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Transcript
Superposition
John T.
Baldwin
Superposition
John T. Baldwin
October 22, 2007
Congruence and Similarity
Superposition
John T.
Baldwin
1
Another take on Congruence
2
Similarity -the role of arithmetic and geometry
Congruence of triangles-Hilbert
Superposition
John T.
Baldwin
Basic notions: point, line, incidence, between, congruence
(angle,segment)
defined terms: segment angle
Axioms included SAS
Congruence of triangles-tranformation approach
Superposition
John T.
Baldwin
A transformation of a geometry is a permutation of the points
and a permutation of the lines that preserves incidence.
If it also preserves congruence of segments and angles then it is
a rigid motion
Now a key axiom is:
Superposition Axiom:
If angle BAC = DEF there is an isometry taking A to E and
such that B 0 A0 (i.e. BE ) lies on DE and C 0 A0 lies on FE .
yields SAS
Exercises
Superposition
John T.
Baldwin
Exercise: Prove SAS from this axiom and Hilbert order and
congruence up to but not including SAS.
Exercise: Prove that if AB ∼
= CD there is rigid motion mapping
A to C and B to D.
Congruence-Weinzweig
Superposition
John T.
Baldwin
Basic notions: point, line, incidence, between, motion
defined terms: segment angle, congruence
Definition. Two figures are congruent if they are mapped to
each other by a motion.
Axioms guarantee that the group of motions are ‘the right
group’:
SAS is a theorem.