general angle properties of geometric figures definitions

Caltec Academy Makerere mathematics department

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


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.


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.


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.


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


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)


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.


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


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


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


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


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


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.


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;


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

general angle properties of geometric figures definitions

Documents