DAY 1 – DEFINITION OF ANGLES

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.

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.

A line

Ray

A line with one endpoint and extending infinitely in

the other direction.

A ray

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

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

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

The common endpoint, intersection point or starting point of two or more rays or lines is called a vertex.

Vertex Q is formed when rays PQ and RQ meet at

the common endpoint Q.

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.

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

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

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

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°.

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

∠𝑃 is also an acute angle.

P

OQ

(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

(c) Obtuse angles

Is an angle that is greater than 90° but less than

180° .

Angle 𝛽 is an obtuse angle.

V

WX

𝛽

(d) Straight angles

These are angles that are exactly 180°.

∠𝑃𝑄𝑅 is a straight angle.

QP R

(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

(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

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.

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.

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.

HOMEWORK

Identify the type of angle and name the angle using

the three methods.

(a)

H J

I

𝜃

(b)

X

Y

Z

𝛽

ANSWERS TO HOMEWORK

(a) Type: reflex angle

Naming: ∠𝜃, ∠𝐽𝐼𝐻 𝑜𝑟 ∠𝐻𝐼𝐽, ∠𝐼

(b) Type: Obtuse angle

Naming: ∠𝛽 , ∠𝑋𝑌𝑍 𝑜𝑟 ∠𝑍𝑌𝑋, ∠𝑌

THE END