelograms or trapeziums

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To find the areas of such figures, we must:
a) Separate or identify it into triangles, parallelograms or trapeziums.
b) Find the area of each shape.
c) Add up all the areas.
--------------------------------------------- Mathematics (Chapter 11) -----------------------------------------

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Mathematics (Chapter 12)
12.1 Geometric Solids
To identify geometric solids
1. Geometric solid are solids that have specific shapes.
2. A solid is a three-dimensional (3D) object that has length, widthand height.
3. Every geometric solid has a fixed number of edges, vertices and surfaces. (Refer figure 14)
4. If a solid has a curved surface, it has no faces.
5. Three example of geometric solid:
a) Cube b) Cuboid c) Cone
6. In figure 14, shown many examples of solid and their number of edges, vertex and face.
Name of solid Diagram of solid Edges Vertex Flat face Curved face
Cube 12 8 6 0
Cuboid 12 8 6 0
Pyramid 8 5 5 0
Cylinder 0 0 2 1
Cone 0 1 1 1
Sphere 0 0 0 1
To state the geometric properties of cubes and cuboids
1. A cube is a solid has six square faces are same size. (Refer figure 15)
2. A cuboid is a solid has sixrectangular faces. (Refer figure 16)
3. The edge of a cube or a cuboid is the line where the faces meet.
4. The vertex of a cube or a cuboid is the point where the edges
meet.
Figure 14

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5. In figure 15&16, shown a cube and a cuboid edges, vertices and surfaces.
12.2 Volume of Cuboids
To find the volume of a solid
1. Volume is the space taken up by a solid.
2. A unit cube (1 unit3
) is volume of a cube of 1 unit length.
3. The volume of a unit cube = 1 unit X 1 unit X 1 unit
= 1 unit3
4. The standard units for measuring volume are cubic centimeter (𝐜𝐦3
) and cubic meter ( 𝐦3
).
To find the volume of cubes and cuboids
Volume of cubes and cuboids (cm3
) = length X width X height
------------------------------------------------ Mathematics (Chapter 12) ---------------------------------------