Geometry investigation - CNX

OpenStax-CNX module: m35640 1

Geometry investigation*

Pinelands High School

This work is produced by OpenStax-CNX and licensed under the

Creative Commons Attribution License 3.0�

Abstract

Terms and de�nitions used in geometry

1 Grade 8: Investigation of geometry

1.1 Study Unit 1: Angles: the basics

1.1.1 Introducing the concept of the angle

The diagram below shows the angle AΘ

BC. AB and BC are called the arms of the angle. AΘ

BC is theangle. B is the vertex, which is the point where the angle is.

*Version 1.1: Sep 20, 2010 5:11 am -0500�http://creativecommons.org/licenses/by/3.0/

http://cnx.org/content/m35640/1.1/


Figure 1

1.1.2 Naming angles

In the diagram below the angle at vertex B is calledΘ

Bor AΘ

BC or CΘ

BA.If we use a capital letter to name an angle we must put a cap on it to distinguish it from a point.



Figure 2

Note that if there are two or more angles that share the same vertex, as in the diagram below, we cannot

simply speak ofΘ

B . We must distinguish between AΘ

BC, MΘ

BC and AΘ

BM .



Figure 3

We may also put a small letter near the vertex and name the angle using the small letter. We do not putcaps on small letters because they are not referring to points. Also note that a small letter can either meanthe angle or the size of the angle. So in the diagram below we have:

AΘ

BM or x.

MΘ

BC or y.

Also: AΘ

BC = x+ y.



Figure 4

1.1.3 Measuring angles (in degrees)

The ancient Babylonians believed that there were only 360 days in a year (the amount of time for the earthto round the sun) and therefore divided the circle up into 360 equal parts, which they called degrees, denotedby the symbol ◦.



Figure 5

It follows that half a circle (a semi-circle) may be divided into 180 ◦ and a quarter of a circle may bedivided into 90 ◦.



Figure 6

1.1.4 Angles on a straight line

If half a revolution is 180 ◦, then angles on a straight line must add up to 180 ◦. In the diagram belowx+ y + z = 180

◦

Key concept: Angles on a straight line add up to 180 ◦.

1.1.5 Test your knowledge 1:

1) Consider the diagram below. If x = 50 ◦, what is the value of y?



Figure 7

2) Consider the diagram below. If x = 50 ◦ and y = 100 ◦, what is the value of z?

Figure 8



1.2 Study unit 2: Polygons

A polygon is a �at, closed shape formed by straight sides. Polygons that you should be familiar withalready include squares, triangles and rectangles.

A regular polygon has equal sides, equal interior angles and equal exterior angles. The followingshapes are examples of regular polygons:

Figure 9


1) Which of the following shapes are polygons? Write down only the letter(s).

Figure 10

2) What is the smallest number of sides that a polygon can have?

1.3 Study unit 3: Interior angles of polygons

Recall from study unit one that an angle is formed by two lines that meet at a point. Also recall that thepoint where two lines meet is called a vertex(plural vertices). The angle formed by two sides of a polygonat a vertex is called an interior angle (because the angle is INSIDE the polygon).



In the diagram below K, L, M and N are vertices of the rectangle and w, x, y, z are the interior anglesof the rectangle. In a rectangle each interior angle is a right angle (size = 90 ◦).

Figure 11


What is the sum of the interior angles of a rectangle?

1.4 Study unit 4: The sum of the interior angles of a triangle.

If you were to print out and cut out the triangles shown below and then fold along the dotted lines youwould �nd that the interior angles of the triangle form a straight line and so add up to 180 ◦.



Figure 12

Key concept: The sum of the interior angles of a triangle is 180 ◦.


1) Determine the value of x in the triangle shown below:



Figure 13

2) Determine the value of x in the triangle given below:

Figure 14

1.5 Study unit 5: Exterior angles of polygons

If one of the sides of a polygon is extended (in other words the side is lengthened or produced) the anglethat the line makes with the adjacent side is called the exterior angle.



Figure 15

In the diagram below, each of the sides has been produced (extended) in order. When this is done, eachside is extended in one direction only. a, b, c and d are each exterior angles of the square.



Figure 16

In the diagram below the four sides have each been produced (extended) in one direction only, but notin order.



Figure 17

In the triangle below each of the sides has been produced in order. X, Y and Z are the verices of thetriangle, a, b, c are the interior angles and d, e, f are the exterior angles.



Figure 18


Extend each side of the pentagon in order:



Figure 19

1.6 Study unit 6: The sum of the exterior angles of a triangle

The sum of the interior angles of an equilateral triangle is 180 ◦, so each interior angle must be 60 ◦ (180 ◦

divided by 3, the number of verices).



Figure 20

The number of degrees on a straight line (called a �straight angle�) is 180 ◦, so 60 ◦ + a = 180 ◦ and a =120 ◦.



Figure 21

In the diagram below the sides of the equilateral triangle have been produced in order, to create exteriorangles of the triangle.



Figure 22

Thus (a + a + a) represents the sum of the exterior angles of a regular (equilateral) triangle formed byproducing each side in order.


What is the sum of the three exterior angles in the triangle above?

1.7 Study unit 7: Diagonals

Diagonals are straight lines drawn from one vertex to the other, as shown in the following diagrams:



Figure 23

As you can see in the following diagram if we draw one diagonal in a quadrilateral it divides thatquadrilateral into two triangles.

Figure 24

The diagram show below shows that, by drawing two diagonals in a pentagon (both originating from thesame vertex A) we may divided it into three triangles.



Figure 25

Similarly by drawing diagonals from one vertex in any polygon we may divide it into triangles.


In the hexagon shown below, draw all the possible diagonals from vertex A to divide it into triangles. Howmany triangles have you made?



Figure 26

Test your knowledge: answers

1.7.2 Section 1:

1) y = 130 ◦

2) z = 30 ◦

1.7.3 Section 2:

1) A, C, F, G, H, J, K, L, M2) Three

1.7.4 Section 3:

360 ◦

1.7.5 Section 4:

1) 40 ◦

2) 60 ◦



1.7.6 Section 5:

Figure 27

Section 6:a = 120 ◦

Therefore:

a+ a+ a = 120 + 120 + 120 = 360◦

(27)



1.7.7 Section 7:

Figure 28

Four triangles.


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Geometry investigation - CNX