angle

1 Definition

In an ordered geometry $S$, given a point $p$ let $\mathrm{\Pi }(p)$ be the family ofall rays emanating from it. Let $\alpha ,\beta \in \mathrm{\Pi }(p)$ such that$\alpha \ne \beta$ and $\alpha \ne -\beta$. The angle between rays (http://planetmath.org/BetweennessInRays)$\alpha$ and $\beta$ at $p$ is

$$\{\rho \in \mathrm{\Pi }(p)\mid \rho \text{ is between }\alpha \text{ and }\beta \}.$$

This angleis denoted by $\mathrm{\angle }\alpha p\beta$. The two rays $\alpha$ and$\beta$ are the sides of the angle, and $p$ the vertex of theangle. Since any point (otherthan the source $p$) on a ray uniquely determines the ray, we mayalso write the angle by $\mathrm{\angle }apb$, whenever we have points$a\in \alpha$ and $b\in \beta$.

The notational device given for the angle suggeststhe possibility of defining an angle between two line segments satisfying certain conditions:let $\overline{pq}$ and $\overline{qr}$ be two open line segments with a common endpoint$q$. The angle between the two open line segments is the angle between therays $\overrightarrow{qp}$ and $\overrightarrow{qr}$. In this case, we may denote the angle by $\mathrm{\angle }pqr$.

Suppose $\mathrm{\ell }$ is a line and $p$ a point lying on $\mathrm{\ell }$. We havetwo opposite rays emanating from $p$ that lie on $\mathrm{\ell }$. Call them$\sigma$ and $-\sigma$. Any ray $\rho$ emanating from a point $p$that does not lie on $\mathrm{\ell }$ produces two angles at $p$, one between$\rho$ and $\sigma$ and the other between $\rho$ and $-\sigma$.These two angles are said to be supplement of one another, orthat $\mathrm{\angle }\sigma p\rho$ is supplementary of $\mathrm{\angle }(-\sigma )p\rho$. Every angle has exactly two supplements.

2 Ordering of Angles

Let $S$ be an ordered geometry and $\rho$ a ray in $S$ with sourcepoint $p$. Consider the set $E$ of all angles whose one side is$\rho$. Define an ordering on $E$ by the following rule: for$\mathrm{\angle }\alpha p\rho ,\mathrm{\angle }\beta p\rho \in E$,

1. $\mathrm{\angle }\alpha p\rho =\mathrm{\angle }\beta p\rho$ if $\alpha =\beta$,

2. if $\alpha \in \mathrm{\angle }\beta p\rho$, and

3. $\mathrm{\angle }\alpha p\rho >\mathrm{\angle }\beta p\rho$ if $\beta \in \mathrm{\angle }\alpha p\rho$.

The ordering relation above is well-defined. However, it is quitelimited, because there is no way to compare angles if the pair (ofangles) do not share a common side. This can be remedied with anadditional set of axioms on the geometry: the axioms of congruence.

In an ordered geometry satisfying the congruence axioms, we have theconcept of angle congruence. This binary relation turns out to be anequivalence relation, so we can form the set of equivalence classeson angles. Each equivalence class of angles is called a freeangle. Any member of a free angle $𝔞$ is called arepresentative of $𝔞$, which is simply an angle of form$\mathrm{\angle }abc$, where $b$ is the source of two rays $\overrightarrow{ba}$ and$\overrightarrow{bc}$. We write $𝔞=[\mathrm{\angle }abc]$. It is easy tosee that given a point $p$ and a ray $\rho$ emanating from $p$, wecan find, in each free angle, a representative whose one side is$\rho$. In other words, for any free angle $𝔞$, it ispossible to write $𝔞=[\mathrm{\angle }\alpha p\rho ]$ for some ray$\alpha$.

Now we are ready to define orderings on angles in general. In fact,this this done via free angles. Let $𝔄$ be the set ofall free angles in an ordered geometry satisfying the congruenceaxioms, and $𝔞,𝔟\in 𝔄$. Write$𝔞=[\mathrm{\angle }\alpha p\rho ]$ and $𝔟=[\mathrm{\angle }\beta p\rho ]$. We say that if ray $\alpha$ isbetween $\beta$ and $\rho$. The other inequality is dually defined.This is a well-defined binary relation. Given the ordering on freeangles, we define if.

Let $\mathrm{\ell }$ be a line and $p$ a point lying on $\mathrm{\ell }$. The point $p$determines two opposite rays $\rho$ and $-\rho$ lying on $\mathrm{\ell }$. Anyray $\sigma$ emanating from $p$ that is distinct from either $\rho$and $-\rho$ determines exactly two angles: $\mathrm{\angle }\rho p\sigma$ and$\mathrm{\angle }(-\rho )p\sigma$. These two angles are said to besupplements of one another, or that one is supplementary of theother.

In an ordered geometry satisfying the congruence axioms,supplementary free angles are defined if each contains arepresentative that is supplementary to one another. Given twosupplementary free angles $𝔞,𝔟$, we may makecomparisons of the two:

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

• if $𝔞>𝔟$, then $𝔞$ is calledan obtuse free angle, or an obtuse angle. Thesupplement 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 otherwords, the law of trichotomy is satisfied by the ordering of freeangles: for any $𝔞$ and $𝔟$, exactly one of

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$𝔞=[\mathrm{\angle }\alpha p\beta ]$ and $𝔟=[\mathrm{\angle }\beta p\gamma ]$. If $\beta$ is between $\alpha$ and $\gamma$, we definean “addition” of $𝔞$ and $𝔟$, written$𝔞+𝔟$ as the free angle $𝔠$ withrepresentative $\mathrm{\angle }\alpha p\gamma$. In symbol, this says that if$\beta$ is between $\alpha$ and $\gamma$, then

$$[\mathrm{\angle }\alpha p\beta ]+[\mathrm{\angle }\beta p\gamma ]=[\mathrm{\angle }\alpha p\gamma ].$$

This is awell-defined binary operation, provided that one free angle isbetween the other two. Therefore, the sum of a pair ofsupplementary angles is not defined! In addition, if $𝔞$and $𝔠$ are two free angles, such that there exists afree angle $𝔟$ with$𝔞+𝔟=𝔠$, then $𝔟$ isunique and we denote it by $𝔠-𝔞$. It is alsopossible to define the multiplication of a free angle by a positiveinteger, provided that the resulting angle is a well-defined freeangle. Finally, division of a free angle by positive integral powersof 2 can also be defined.

4 Angle Measurement

An angle measure $𝒜$ is a function defined on free anglesof an ordered geometry $S$ with the congruence axioms, such that

1. $𝒜$ is real-valued and positive,

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${𝔞}^s$. Then $𝒜(𝔞)+𝒜({𝔞}^s)$ is a positive constant$r_{𝒜}$ that does not depend on $𝔞$.

• $𝒜$ is bounded above by $r_{𝒜}$.

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

• if $𝒜$ is an angle measure, then for any positivereal number $r$, $r𝒜$ defined by$(r𝒜)(𝔞)=r(𝒜(𝔞))$ is alsoan 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 fora function to be an angle measure:

1. for any real number $r$ with , there is afree angle $𝔞$ such that$𝒜(𝔞)=r$.

Once the measure of a free angle is defined, one can next define themeasure of an angle: let $𝒜$ be a measure of the freeangles, define ${𝒜}^{\prime }$ on angles by${𝒜}^{\prime }(\mathrm{\angle }\alpha p\beta )=𝒜([\mathrm{\angle }\alpha p\beta ])$. This is a well-defined function. It is easy tosee that ${𝒜}^{\prime }(\mathrm{\angle }\alpha p\beta )={𝒜}^{\prime }(\mathrm{\angle }\gamma q\delta )$ iff $\mathrm{\angle }\alpha p\beta \cong \mathrm{\angle }\gamma q\delta$, and ${𝒜}^{\prime }(\mathrm{\angle }\alpha p\beta )>{𝒜}^{\prime }(\mathrm{\angle }\gamma q\delta )$ iff$\mathrm{\angle }\alpha p\beta >\mathrm{\angle }\gamma q\delta$.

Two popular angle measures are the degree measure and the radianmeasure. In the degree measure, $r_{𝒜}={180}^{\circ }$. Inthe radian measure, $r_{𝒜}=\pi$.