1 Definition

In an ordered geometryMathworldPlanetmath S, given a point p let Π(p) be the family of all rays emanating from it. Let α,βΠ(p) such that αβ and α-β. The angle between rays ( α and β at p is

{ρΠ(p)ρ is between α and β}.

This angle is denoted by αpβ. The two rays α and β are the sides of the angle, and p the vertex of the angle. Since any point (other than the source p) on a ray uniquely determines the ray, we may also write the angle by apb, whenever we have points aα and bβ.

The notational device given for the angle suggests the possibility of defining an angle between two line segments satisfying certain conditions: let pq¯ and qr¯ be two open line segments with a common endpoint q. The angle between the two open line segments is the angle between the rays qp and qr. In this case, we may denote the angle by pqr.

Suppose is a line and p a point lying on . We have two opposite rays emanating from p that lie on . Call them σ and -σ. Any ray ρ emanating from a point p that does not lie on produces two angles at p, one between ρ and σ and the other between ρ and -σ. These two angles are said to be supplement of one another, or that σpρ is supplementary of (-σ)pρ. Every angle has exactly two supplements.

2 Ordering of Angles

Let S be an ordered geometry and ρ a ray in S with source point p. Consider the set E of all angles whose one side is ρ. Define an ordering on E by the following rule: for αpρ,βpρE,

  1. 1.

    αpρ=βpρ if α=β,

  2. 2.

    αpρ<βpρ if αβpρ, and

  3. 3.

    αpρ>βpρ if βαpρ.

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 geometryMathworldPlanetmathPlanetmathPlanetmath: the axioms of congruence.

In an ordered geometry satisfying the congruence axioms, we have the concept of angle congruencePlanetmathPlanetmathPlanetmathPlanetmath. This binary relationMathworldPlanetmath turns out to be an equivalence relationMathworldPlanetmath, so we can form the set of equivalence classesMathworldPlanetmath on angles. Each equivalence class of angles is called a free angle. Any member of a free angle 𝔞 is called a representative of 𝔞, which is simply an angle of form abc, where b is the source of two rays ba and bc. We write 𝔞=[abc]. It is easy to see that given a point p and a ray ρ emanating from p, we can find, in each free angle, a representative whose one side is ρ. In other words, for any free angle 𝔞, it is possible to write 𝔞=[αpρ] for some ray α.

Now we are ready to define orderings on angles in general. In fact, this this done via free angles. Let 𝔄 be the set of all free angles in an ordered geometry satisfying the congruence axioms, and 𝔞,𝔟𝔄. Write 𝔞=[αpρ] and 𝔟=[βpρ]. We say that 𝔞<𝔟 if ray α is between β and ρ. The other inequality is dually defined. This is a well-defined binary relation. Given the ordering on free angles, we define αpβ<γqδ if [αpβ]<[γqδ].

Let be a line and p a point lying on . The point p determines two opposite rays ρ and -ρ lying on . Any ray σ emanating from p that is distinct from either ρ and -ρ determines exactly two angles: ρpσ and (-ρ)pσ. 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 𝔞,𝔟, we may make comparisons of the two:

  • if 𝔞=𝔟, then we say that 𝔞 is a right free angle, or simple a right angleMathworldPlanetmath. Clearly 𝔟 is a right angle if 𝔞 is;

  • if 𝔞>𝔟, then 𝔞 is called an obtuse free angle, or an obtuse angle. The supplement of an obtuse angle is called an acute free angle, or an acute angle. Thus, 𝔟 is acute if 𝔞 is obtuse.

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 𝔞 and 𝔟, exactly one of

𝔞>𝔟    𝔞=𝔟    𝔞<𝔟

is true.

3 Operations on Angles

Let S be an ordered geometry satisfying the congruence axioms and 𝔞 and 𝔟 are two free angles. Write 𝔞=[αpβ] and 𝔟=[βpγ]. If β is between α and γ, we define an “additionPlanetmathPlanetmath” of 𝔞 and 𝔟, written 𝔞+𝔟 as the free angle 𝔠 with representative αpγ. In symbol, this says that if β is between α and γ, then


This is a well-defined binary operationMathworldPlanetmath, 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 𝔞 and 𝔠 are two free angles, such that there exists a free angle 𝔟 with 𝔞+𝔟=𝔠, then 𝔟 is unique and we denote it by 𝔠-𝔞. 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 𝒜 is a function defined on free angles of an ordered geometry S with the congruence axioms, such that

  1. 1.

    𝒜 is real-valued and positive,

  2. 2.

    𝒜 is additive; in other words, 𝒜(𝔞+𝔟)=𝒜(𝔞)+𝒜(𝔟), if 𝔞+𝔟 is defined;

Here are some properties:

  • if 𝒜(𝔞)=𝒜(𝔟), then 𝔞=𝔟.

  • 𝔞>𝔟 iff 𝒜(𝔞)>𝒜(𝔟).

  • for any free angle 𝔞, denote its supplement by 𝔞s. Then 𝒜(𝔞)+𝒜(𝔞s) is a positive constant r𝒜 that does not depend on 𝔞.

  • 𝒜 is bounded above by r𝒜.

  • if 𝒜 and are angle measures, then 𝒜+ defined by (𝒜+)(𝔞)=𝒜(𝔞)+(𝔞) is an angle measure too.

  • if 𝒜 is an angle measure, then for any positive real number r, r𝒜 defined by (r𝒜)(𝔞)=r(𝒜(𝔞)) is also an angle measure. In the event that r is an integer such that r𝔞 makes sense, we also have r(𝒜(𝔞))=𝒜(r𝔞).

If S is a neutral geometry, then we impose a third requirement for a function to be an angle measure:

  1. 3.

    for any real number r with 0<r<r𝒜, there is a free angle 𝔞 such that 𝒜(𝔞)=r.

Once the measure of a free angle is defined, one can next define the measure of an angle: let 𝒜 be a measure of the free angles, define 𝒜 on angles by 𝒜(αpβ)=𝒜([αpβ]). This is a well-defined function. It is easy to see that 𝒜(αpβ)=𝒜(γqδ) iff αpβγqδ, and 𝒜(αpβ)>𝒜(γqδ) iff αpβ>γqδ.

Two popular angle measures are the degree measure and the radian measure. In the degree measure, r𝒜=180. In the radian measure, r𝒜=π.


  • 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, EuclideanMathworldPlanetmath and Non-Euclidean Geometries, Development and History, W. H. Freeman and Company, San Francisco (1974)
Title angle
Canonical name Angle
Date of creation 2013-03-22 15:32:36
Last modified on 2013-03-22 15:32:36
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 22
Author CWoo (3771)
Entry type Definition
Classification msc 51F20
Classification msc 51G05
Synonym supplement
Related topic PaschsTheorem
Related topic BetweennessInRays
Related topic SupplementaryAngles
Defines supplementary
Defines right angle
Defines between rays
Defines crossbar theorem
Defines free angle
Defines acute angle
Defines obtuse angle
Defines angle measure
Defines side
Defines vertex