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GENERAL ANGLE PROPERTIES OF GEOMETRIC FIGURES
Definitions
Acute angles
These are angles that are less than 900.
Obtuse angles
These are angles which lie between 900 and 1800
Reflex angles
These are angles greater than 1800 but less than 3600
Right angles
They are angles with exactly 900
They are also called perpendicular angles
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Exercise
1. Identify which of the following are acute, obtuse, and reflex angles.
i. 350 v. 1250
ii. 1750 vi. 950
iii. 2000 vii. 500
iv. 1100 viii. 3100
2. Measure the following angles using a protractor and indicate the type of angle.
3. Draw the following angles.
i. 200 iv. 1050 vii. 2750
ii. 420 v. 1700 viii. 3050
iii. 800 vi. 2000
Activity
Immaculate finds out the favourite sports for members of her class. She works out the angles in
the list shown below for a pie chart.
Draw the pie chart.
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Sports Angles
Football 1100
Swimming 700
Tennis 800
Rugby 400
Hockey 300
Badminton 100
Others 200
Exercise
The table below shows marks obtained by 5 students
Students Marks
Rose 50
Trace 80
Jane 90
Rogers 43
James 19
Draw a pie chat to represent the above information.
Angles on a straight line
Activity
i. Draw a straight line
ii. Draw any angle on the line without using a protractor
iii. Measure the sizes of the two angles
iv. Do they add up to 1800?
Conclusion
Angles on a straight line will always add up to 1800
Two angles that add up to 1800 are called supplementary angles.
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Complementary angles
Two angles which add up to 900 are called complementary angles
Activity
Exercise
By giving reasons, find the missing angles.
Angles at a point
These arise when two or more lines meet.
Activity
Draw any two intersecting lines such as the one given below.
Note:
The angle that are equal are called vertically opposite angles
When two lines intersect, the opposite angles are equal.
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Exercise
Find the unknown angles.
Give a reason for each answer.
Activity
What angle does a wheel go through after turning?
i. 2 revolutions iii. 3
5 of a revolution
ii. 1
4 of a revolution 1v. 3 revolutions
Parallel lines
Parallel lines are lines that never meet and the perpendicular distance between them is always
the same (constant)
If two lines AB and CD are parallel, then we can express this mathematically as AB CD and
geometrically as
Angle formed on parallel lines and a transversal
Activity
Draw any two horizontal parallel lines
Draw a straight line cutting across the two lines using a straight ruler
The line that cuts the two lines is called a transversal line
Definition a transversal line is a line that intersects parallel lines
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On your diagram several important angle properties arise
Measure the following angles as indicated below.
The above angles are called alternate angles
What do you notice about angles?
i. b and f ii. d and h
ii. a and e iv. c and g
The above angles are called corresponding angles
What do you notice when you add the following angles
i. c and e
ii. d and f
The angles add up to 1800 and are called co-interior angles
In summary
The corresponding angles
These angles are in corresponding position; they are all to the right of the transversal and
below the parallel lines. (The simplest way to recognize corresponding angles is to look for F
shape)
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The alternate angles
The pair of shaded angles between the parallel lines and on alternate sided of the transversal.
(The simplest way to recognize alternate angles is to look for a Z or N shape)
Co-interior angles
In the diagram the shaded angles are on the same side of a transversal and “inside” the parallel
lines.
a +b = 1800
The simplest way to recognize a pair of Co-interior angles is to look for U or C shape.
Exercise
While giving reasons, find the missing angles.
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Polygons
This comes from the two Greek works, poly that means many and gon meaning angle.
These are plane figures bounded by line segments.
A plane is a flat surface in 2 dimensions.
Common polygons include
Triangle (3 sides)
Quadrilateral (4 sides)
Pentagon (5sides)
Hexagon (6 sides)
Note:
Polygons can be regular or irregular
A regular polygon has all its sides and all its angles equal
Interior and exterior angles
Interior angles are those found inside a polygon while exterior angles are those got by
extending the sided of a polygon; they are found outside the polygon.
General properties of polygons
Triangle
A triangle is a 3 sided polygon with three interior angles and 3 exterior angles at each of three
vertices.
A vertex is a point where two or more lines meet
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Vertices is the plural of vertex
Activity
i. Draw any 3 triangles
ii. Measure the size of the angles and find the interior angle sum of each of the triangle
iii. What do you notice?
iv. Does the interior angle sum in each triangle equal to 1800
Conclusion
The sum of the interior angle sum of a triangle is 1800
Types of triangles
Acute angled triangle
Where all angles in the triangle are acute
Obtuse angled triangle
One of the angles is greater than 900 but bless than 1800
Right angled triangle
One of the angles is 900
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Special triangles
Equilateral triangle
All its sides and interior angles are equal
Have three lines of symmetry
Activity
Construct an equilateral triangle of side 6cm on a piece of paper
Measure the size of its interior angle. Are they all equal?
Cut your triangle out.
Fold to see how many lines of symmetry it has
Isosceles triangle
It has two opposite equal sides and the angles opposite these two sides are also equal to each
other.
Has one line of symmetry.
Activity
Construct a triangle ABC, AB= AC = 6cm. angle CAB = 300. Measure angles
i. ABC
ii. BCA
What do you notice about the angles?
Which angles are equal?
Which sides are equal?
Cut out your triangle
Fold to see how many lines of symmetry it has
Rhombus
All sides are equal and opposite sides are parallel.
It has two lines of symmetry which are its diagonals
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Angle BAD + angle ADC =1800
Activity
Construct a rhombus ABCD whose sides are 6cm and angle BAD = 600
What angle do the diagonals form?
Measure the distance from the various corners to where the diagonals intersect.
What do you notice?
Parallelogram
Opposite sides are equal and parallel
Have two lines of symmetry
Activity
Construct a parallelogram PQRS of side 6cm and 4cm and angle SPQ = 700
Measure the distance from where the diagonals intersect.
What do you notice?
Square
All sides are equal
Diagonals intersect at right angles
It also has 4 lines of symmetry
Rectangle
Opposite sides are equal
Have two lines of symmetry
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Kite
Consists of two isosceles triangles which share the same base
Has one line of symmetry
Exercise
With reasons for your calculations, find the missing angles marked with letters
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Interior angles
Activity
Draw a pentagon ABCDE
Draw diagonals from one vertex to the opposite vertices
How many triangles are formed?
What is the interior angle sum of a triangle?
Find the interior angle sum of the pentagon.
A pentagon has 3 triangles and therefore; interior angle sum
=3 x 1800
=7200
Exercise
Find the interior angle sum of a hexagon.
Using the formula to find interior angle sum of polygons
The sum of the interior angles of a polygon of n sides is (2n-2) x 1800 or (2n -4) x 900
Where n is the number of sides of the polygon.
Exterior angles
The angle between the extended side of the polygon and the polygon itself is called the
exterior angle.
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For any regular polygon the sum of exterior angles is 3600
The sum of exterior angles can be used to find the size of an exterior angle of any regular
polygon.
Exterior angle =3600
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑖𝑑𝑒𝑠 𝑜𝑓 𝑝𝑜𝑙𝑦𝑔𝑜𝑛
Interior angle + exterior angle = 1800 (angle on a straight line)
Exercise
1. Copy and complete the table below
No. of sides No. of triangles Sum of interior angles Size of interior angle
3 1 1800 1200
4
5
6
7
8
10
12
2. Calculate the number of sides a regular polygon whose interior angles are each;
i. 165.60
ii. 1400
iii. 1080
3. Calculate the number of sides of a regular polygon whose exterior angles are each:
i. 90
ii. 300
iii. 22.50
4. Calculate the size of each exterior angle of each of the following regular polygons with
the given number of sides;
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i. 14
ii. 25
iii. 50
iv. 7
5. Calculate the size of the interior angle of each of the following regular polygons with the
number of sides:
i. 8
ii. 12
iii. 24
iv. 32
Activity of integration