I've started to read "Coxeter - Introduction to Geometry" and I've encoutered a problem at page 5 (link to the book: https://www.amazon.com/gp/reader/0471504580/ref=sib_dp_pt#reader-link ), where Coxeter tells about the principle of superposition. Coxeter says that this principle is nowdays replaced with an axiom: the axiom of the rigidity of a triangle with a tail, numerated with 1.26.

Coxeter says that this axiom can be used to extend the notion of congruence to more complex figures than line segments, such as angles: in fact, previously he tells about congruence only for line segment (line 10-11-12).

Coxeter also says that this axiom can be used to prove the first law of congrunce for triangle (i.e., Euclid I.4), avoiding to use the principle of superposition (line 26).

Well... my questions are:
1) how can we define the congruence for angles using 1.26?
2) how can we prove Euclid I.4 using 1.26?
3) statement 1.26 can be proved using the third and first law of congruence for triangle: despite this possibility, it is really an axiom?

Thanks.

P.S.
I'm very sorry for my very bad english!

Saying "nowadays" implies a generality of sorts as if everyone universally is using Coxeter's I.26. I, for one, am unaware of any source, besides, Introduction to Geometry where I.26 is mentioned explicitly. It is certainly not among Hilbert's axioms.

>Coxeter says that this axiom can be used to extend the
>notion of congruence to more complex figures than line
>segments, such as angles: in fact, previously he tells about
>congruence only for line segment (line 10-11-12).

We may always try to guess what he meant.

>Coxeter also says that this axiom can be used to prove the
>first law of congrunce for triangle (i.e., Euclid
>I.4), avoiding to use the principle of superposition (line
>26).

This may be a good exercise.

>Well... my questions are:
>1) how can we define the congruence for angles using 1.26?

Assuming that angles are defined as a pair of rays (AU, AV) with a common point A, I can imagine the following definition (but my guess may be as good as yours):

(AU, AV) = (A'U', A'V') iff, for any B on AU, C on AV, B' on A'U', C' on A'V', AB = A'B' & AC = A'C' implies B'C'.

I propose that I.26 is used to prove that the definition of the angle congruence does not depend on a particular choice of B, C, B', C'.

>2) how can we prove Euclid I.4 using 1.26?

This is rather a direct consequence of the definition I just gave.

>3) statement 1.26 can be proved using the third and first
>law of congruence for triangle: despite this possibility, it
>is really an axiom?

I am sorry to say that this question does not make sense. It does not make sense to talk of a single axiom, but rather of a system of axioms. When one comes to putting together such an axiomatic system, his/hers choices are governed by considerations of consistency, frugality, esthetics, etc. In principle, math statements may be interchangeable. A theorem in an axiomatic system may be included as an axiom in another. In the latter, some or all axioms of the former will become theorems.

This is also discussed in:
Peter R. Cromwell, Polyhedra, p.219
(Cambridge University Press, 1997)