Are Negative Angles Important in Geometry?

[Zalajbeg]

Hi everybody;

I am a bit stuck with negative angles. What I know is that: If we measure an angle from x-axis clockwise direction, the angle is negative.

However, I am a bit dissapointed because while I researched the web I see the negative agles only in topics subject to trigonometry. So I need to integrate my angle measurement knowledge from trigonometry to geometry.

As far as I remember we never encountered negative angles in geometry lessons while I am at high school. For example we always assumed angles of a triangle are positive. Is there a rule about this?

For example, let's say ABC is a triangle. Then is it right or not: ∠ABC = ∠CBA

There is a sentence: "The sum of the angles in a triangle is 180 degrees." Can it be deficit? "The sum of the magnitudes of the angles in a triangle is 180 degrees." It sounds better to me.

Is negative angles only defined on Cartesian coordinates? If yes, then if I locate a triangle on a Cartesian plane, can an angle have a negative value?

I know it is a small detail. But I really need a clarification. If you prefer to offer a geometry book for this kind of definitions I will be pleased to read.

 

[Mark44]

Hi everybody;

I am a bit stuck with negative angles. What I know is that: If we measure an angle from x-axis clockwise direction, the angle is negative.

However, I am a bit dissapointed because while I researched the web I see the negative agles only in topics subject to trigonometry. So I need to integrate my angle measurement knowledge from trigonometry to geometry.

As far as I remember we never encountered negative angles in geometry lessons while I am at high school. For example we always assumed angles of a triangle are positive. Is there a rule about this?

As far as I know, there is no rule about angles being negative in geometry. I suppose the reason for this is that figures in geometry are not usually placed in a coordinate plane, as they are in analytic geometry, which combines ideas from algebra with those from plane geometry.

For example, let's say ABC is a triangle. Then is it right or not: ∠ABC = ∠CBA

Yes, I believe the orientation of the angle doesn't make any difference in this context.

There is a sentence: "The sum of the angles in a triangle is 180 degrees." Can it be deficit? "The sum of the magnitudes of the angles in a triangle is 180 degrees." It sounds better to me.

Or, the sum of the measures of the angles in a triangle ...

Is negative angles only defined on Cartesian coordinates? If yes, then if I locate a triangle on a Cartesian plane, can an angle have a negative value?

No. You can define negative angles in polar coordinates in the plane, and in cylindrical or spherical coordinates in space.

I know it is a small detail. But I really need a clarification. If you prefer to offer a geometry book for this kind of definitions I will be pleased to read.

You're probably better off looking for a book that covers analytic geometry.

 

[berkeman]

Have you been introduced to the basics of complex numbers yet? If not, then ignore the rest of this.

But if you have, then thinking about the complex plane can help motivate the need for negative angles.

http://en.wikipedia.org/wiki/Complex_plane

In the complex 2-D plane shown on the right at the start of that article, the Real axis points to the right (the "x" axis), and the Imaginary axis points up. Points in the complex plane are represented by a vector from the origin to the point, and the components of that vector are the Real and Imaginary parts of the vector.

This representation is very valuable in many areas of science and engineering, since complex numbers and their complex exponential representation can be used in calculations of periodic signals (like in communications, etc.).

Anyway, when you mentioned negative angles, complex numbers and the complex plane was the first thing that came to mind...

 

[Studiot]

Hello Zalajbeg,

I think perhaps we are all making things too complicated. It really is very simple, and nothing whatsoever to do with triangles.

Since all the angles put together in a triangle add to 180° you cannot have a bigger angle than this in a triangle.

Yet we regularly use larger angles in other contexts for instance in navigation and surveying where the value can run up to 360°.

There are two things about angles and both are conventions.

First you have to agree where to start measuring. This gives you one line.

Then you have to agree in which direction to measure. This gives you the second line.

The angle is defined between these two lines.

Mathematicians choose to measure from the horizontal (x) axis.
And they choose to measure in an anticlockwise sense.

Angles measured this way are called positive to distinguish them from angles measured in the other (clockwise) direction.

There is nothing more to it than that.

Except that surveyors and navigators (who actually got there first) choose to meadure from the vertical (y) axis and in a clockwise direction.

As you study you will find there are many situations where we need to make a 'sign convention' which is nothing more than an agreement to identify one of two possible directions. You probably already know this on graphs where directions towards the right and upwards are called positive and towards the left and down are called negative.

 

[coolul007]

BTW, in Euclid's day there were no negative numbers...

 

[Zalajbeg]

No. You can define negative angles in polar coordinates in the plane, and in cylindrical or spherical coordinates in space.

Sorry, I intended to mean in coordinates plane. If I don't locate an angle on a coordinates plane then I think there is no need to define negative angles.

You're probably better off looking for a book that covers analytic geometry.

I was thinking of that. After your confirmation I will try to find a good analytic geometry book

Have you been introduced to the basics of complex numbers yet? If not, then ignore the rest of this.

Yes, I have. Now I am trying to learn more about trigonometry, then I have plans to study on complex numbers.

There are two things about angles and both are conventions.

First you have to agree where to start measuring. This gives you one line.

Then you have to agree in which direction to measure. This gives you the second line.

The angle is defined between these two lines.

I agree with these. But if the direction matters. Shouldn't it be right? ∠ABC = -∠CBA

( I assume ∠ABC --> Starting line is AB
∠CBA --> Starting line is CB

The measuring direction will be counter of each other)

By the way, I thank you all for your nice replies. I am learning much in this forum. I am a bit obsessed with definitions. I hope I can handle this one.

 

[Diffy]

Sorry, I intended to mean in coordinates plane. If I don't locate an angle on a coordinates plane then I think there is no need to define negative angles.



I was thinking of that. After your confirmation I will try to find a good analytic geometry book



Yes, I have. Now I am trying to learn more about trigonometry, then I have plans to study on complex numbers.



I agree with these. But if the direction matters. Shouldn't it be right? ∠ABC = -∠CBA

( I assume ∠ABC --> Starting line is AB
∠CBA --> Starting line is CB

The measuring direction will be counter of each other)

By the way, I thank you all for your nice replies. I am learning much in this forum. I am a bit obsessed with definitions. I hope I can handle this one.

I think about it this way. There is only ONE angle here, but there are two names for it, ∠ABC and ∠CBA.

If you have a triangle, with none of the vertex labeled, drawn on a piece of paper, then how are we to define the orientation?

You are defining the orientation. YOu said it yourself, that you are assuming the starting line is AB, as you call it. Why should it be so?

Sometimes we define things in mathematics for convenience. You say " if the direction matters", but in pure geometry it doesn't. And it shouldn't otherwise you run into problems like ∠ABC = -∠CBA.

I hope that helps.

 

[Zalajbeg]

I think about it this way. There is only ONE angle here, but there are two names for it, ∠ABC and ∠CBA.

If you have a triangle, with none of the vertex labeled, drawn on a piece of paper, then how are we to define the orientation?

You are defining the orientation. YOu said it yourself, that you are assuming the starting line is AB, as you call it. Why should it be so?

Sometimes we define things in mathematics for convenience. You say " if the direction matters", but in pure geometry it doesn't. And it shouldn't otherwise you run into problems like ∠ABC = -∠CBA.

I hope that helps.

Thanks for your answer.

I thought about the matter later and saw this it was similar to length and position.

For example if I try to measure length of an object, it is always positive, but If I want to state a position, I need to define negative and positive directions.

It is kind of the same, one of the angle of a triangle is similar to the length, and when we define angles on coordinates it is similar to (even may be the same with) position.