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I saw two definitions of an angle. Are those equivalent or is another wrong in some axiomatic system?

  1. An angle is the union of two rays.

  2. An angle is a subset of a plane restricted by two rays.

I guess that they are not equivalent as the second definition allows me to say that an angle might have positive area but the first gives than any angle has zero area. Which one is correct?

curious
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4 Answers4

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Intuitive ways of defining the term angle are discussed in the book Experiencing Geometry by David Henderson and Daina Taimina. As a geometric shape, an angle is defined in this book as the delineation of space by two intersecting lines. This may seem like an ambiguous definition if we consider the angles $90^{\circ}$ and $270^{\circ}$. However, the term directed angle is defined separatedly in this book: a directed angle is an angle with one of its sides designated as the initial side, and the other side designated as the terminal side. It is customary to indicate the direction with an arrow. The following discussion concerning intuitive definitions of the term angle is taken from this book:

"There are at least three different perspectives from which we can define 'angle,' as follows:

$ \ $ $\bullet$ a dynamic notion of angle - angle as movement;

$ \ $ $\bullet$ angle as measure; and,

$ \ $ $\bullet$ angle as geometric shape.

A dynamic notion of angle involves an action: a rotation, a turning point, or a change in direction between two lines. Angle as measure may be thought of as the length of a circular arc or the ratio between areas of circular sectors. Thought of as a geometric shape, an angle may be seen as the delination of space by two intersecting lines. Each of these perspectives carries with it methods for checking angle congruency."

John M. Campbell
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These definitions are ambiguous.

Think of the two rays given by the positive $x$-axis and the positive $y$-axis.

What angle do they make?

If you go from the $x$-axis to the $y$-axis is an anti-clockwise direction then you get $90^{\circ}$

If you go from the $x$-axis to the $y$-axis is an clockwise direction then you get $270^{\circ}$

You need to specify a direction/orientation.

Fly by Night
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  • I was thinking just definitions, not how to measure the angles. I think one can first define the angle before one constructs a measure to say how big is an angle. – curious Mar 18 '16 at 18:54
  • Okay, so tell me how to define the subset of the plane restricted by the two rays. Is it the subset you get by going anti-clockwise from the positive $x$-axis to the positive $y$-axis? (Top right quadrant) Or is it the subset you get from going clockwise from the positive $x$-axis to the positive $y$-axis? You have the same ambiguity. You need a direction/orientation. – Fly by Night Mar 18 '16 at 18:58
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An angle is a part of the plane enclosed by two rays which share a common beginning.

Note that two rays with a common beginning define two angles.
If you have to resolve that ambiguity, you can order the rays as 1st and 2nd,
then the angle which the two rays define is defined in an unique way.

peter.petrov
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A good definition must be uniquely precise. In 1. "The union of two rays" define two possible angles (for example $90^{\circ}$ and $270^{\circ}$). In 2. "Subset of a plan restricted by two rays" is also ambiguous (the same example fits).

An elementary way can be done by the following: first at all define a fixed "ray" (a good one is the positive x-axis) and the other "ray" can be taken in the positive or negative direction (counterclockwise and clockwise respectively). In this case you have to assume that the angle $\alpha$ coincide in position with $\alpha +360^{\circ}k$ where $k\in\mathbb Z$.

Looking finer at this you can also exhibit "deficiency" in the definition, however you have to learn that the "measure of angles" is some enough delicate topic we avoid here.

Piquito
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