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I am reading the book "Foundations of Geometry" by Gerard A. Venema (2nd ed). A draft of the axiomatic setting of the book is the following (I will expand if required):

  1. Incidence postulate: For every pair of distinct points $A$ and $B$ there is exactly one line $l$ such that $A$, $B\in l$.
  2. Ruler postulate: For every pair of points $P$ and $Q$ there exists a number $PQ$ called distance from $P$ to $Q$. For each line $l$ there is a one to one correspondence from $l$ to $\mathbb{R}$ such that if $P$, $Q\in l$ correspond to the real numbers $x$ and $y$, respectively, then $PQ=|x-y|$.
  3. Plane separation postulate: For every line $l$, the points that do not lie on $l$ form two disjoint, nonempty sets $H_1$ and $H_2$ called half-planes bounded by $l$ such that $H_1$ and $H_2$ are convex and if $P\in H_1$ and $Q\in H_2$ then the segment $\overline{PQ}$ intersects $l$.
  4. Protractor postulate: For every angle $\angle BAC$ there is areal number $\mu\angle BAC$ called measure of $\angle BAC$ such that $0\leq \mu\angle BAC<180$, $\mu\angle BAC=0$ if and only if the rays $\overrightarrow{AB}=\overrightarrow{AC}$, for each $0<r<180$ and each half-plane $H$ bounded by the line $\overleftrightarrow{AB}$ there is an unique ray $\overrightarrow{AE}$ such that $E\in H$ and $\mu\angle BAE=r$, and if the ray $\overrightarrow{AD}$ is between rays the $\overrightarrow{AB}$ and $\overrightarrow{AC}$, then $\mu\angle BAD+\mu\angle DAC = \mu\angle BAC$.
  5. Side-Angle-Side postulate: If $\triangle ABD$ and $\triangle DEF$ are two tiangles such that $\overline{AB}\cong\overline{DE}$, $\angle ABC\cong \angle DEF$, and $\overline{BC}\cong\overline{EF}$, then $\triangle ABC\cong \triangle DEF$.

In the section C.3 of the book it is said that

"The protractor postulate is almost entirely unnecessary [...] in fact, the reflection postulate [see below] can replace both SAS and the Protractor postulate".

This quote lead me to believe that we can construct an angle measurement function using only the postulates 1, 2, 3 and the reflection postulate [see below]. However, the reflection postulate sais the following:

Reflection postulate: For every line $l$ there exists a bijective function $\rho_{l}$ such that it's only fixed points are the ones in $l$ and preserves collinearity, distance and angle measure.

So I don't really know how to prove that an angle measurement function exists using a postulate that requieres the existence of an angle measurement function.

The book references the paper "2-dimensional axiomatic geometry" by Frederic Ancel from "Abstracts of the American Mathematical Society 25 (2004), pages 329-345, which I am unable to find anywhere. The closest I can get is this, which is far from satisfactory.

Do you have some reference in which it could be found a construction of an angle measure based on a similar axiomatic setting? Any idea?

Some ideas

  1. We may state that two angles are congruent is they are isometric, i.e., there is a distance-preserving function between them.

  2. I am pretty sure that every function that preserves distance also preserves collinearity, so we can crop the asumption of collinearity, and, by definition, the asumption of angle measure.

  3. We may have to define what we mean by "right angle", a right angle may be any angle $\angle BAC$ such that $\rho_{\overleftrightarrow{AC}}(\overleftrightarrow{AB})=\overleftrightarrow{AB}$.

Defining angle measure the Archimedes way

In order to define an angle measure I was thinking about, for each angle $\angle BAC$, drawing a unit circle $\mathcal{C}$ around $A$, taking a finite set of points $\{D_0,\dotsc,D_n\}$ in $\overline{BC}$, the corresponding intersections $\{D_0',\dotsc,D_n'\}$, where $D_i':=\overrightarrow{AD_i}\cap\mathcal{C}$, and considering the broken line perimeter $\sum_{i=0}^{n-1}D_iD_{i+1}$. Then, somehow proving that the set of broken line perimeters is bounded above and finally defining the angle measure as the supremum, being hopefully easy to prove the remaining properties of the angle measure function.

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    $\begingroup$ I don't know this exact axiomatic setup but you can have a primitive notion of angle congruence separate from angle measure. Perhaps that is what is intended: the reflection postulate would then say $\rho_l$ takes angles to congruent angles. $\endgroup$ Jun 15, 2021 at 23:19
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    $\begingroup$ Given a notion of angle congruence and sufficient other axioms, you can then define an addition operation on the set of congruence classes of angles, and then you can use this to define angles that are rational multiples of a right angle, and then you can extend this to real measures of arbitrary angles by considering the Dedekind cut formed among the rational multiples of a right angle. $\endgroup$ Jun 15, 2021 at 23:23
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    $\begingroup$ Or perhaps the reflection postulate is meant to be modified to just not mention angles at all. Then you can define two angles as being congruent if one can be turned into the other by a sequence of reflections, and then use this to define angle measure as in my previous comment. (But as I said, I'm not familiar with the exact details of this axiomatic setup so I don't know if the axioms are actually strong enough to carry this out.) $\endgroup$ Jun 15, 2021 at 23:44
  • $\begingroup$ @EricWofsey I updated the post with the full axiomatic setup. I was having a different idea about the construction of an angle measure function as you can see. However, I'm obviously open to your proposal, but I don't really know how to define rational multiples of a right angle. $\endgroup$ Jun 16, 2021 at 10:37
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    $\begingroup$ You can bisect a right angle, then bisect the two 45-degree angles, and keep on bisecting, thus obtaining the set A of all angles with measure equal to 1/2^n of a right angle (n = 0, 1, 2, ...). Because any real number has a binary expansion, you can add an infinity of appropriate elements of A (angle adition and limit of an angle sum defined in the obvious way) to obtain an angle whose measure (in terms of right angles) is any real number between 0 and 2. $\endgroup$ Feb 16, 2022 at 16:47

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