Axiom 1.1 SAS Congruence: If two corresponding sides and the included angle of a pair of triangles are congruent, then the triangles are congruent.

N.B.: Considerable consternation has been expressed about SAS of an axiom whereas ASA and SSS are theorems. Euclid, in fact, treats all three as theorems but he relys on a proof based on superposition (moving one triangle without changing its size or shape and placing it on top of the other). But superposition is not justified in Euclids axioms (or postulates). Efforts to ‘correct’ Euclid have found alternative axiom systems and when we carefully develop transformational geometry, this difficulty is addressed via the concept of isometry.

Hilbert and Birkoff take SAS as an axiom and prove ASA and SSS as theorems. Some textbooks call all three of them postulates (axioms). One idea of a ‘good’ axiomatic system is to minimize the number of postulates (axioms) — that is, don’t assume anything that can be proved by previous axioms or theorems.

Some textbooks assume many postulates because many of the early ones are “obviously true” and tedious to prove as Theorems. That is why some school textbooks may present all of SAS, SSS, ASA, and HL as postulates. The goal is to move on to more significant theorems rather than worry about an axiomatic system with a minimal number of postulates.

## Answers ( )

Answer:Axiom 1.1 SAS Congruence: If two corresponding sides and the included angle of a pair of triangles are congruent, then the triangles are congruent.

N.B.: Considerable consternation has been expressed about SAS of an axiom whereas ASA and SSS are theorems. Euclid, in fact, treats all three as theorems but he relys on a proof based on superposition (moving one triangle without changing its size or shape and placing it on top of the other). But superposition is not justified in Euclids axioms (or postulates). Efforts to ‘correct’ Euclid have found alternative axiom systems and when we carefully develop transformational geometry, this difficulty is addressed via the concept of isometry.

Hilbert and Birkoff take SAS as an axiom and prove ASA and SSS as theorems. Some textbooks call all three of them postulates (axioms). One idea of a ‘good’ axiomatic system is to minimize the number of postulates (axioms) — that is, don’t assume anything that can be proved by previous axioms or theorems.

Some textbooks assume many postulates because many of the early ones are “obviously true” and tedious to prove as Theorems. That is why some school textbooks may present all of SAS, SSS, ASA, and HL as postulates. The goal is to move on to more significant theorems rather than worry about an axiomatic system with a minimal number of postulates.

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