Angles

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Introduction

We measure things or we refer to measurement all the time: our height, the height of a mountain, our weight, the weight of potatoes, our temperature, the air temperature, the speed of our car, the speed of an aeroplane. But there are some things we don’t often think about measuring – turn, for example. How do you measure turn and what units do you measure it with?

Well, of course, turning is all about angling. And the amount that we turn through is just the size of the angle of the turn and we measure that in degrees.

As in most things mathematical, there is a progression in the learning of turning. We don’t have research backing to say exactly what the progression is but it must look something like the following:

Realisation of different amounts of turning;

Full turns, half turns, quarter turns;

Clockwise and anticlockwise;

Quarter turns are 90º; half turns are 180º; full turns are 360º;

Other simple fractions of a full turn – 45º as half a quarter turn; 30º as a third of a quarter turn; 60º as a third of a half turn (or two thirds of a quarter turn);

Use of a protractor to measure angles; use of a protractor to make given angles; first less than 90º and then less than 180º and then less than 360º;

Application of the use of angle in simple shapes and constructions;

The sum of the angles of a triangle is 180º;

The sum of the interior angles of a quadrilateral;

The use of angle in simple geometric proofs;

The sum of the angles in an n-sided polygon;

Basic trigonometry to measure sides of right angled then non-right angles triangles;

Further trigonometry in 3-dimensional situations;

Application of trigonometry in abstract settings such as periodic graphs; differentiation; and integration.

Even more abstract applications.

As this progression goes from Level 1 to Level 8 and beyond we won’t cover it all in this session. But what we will do is to show an application of turning to prove (not just justify), some things about the angle sums of triangles and polygons.

360º, right angles and left angles

But first some questions. Why are there 360º in a complete turn? Why is a right angle ‘right’? Why is there no left angle?

So why 360º? I think that the answer to that is that we’re not quite sure. An attempt at an answer is made in the March Newsletter 2003. It goes roughly like this. The Mesopotamians used a base 60 number system (as did other ancient people like the Babylonians). So this meant that they were interested in multiples of 60. Now they realised that the year was about 365 days long and 360 was the nicest number close to 365. It was nicest in the sense that it had lots of factors. They also equated the year in some sense to be a cycle – there was a regular sequence of seasons. So they went for 360º in a circle. Now they could have taken up an angle measurement of 365 but aren’t we glad that they didn’t? After all, instead of 90º we’d have had 91.25º. That would have been a mess to calculate with.

Of course, you might think that they would have been better to have used 100º for a complete turn, but the decimal system was still many hundred years off. So just be grateful we got 360º rather 365º.

There is also an argument that suggests that up until the 8th century BC the earth's year was actually 360 days long, and that at that time a cosmic event altered the earth's orbit, adding 5 days to the length of the year.

Anyway, that gives us a half turn (either clockwise or anti-clockwise) of 360º/2 = 180º and a quarter turn of 360º/4 = 90º. And you can work out the number of degrees in various other nice fractions of a complete turn, as well as the fraction of a complete turn that a specific number of degrees makes.

Well, that brings us to right angles. So what is so right about a quarter of a turn? Does that mean that every other angle is a wrong angle? Like many English words, right has several meanings. Look them up in your dictionary. There’s morally good; proper, correct; a wing of a political party; and so on. There is also ‘to make upright’ as in righting a boat. Now it’s this upright meaning that applies to the right angle.

Right in angle comes to us from the Latin ‘rectus’. It’s the angle that you would want to make when you are putting up the wall of a building. The wall would need to be vertical so the angle made between the wall and the (horizontal) ground would be an upright angle.

 diagram.

Now, of course, you shouldn’t be tempted to think, as some children seem to, that there are left angles. We show what we have seen in the diagram below. Clearly if the first one is a right angle the other one has to be a left angle.

 

diagram.

180º in a triangle – a demonstration

So now we get down to a proof by turning that the interior angles of a triangle add up to 180º. We’ll start by just demonstrating that it might be true. First cut out a triangle from a piece of paper. You can do it using the newspaper but it may be easier to see if you use a plain piece of paper. Then tear off the three corners. Now put the three angles A, B and C together and they will make a straight line. (This is shown below.)

 diagram.

No matter what triangle you draw, the same thing always happens. Now a straight line is a half turn; and a half turn is 180º. So the sum of the three angles in the triangle we have here is 180º or a half turn.

Why is this not a proof? Well it only ‘proves’ it for the one triangle that you have cut out. Admittedly it works for every triangle that you will ever cut out. But we can’t be sure of that. There may be some triangle somewhere that doesn’t play ball. So how can we actually prove that the interior angles of any triangle add up to 180º? And how can we use turning to do it?

 

diagram.

180º in a triangle – a proof

Extend the sides from A to B and from C to B in the above diagram. Now the first thing that you need to know is that the angle we get on the other side of B is the same size as B.  You can see this by turning. Imagine that you are standing on the corner where B is, your arms extended 180º apart, your left arm pointing to A along the side from B to A. Then turn from A to C. Whatever angle your left arm has turned through, that’s the angle that your right arm has turned through. If it isn’t then eventually one arm will catch up with the other.

 diagram.

Now imagine you are standing at A and facing along the side of the triangle to C with your left arm pointing to C. Then rotate in an anti-clockwise direction until you are facing B along the side of the triangle.  So far your left arm has turned through the angle A.

Walk along the side AB until you get to B. Bear in mind that you haven’t turned at all while doing this. At this stage your left arm is pointing to X. Now turn anti-clockwise through the angle B that is outside the triangle. So far you have turned an amount equal to the angles A plus B.

Walk backwards along the side of the triangle till you get to the C corner. All this time your left arm hasn’t turned at all. Now turn anti-clockwise from the side from B to C to the side from B to A.

There are now two things to consider. First your left arm has turned a total of A plus B plus C. But second, how far is this? Remember you started facing along the side from A to B and you ended along the same line facing from B to A. So you had to have turned a total of 180º.

So the angle A plus B plus C must equal 180º. The sum of the interior angles of a triangle is 180º.

Polygons

If you managed to come through that unscathed you might be able to use exactly the same two arguments to show that the interior angles of any four-sided figure sum to 360º. You can cut out a four-sided figure, cut off the corners and put them together to make a full turn. You can also go around the quadrilateral ABCD in the same way we did round the triangle. This time you will need to use two opposite angles – the ones at B and D.

And if you can do a four-sided figure how much harder can it be to do five- and six- and … any-sided figures?

Actually it’s easiest to find the sum of the interior angles of a quadrilateral by cutting the shape into two triangles. Just draw a line from one vertex to an opposite vertex as we have done in the diagram.

 diagram.

Then the sum of the interior angles of a quadrilateral, add up to twice the sum of the interior angles of a triangle. And this is 360º.

By the same method you can Take any polygon with n sides and cut it up into n – 2 triangles. So the sum of the interior angles of any n-gon is (n-2) x 18º.

But why is that a Proof?

You don’t need to look at this on a first reading unless you are interested in the finer points of mathematics.

We have given a plausible argument and a proof here. The plausible argument uses the torn off angles as a demonstration that for ONE TRIANGLE the interior angles add to 180º. Proofs have to hold for ALL triangles. You can certainly do this demonstration for a large number of triangles but you can never do it for EVERY POSSIBLE triangle.

On the other hand you can see with the turning argument that you must start facing along a side and away from a given corner, and end up facing along the same side and into the given corner. That means that you have turned through 180º. (You can’t have gone through 180º + 360º. Why?) But the angle you have turned through is the sum of the interior angles of the triangle. So that sum is 180º.

In this argument we are not restricted to any one triangle. What we do is true for ANY triangle. Hence we have a proof.

Seminar

The things that need to be emphasised throughout this seminar are

  1. the simple fact that the size of an angle is an amount of turning; and
  2. the unit of measurement of turn is degrees.  

The other things that we’ve listed are just vehicles for these two ideas.

Here is a suggestion for the seminar.

1. Open with a series of questions.

How many degrees in a complete turn? (360)

How many degrees in a half turn clockwise? (180)

How many degrees in a half turn anti-clockwise? (180)

What fraction of a complete turn is 120º? (360/120 = 1/3)

How many degrees in a quarter turn anti-clockwise? (90)

What fraction of a complete turn is 60º? (1/6)

How many degrees in a fifth of a turn clockwise? (72)

What fraction of a complete turn is 20º? (1/18)

How many degrees in a tenth of a turn anti-clockwise? (36)

What fraction of a complete turn is 270º? (3/4)

 

2. But why are there 360º in a complete turn?

 

Discuss what they and you know.

Can you think of a better number of ‘degrees’ to have in a complete turn?

Why would that be better?

Note that there is a European scheme that has 400 grads to a complete turn.

Why might this be better than 360?

Does anyone know what a radian is? (1 radian is the angle that sweeps out an arc of 1 radius on any circle. This means that one complete turn is 2π radians. Radians are used a lot in higher maths.) Why would this be of any use?

3. Angles in a triangle?

This brings us to the triangle.

When you add up the angles in any triangle how many degrees do you get?

How do you know that’s right?

Can you demonstrate it somehow?

Go through the two methods that we used in the sections on ‘180º in a triangle.’ If one of your staff members knows these (or any other method) let them show the rest of the group. You don’t have to be the only one to know all the answers.

There is actually another method based on the diagram below. If the line that goes up to X is parallel to the side next to angles A and B, why are the angles marked A* and B* equal to A and B, respectively? (It’s a property of corresponding angles, for A, and alternate angles, for B.) Clearly (?) A* + B* + C = 180º.

 diagram.

4. Other polygons

Go on to explore the sums of the interior angles of other polygons. As a result answer the following.

What is the sum of the internal angles of any quadrilateral?

Why are the interior angles of a rectangle equal to 90º?

How big are the interior angles of an equilateral triangle? Why?

How big are the interior angles of a regular hexagon? Why?

How big are the interior angles of a regular octagon? Why?

Suppose you were an ant sitting on a corner of a regular decagon and facing along one of the edges next to that corner. How small an angle would you have to turn through to face along the other edge next to that corner?