The document discusses fractions of revolutions and measuring angles. It explains that a full revolution is divided into 360 degrees. When the second hand of a clock moves from one hour to another, it turns through a fraction of a revolution or number of degrees. Examples show calculating fractions of revolutions and measuring angles in degrees using a protractor. Different types of angles such as acute, obtuse, right and reflex angles are also defined.

1. Introduction to Geometry
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2. Fraction of a revolution
When the second hand of a clock starts at 12 and moves
round until it stops at 12 again, it has gone through one
complete turn. One complete turn is called a revolution.
When the seconds hand starts at 12 and stops at 3, it has
turned through
1
4
of a revolution.
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3. Worked Example 10a
What fraction of a revolution does the seconds hand of a clock turn through when
It starts at 3 and stops at 12 It starts at 4 and stops at 8
If the seconds hand starts at 3 and stops at
12, it turns through
3
4
of a revolution.
If the seconds hand starts at 4 and stops at
8, it turns through
1
3
of a revolution.
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4. Questions
What fraction of a revolution does the second hand of a clock turn through when it (show by
drawing second hand in the circle)
a) Starts at 12 and stops at 6
Ans: Half (
1
2
) of a revolution
b) Start at 7 and stops at 10
Ans:
1
4
of a revolution
c) Start at 5 and stops at 2
Ans:
3
4
of a revolution
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5. Worked example
Where does the hand stop if it starts at 12 and turns through
1
4
of a revolution
Ans: It stops at 3
A revolution has 12 steps
1
4
of 12 is 3. 3 steps from 12 is 3.
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6. Questions
Where does the hand stop if
a) It starts at 12 and turns through
3
4
of a complete turn
Ans: It stops at 9
b) It starts at 9 and turns trough a
1
2
a complete turn
Ans: It stops at 3
c) It starts at 12 and turns through
2
3
of a revolution
Ans: It stops at 8
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7. Compass directions
The four main compass directions are North, South,
East and West.
Question
If you are facing West and turn through 1
2
3
revolutions, in Which direction are you facing?
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If you stand facing North, and turn clockwise through
1
2
a revolution, you are then facing South

8. Angles
• When the hand of a clock moves from one position to another it has turned trough an angle
• A quarter of a revolution is called a right angle
• Half of a revolution is two right angle
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Right angle Two right angles

9. Worked example
How many right angles does the second hand of a clock turn through when it starts at 3 and stops
at 12
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Ans: It turns trough three right
Questions
How many right angle does the seconds hand of a clock turn through when
a) Starts at 6 and stops at 9
b) Start at 8 and stops at 2
c) How many right angles do you turn trough if you face North and turn anticlockwise to face East?

10. Measuring angles
One complete revolution is divided into 360 parts. Each part is called a degree. 360 degrees is
3600
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Worked example
How many degrees has the seconds hand of a clock turned through
when it moves from 6 to 8?
When the hand moves from 6 to 8 it
turns through
2
12
revolution
Ans; The hand turns through
2
12
of 3600
which is 600
8
6

11. Using protractors to measure angles
A protractor looks like this
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It has the following
- Base line
- Center
- scale
From the base line, you read
anticlockwise on the scale from
00
to 700
. The angle is 700
Measure this angle

12. Questions
Measure the following angles;
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𝟓𝟎 𝟎
𝟏𝟐𝟎 𝟎

13. Worked example
a) Find the size of the angle marked p
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Angle p and 400 together make 3600
So angle p is 3600- 400 = 3200 400
P
b) Measure the following angle
This is a reflex angle and it is bigger than 3
right angles, i.e it is greater than 2700
To find this angle, we need to measure the smaller
angle, marked p
Angle p is 550
, so the reflex angle is
3600
- 550
= 3050
P

14. Drawing angles using a protractor
To draw an angle of 1200, start by drawing one arm and mark
the corner of your angle.
The corner, or point, of an angle is called vertex.
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Place the protractor as shown in the diagram. Make sure that
the vertex is at the center of the base line.
Choose the scale that starts at 00
on the drawn line, and mark the
paper next to the 1200
mark on the scale.
Remove the protractor and join your mark to the vertex.

15. Types of Angles
Any angle that is smaller than a right angle is called acute angle
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Any angle that is greater than one right angle and less than two
right angles is called an obtuse angle
Any angle that is greater than two right angles is called a reflex
angle.
A right is 900

16. Related angles
Vertically opposite angles
When two straight lines cross, four angles are formed.
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The two angles that are opposite each other are called vertically opposite angles .
Vertically opposite angles are equal
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