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DAY 1 – DEFINITION OF ANGLES
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INTRODUCTION
In daily life we encounter patterns, designs and
a variety of shapes. Roads, furniture, vehicles and
houses, among others, are designed by accurate use
of angles and combining different shapes. For
instance, a carpenter uses angles to make furniture
and architects use angles to draw designs of
buildings.
In this lesson we are going to define an angle
based on the undefined terms of a point, a line,
distance along a line and distance around a circular
arc. Undefined terms imply that the meaning is
accepted without definition.
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VOCABULARY
Point
An exact position or location on a plane or in space
that has no size and dimension. It is usually
represented by a dot and named by a capital letter.
A point
Line
A collection of points along a path that extends
infinitely in both directions. It is has no thickness
and no height.
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A line
Ray
A line with one endpoint and extending infinitely in
the other direction.
A ray
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A ray is named starting with the endpoint and
another point on the ray.
The endpoint is written first, then any other point
on the ray is written next.
This is shown below.
Ray 𝑄𝑅
Q R
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Example 1
The figure below represents line AB, written as
AB
Example 2
The figure below represents ray AB, written as
AB
A B
A B
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Angles and parts of an angle
When two rays diverge from a common point or
meet at the same endpoint, the space which is a
amount of turn between them is called an angle.
rays (arms)
Vertex
Angle
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The common endpoint, intersection point or starting point of two or more rays or lines is called a vertex.
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Vertex Q is formed when rays PQ and RQ meet at
the common endpoint Q.
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NAMING ANGLES
Angles are named based on the vertices and the
rays, also called arms of the angle. The following
are ways of naming them
1. Use of three capital letters
When an angle is marked by three points, we first
name one point, then the vertex and finally the
other point, in that order. This method is also
called point‐vertex‐point method due to the
order followed.
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Example 3
This is angle ABC or angle CBA. Note that the
vertex is at the center.
We can replace the word angle with the symbol ∠.
A
BC
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Therefore, we name the above angle as ∠ABC or ∠CBA.
2. Use of the vertex
In unambiguous cases, angles can also be named using the vertex only where. This method is also called the vertex method of naming angles
Example 4
This angle is ∠L.
K
ML
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3. Use of a number or a symbol at the vertex.
A number or symbol can be written at the vertex to denote an angle. In most cases lower case letters and Greek letters are used.
Example 5
The angles are named as angle 𝜃 and angle 𝛼.
𝜃
𝛼
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MEASURING ANGLES
In geometry angles are commonly measure using
two units, in degrees or radians.
The symbol for degree is °.
This means that 90 degrees can be written as 90°.
A full revolution or rotation is taken be 360°.
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BASIC TYPES OF ANGLES
The following are basic types of angles.
(a) Acute angles
Is an angle which is less than 90°
∠𝐸𝐹𝐺 is an acute angle
E
GF
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∠𝑃 is also an acute angle.
P
OQ
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(b) Right angles
Is an angle that is exactly 90°. A right angle is
represented below. We use a symbol similar to a section
of a square to show that an angle is a right angle.
∠𝐾𝐿𝑀 is a right angle.
K
L M
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(c) Obtuse angles
Is an angle that is greater than 90° but less than
180° .
Angle 𝛽 is an obtuse angle.
V
WX
𝛽
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(d) Straight angles
These are angles that are exactly 180°.
∠𝑃𝑄𝑅 is a straight angle.
QP R
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(e) Reflex angles
Is an angle that is greater than 180° but less than
360°.
The angle on the outer side is the reflex angle.
∠𝐽𝐾𝐿 is reflex angle.
J
K
L
Acute ∠𝐽𝐾𝐿 Reflex ∠𝐽𝐾𝐿
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(f) Round angles
Is an angle which measures exactly 360°. It is also
called a full angle or a full circle.
Angle 𝑎 is a round angle.
𝑎
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Angles in real life
There are many examples of angles in real life
situations. Some of these include:
1.The vertical edge of a corner of the classroom
forms a right angle with the floor.
2. The edges of most tables meet at the vertex of
the table to form a right angle.
3. A ladder forms an acute angle when leaned on a
wall.
4. A ramp or an inclined plane forms an acute
angle with the ground.
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5. A corner on the outside of a building forms a
reflex angle.
6. On house rooftops where the two roof surfaces
slope down from the apex (ridge) form an reflex or
obtuse angle.
8. At exactly six o’clock, the minute hand and hour
hand of an analogue wall clock form a straight
angle.
9. A door makes angles with the wall when it is
opened to a certain point.
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10. Latitudes and longitudes are used to locate
points on the surface of the earth by measure of
angles from the equator and the Prime meridian.
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HOMEWORK
Identify the type of angle and name the angle using
the three methods.
(a)
H J
I
𝜃
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(b)
X
Y
Z
𝛽
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ANSWERS TO HOMEWORK
(a) Type: reflex angle
Naming: ∠𝜃, ∠𝐽𝐼𝐻 𝑜𝑟 ∠𝐻𝐼𝐽, ∠𝐼
(b) Type: Obtuse angle
Naming: ∠𝛽 , ∠𝑋𝑌𝑍 𝑜𝑟 ∠𝑍𝑌𝑋, ∠𝑌
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THE END