This angle is denoted by . The two rays and are the sides of the angle, and the vertex of the angle. Since any point (other than the source ) on a ray uniquely determines the ray, we may also write the angle by , whenever we have points and .
The notational device given for the angle suggests the possibility of defining an angle between two line segments satisfying certain conditions: let and be two open line segments with a common endpoint . The angle between the two open line segments is the angle between the rays and . In this case, we may denote the angle by .
Suppose is a line and a point lying on . We have two opposite rays emanating from that lie on . Call them and . Any ray emanating from a point that does not lie on produces two angles at , one between and and the other between and . These two angles are said to be supplement of one another, or that is supplementary of . Every angle has exactly two supplements.
2 Ordering of Angles
Let be an ordered geometry and a ray in with source point . Consider the set of all angles whose one side is . Define an ordering on by the following rule: for ,
if , and
The ordering relation above is well-defined. However, it is quite
limited, because there is no way to compare angles if the pair (of
angles) do not share a common side. This can be remedied with an
additional set of axioms on the geometry: the axioms of congruence.
In an ordered geometry satisfying the congruence axioms, we have the concept of angle congruence. This binary relation turns out to be an equivalence relation, so we can form the set of equivalence classes on angles. Each equivalence class of angles is called a free angle. Any member of a free angle is called a representative of , which is simply an angle of form , where is the source of two rays and . We write . It is easy to see that given a point and a ray emanating from , we can find, in each free angle, a representative whose one side is . In other words, for any free angle , it is possible to write for some ray .
Now we are ready to define orderings on angles in general. In fact, this this done via free angles. Let be the set of all free angles in an ordered geometry satisfying the congruence axioms, and . Write and . We say that if ray is between and . The other inequality is dually defined. This is a well-defined binary relation. Given the ordering on free angles, we define if .
Let be a line and a point lying on . The point determines two opposite rays and lying on . Any ray emanating from that is distinct from either and determines exactly two angles: and . These two angles are said to be supplements of one another, or that one is supplementary of the other.
In an ordered geometry satisfying the congruence axioms, supplementary free angles are defined if each contains a representative that is supplementary to one another. Given two supplementary free angles , we may make comparisons of the two:
if , then we say that is a right free angle, or simple a right angle. Clearly is a right angle if is;
Given any two free angles, we can always compare them. In other words, the law of trichotomy is satisfied by the ordering of free angles: for any and , exactly one of
3 Operations on Angles
Let be an ordered geometry satisfying the congruence axioms and and are two free angles. Write and . If is between and , we define an “addition” of and , written as the free angle with representative . In symbol, this says that if is between and , then
This is a well-defined binary operation, provided that one free angle is between the other two. Therefore, the sum of a pair of supplementary angles is not defined! In addition, if and are two free angles, such that there exists a free angle with , then is unique and we denote it by . It is also possible to define the multiplication of a free angle by a positive integer, provided that the resulting angle is a well-defined free angle. Finally, division of a free angle by positive integral powers of 2 can also be defined.
4 Angle Measurement
An angle measure is a function defined on free angles of an ordered geometry with the congruence axioms, such that
is real-valued and positive,
is additive; in other words, , if is defined;
Here are some properties:
if , then .
for any free angle , denote its supplement by . Then is a positive constant that does not depend on .
is bounded above by .
if and are angle measures, then defined by is an angle measure too.
if is an angle measure, then for any positive real number , defined by is also an angle measure. In the event that is an integer such that makes sense, we also have .
If is a neutral geometry, then we impose a third requirement for a function to be an angle measure:
for any real number with , there is a free angle such that .
Once the measure of a free angle is defined, one can next define the
measure of an angle: let be a measure of the free
angles, define on angles by
. This is a well-defined function. It is easy to
see that iff , and iff
Two popular angle measures are the degree measure and the radian measure. In the degree measure, . In the radian measure, .
- 1 D. Hilbert, Foundations of Geometry, Open Court Publishing Co. (1971)
- 2 K. Borsuk and W. Szmielew, Foundations of Geometry, North-Holland Publishing Co. Amsterdam (1960)
- 3 M. J. Greenberg, Euclidean and Non-Euclidean Geometries, Development and History, W. H. Freeman and Company, San Francisco (1974)
|Date of creation||2013-03-22 15:32:36|
|Last modified on||2013-03-22 15:32:36|
|Last modified by||CWoo (3771)|