# 1 Angles

1  Easels

Four art school students sketched the same model. Yet all drawings are different. That is because they were viewing the model from different directions.

 Go to the applet now and put all sketches on the right easels.

You look along straight lines. Those lines are called view-lines. On top of a watch tower you can look very far in all directions. Your field of vision is unrestricted from up there. From a lower viewpoint obstacles usually block your view.

2  Rabbits

Rabbits are playing in the grass behind the lighthouse. The lighthouse keeper walks towards the lighthouse.

 a Will the number of rabbits he sees increase or decrease? increase decrease b Continue with the applet.

3  Island

It is possible to determine one's position at sea using landmarks ashore. In this exercise these landmarks are a lighthouse and a church-tower. There is only one point at sea from which the lighthouse is due east and the church northeast.

 Continue with the applet.

4  Surveying

It's easy to get straight lines on paper, using a ruler or by folding. In the field this doesn't work. Surveyors use red and white poles, called levelling rods or surveyor's staffs. Road-builders would like to construct a straight road through the meadows. In the applet two surveyor's staffs have been placed already.

 a Do the first applet.
By looking along several poles, you can check if they are in line. This way you can find the intersection of two lines.

 b Do the second applet.

5  A difficult puzzle

Nine surveyor's staffs are neatly arranged in a field in three equal rows of three at the same distance apart (see figure).

 a How many lines run through three poles? b Move two poles in such a way that there will be 10 lines running through 3 poles each.

6  Intersections

At intersections your view is often restricted. This may cause dangerous situations. A car and a bicycle approach an intersection. The car is on Main Street and the cyclist on Church Street. Both are going straight ahead.

 Do the applet.

Point, line, and plane are three common notions from geometry. That is why we want to explain their exact meaning. Such precise explanation is called a definition. To define means: "to give the precise meaning of a word or expression". A point can be thought of as a very small dot, smaller than you can imagine. Draw a dot and it is already too big.
A line is something that goes on forever, in a straight fashion, thinner than thin. You could also think of a sharp fold in a piece of paper, but much longer: extending indefinitely in both directions. A line is completely determined by two points.
A plane is something like a tabletop: entirely smooth, without unevenness, and unbounded in all directions.

Take a line and a point on it. The point divides the line in two parts that are unbounded at one end and bounded by the point at the other end. Each part, including the endpoint, is a half-line. Taking a line with two points on it, we call the part of the line in between and including the two points, a line segment.

A view-line is only part of a line: it starts somewhere and runs in one direction. That is why we sometimes call a view-line a half-line.

7 A line has three points on it.

 a How many half-lines have one of these points as endpoint? b How many line segments have two of these points as endpoints?
A line has n points on it (n = 2, 3, 4, ...)

 c How many half-lines have one of these points as endpoint? d How many line segments have two of these points as endpoints? Write down your calculation.

8  The relative position of three lines in a plane

The picture shows the possible relative positions of three lines.
1. They intersect at one point (1).
2. Two lines are parallel and the third intersects both (2).
3. The three lines are parallel. (3)
4. They intersect pairwise in different points (4).
 a How many half-lines are formed by the lines and their intersection points in Figure (2)?
 b How many line segments are formed by the three lines and their intersection points in the picture on the right?

9 Suppose you had drawn four points on paper. You have done it in such a way that no three points are on the same line.

 a How many lines can you draw through any two of these points?
Now you have drawn n points (n = 2, 3, 4, ...) in such a way that no three points are on the same line.

 b How many different lines can you draw through two of these points? Write down your calculation as well.

In exercise 6 we discussed one's field of vision. The picture shows the field of vision of a cyclist. It is bounded by two half-lines. The area between those half-lines is an angle.

An angle is part of the plane bounded by two half-lines with a common endpoint. The endpoint is called the vertex. The two half-lines are the arms of the angle. The arms and the vertex are part of the angle.

When we name the vertex A, the angle is sometimes called angle A.

Remark

The length of the arms doesn't matter; they are infinite in one direction. The area enclosed also extends indefinitely.

10  Jack says: the angle on the left is the bigger one, because its opening is wider. Mike says: the angle on the right is larger, because its area is bigger.
 Who do you think is right? Jack Mike

We say that two angles are the same if they fit exactly over each other. We say that one angle is larger than another if it can cover the other angle completely, but not conversely.

 11 Arrange these angles in order of size, smallest first.

Remark

Dick draws an angle. It is not clear which angle he has in mind, angle 1 or angle 2. In such cases you have to indicate which angle is meant. When there is no further indication we will always assume that it is the smaller angle. In Dick's drawing we take angle 1.

 12 Tear off a piece of paper and fold it in two. Fold it once more, as indicated: the first fold on top of itself. Next unfold completely. The folds form a cross. The four angles are equal. You have four right angles.

Two lines that divide the plane into four equal angles are perpendicular. They make four right angles.

A straight angle is an angle of which the two arms form a straight line. Any angle that is smaller than a right angle is called an acute angle. Any angle that is larger than a right angle but smaller than a straight angle is an obtuse angle. Any angle that is larger than a straight angle is called a reflex angle.

13  The figure shows an octagon. State the type of each angle.
a    b  c  d  e  f  g  h
 acute right obtuse straight reflex

14  In the applet you will learn how to use your protractor to draw a right angle.

 a Study the applet. b In your workbook, draw a line l with a point A on it. Use your protractor to draw a line through A that is perpendicular to line l. c Choose a point B that is not on l. Draw a line through B perpendicular to l.

15  The hands of a clock define an angle. As time goes on, the angle keeps changing.

 Wat kind of angle do the hands of a clock define when it is three o'clock? And at six o'clock? And at two?

Remark

In the following diagrams capitals are used to indicate vertices. That is what we usually do in mathematics.

16
 a What is the smallest angle in quadrilateral 1? And which one is the largest? List the correct vertex. b What is the smallest angle in quadrilateral 2? Which one is the largest? c What is the smallest angle in the triangle? Which one is the largest? d What is the smallest angle in quadrilateral 2? Which one is the largest?

17
 a Draw a quadrilateral in your workbook with one obtuse angle and three acute angles. b Also draw a quadrilateral having two obtuse angles and two acute angles. c Draw another one having three obtuse angles and one acute angle.