CHAPTER EIGHT

Chapter 8: GENERAL AND ANGLE PROPERTIES OF GEOMETRIC FIGURES

Learning objectives

By the end of this topic, the learners should be able to

• Classify angles

• Identify different angles

• Solve problems involving angles at a point ,on a straight line, angles on a transversal and parallel lines

• Know and use the angle sum of a triangle

• State and use angle properties of polygons when solving problems

Introduction

In this topic you will study angles on the straight line, parallel lines and angle properties of polygons. Equipped with the knowledge from this topic, you will be able to solve problems related with angle properties.

8.1 Classifying angles

• An angle is the space (usually measured in degrees) between two intersecting lines or surfaces at or close to the point where they meet. In geometry, angles are measured in degrees using a protractor

Types of angles

• Acute angle: Angle less than 90° e.g. 30° 

• Right angle: An angle that is exactly 90° 

• Obtuse angle: An angle that measures between 90° and 180° 

• Straight angle: An angle that is exactly 180° 

• Reflex angle: An angle that is measures between 180° and 360° 

• Full angle: An angle that is equal to 360°.

8.1 Exercise Set

State whether each of the following angles is acute, obtuse or reflex

8.2 Identify Different Angles

Activity : Identifying objects that form angles.

Identify objects in your class, which make 90°; 180°; 360°

Activity : Identifying angles.

Draw two intersecting lines. Use your mathematical instruments to measure the angles formed at the intersecting point.

• How many angles have been formed at the point of intersection?

• What is the size of each angle formed?

8.2 Exercise Set

1. For each of the following angles, first estimate the angles and then measure the angle marked 0 to see how good your estimate was.

2. Draw the following angles

3. Draw a triangle with one obtuse angle.

4. Draw a triangle with no obtuse angles.

5. Draw a four-sided shape with:

(a) one reflex angle.

(b) two obtuse angles.

8.3 Angle Relationships

In this section we look at angle relationships and their measures. In Geometry, there are five fundamental angle pair relationships:

1. Complementary Angles: These are two positive angles whose sum is 90 degrees.

2. Supplementary Angles: These are two positive angles whose sum is 180 degrees.

3. Adjacent Angles: These are two angles in a plane that have a common vertex and a common side but no common interior points.

4. Angles on a line add up to 1800.This is because there are 1800 in a half turn

5. Angles around a point add up to 3600.This is because there are 3600 in a full turn

6. Vertical Angles(Vertically opposite angles): These are two nonadjacent angles formed by two intersecting lines or opposite rays.

EXAMPLES

1. In the figure below, find the value of x

2. In the figure below , find the value of y

3. The figure below shows two intersecting lines. Find the value of x

8.3 Exercise Set

1. Two angles are supplementary. One angle measures 120 more than the other. Find the measures of the angles.

2. Find the value of a

3. Find the size of two complementary angles that are such that the size of one of them is four times the size of the other.

4. Find the value of y

5. Find the size of the angle marked x

6. Find the value of x for which the angles (2x + 10)° and (130 - x)°  are vertically opposite.

8.4 Parallel and Intersecting Lines

When a line intersects (or crosses) a pair of parallel lines, there are some simple rules that can be used to calculate unknown angles.

2. Find the size of the unknown angles in the gure below.

3. Find the size of the unknown angles in the parallelogram shown in this diagram:

8.4 Exercise Set

1. Find the value of y

2. Find the size of the three unknown angles in the parallelogram.

3. One angle in a parallelogram measures 40° . What is the size of each of the other three angles?

6. One angle in a rhombus measures 133° . What is the size of each of the other three angles?

7. One angle in a parallelogram measures 60° . What is the size of each of the other three angles?

Activity : Identifying the polygons

• Find the number of sides of different polygons and their corresponding names.

• Determine the size of each interior and exterior angles of the regular polygons.

• Determine the sum of the angles in the regular polygons

NOTE

1. In any polygon with n- sides, the following properties apply

• Interior angle sum =(n - 2)180° 

• Exterior angle sum = 360°.

• Each interior angle + each exterior angle =180°.

• Number of diagonals =n(n-3)/2

2. In a regular polygon with n- sides, the following properties apply

• One interior angle=(n-2)180°/n 0

• One exterior angle =360°/n

EXAMPLES

1. Find the interior angle sum of a pentagon.

2. Find the size of the interior angle of a regular pentagon.

3. Find the size of each exterior angle of a regular pentagon.

4. Find the number of sides of a polygon whose interior angle sum is 126°

8.5 Exercise Set

1. Find the sum of the interior angles of a polygon with 22 sides.

2. The interior angle of a regular polygon is 162° . How many sides does the polygon have ?

3. The interior angle sum of a regular polygon is 1800°  . How many sides has the polygon? Name the polygon

4. Find the interior angle sum of a decagon.

5. Find the size of each interior angle of a regular hexagon.

6. Find the number of sides of a regular polygon whose each interior angle is 135° 

7. Find the size of each exterior angle of a regular octagon.

8. If the vertices of a regular hexagon are joined to the centre of the hexagon, what is the size of each of the six angles at the centre? Use your answer to construct a regular hexagon ABCDEF of side 3cm. Start with a circle of radius 3cm. Measure the length of the diagonal AC.

Activity of Integration

The table below shows the Covid 19 active cases discovered in some districts of Uganda in a year 2020.