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 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 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 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 B C

Example 4 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 M L

3. Use of a number or a symbol at the vertex 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 G F

∠𝑃 is also an acute angle. P O Q

(b) Right angles Is an angle that is exactly 90° (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 𝛽 X W

(d) Straight angles These are angles that are exactly 180° (d) Straight angles These are angles that are exactly 180°. ∠𝑃𝑄𝑅 is a straight angle. Q P 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 Reflex ∠𝐽𝐾𝐿 Acute ∠𝐽𝐾𝐿

(f) Round angles Is an angle which measures exactly 360° (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 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

Z (b) Y 𝛽 X

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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