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# Geometry investigation

Module by: Pinelands High School. E-mail the author

Summary: Terms and definitions used in geometry

## Grade 8: Investigation of geometry

### Study Unit 1: Angles: the basics

#### Introducing the concept of the angle

The diagram below shows the angle ABˆCABˆCA hat{B}C . AB and BC are called the arms of the angle. ABˆCABˆCA hat{B}C is the angle. B is the vertex, which is the point where the angle is.

#### Naming angles

In the diagram below the angle at vertex B is called BˆBˆhat Bor ABˆCABˆCA hat{B} C or CBˆACBˆAC hat{B} A.

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

Note that if there are two or more angles that share the same vertex, as in the diagram below, we cannot simply speak of BˆBˆhat{B} . We must distinguish between ABˆCABˆCA hat{B} C, MBˆCMBˆCM hat{B} C and ABˆMABˆMA hat{B} M.

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

ABˆMABˆMA hat{B}M or x.

MBˆCMBˆCM hat{B}C or y.

Also: ABˆC=x+yABˆC=x+yA hat{B}C = x + y.

#### Measuring angles (in degrees)

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

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

#### 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 below x+y+z=180°x+y+z=180°x + y + z = 180^{circ}

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

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

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

### Study unit 2: Polygons

A polygon is a flat, closed shape formed by straight sides. Polygons that you should be familiar with already include squares, triangles and rectangles.

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

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

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

### 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 the point where two lines meet is called a vertex(plural vertices). The angle formed by two sides of a polygon at 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 angles of the rectangle. In a rectangle each interior angle is a right angle (size = 90°).

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

### 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 you would find that the interior angles of the triangle form a straight line and so add up to 180°.

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

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

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

### 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 angle that the line makes with the adjacent side is called the exterior angle.

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

In the diagram below the four sides have each been produced (extended) in one direction only, but not in order.

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

Extend each side of the pentagon in order:

### 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).

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

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

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

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

### Study unit 7: Diagonals

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

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

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

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. How many triangles have you made?

1) y = 130°

2) z = 30°

#### Section 2:

1) A, C, F, G, H, J, K, L, M

2) Three

360°

1) 40°

2) 60°

#### Section 5:

Section 6:

a = 120°

Therefore:

a + a + a = 120 + 120 + 120 = 360 ° a + a + a = 120 + 120 + 120 = 360 ° a + a + a = 120 + 120 + 120 = 360^{circ}
(1)

Four triangles.

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