Unizor - Geometry2D - Triangles - Congruence

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Let's discuss three important statements about congruence of any pair of triangles.
1. Side-angle-side (SAS): if two sides and an angle they form of one triangle are congruent to corresponding sides and an angle they form of another triangle, then these two triangles are congruent to each other. It is important that in both triangles the corresponding congruent angles are those that are formed by congruent sides. This statement was accepted by Hilbert as an axiom III.5 of geometry. There are illustrative explanations of why this is a true statement (e.g. using non-deforming transformation), but, strictly speaking, they cannot be regarded as rigorous. That is why this statement is included in Hilbert's system of axioms.
2. Angle-side-angle (ASA): if two angles and a side common to both of one triangle are congruent to corresponding angles and their common side of another triangle, then these two triangles are congruent to each other. This statement can be proven based on SAS axiom.
3. Side-side-side (SSS): if all three sides of one triangle are congruent to corresponding three sides of another triangle, then these two triangles are congruent to each other. This statement can also be proven based on SAS axiom.

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• Triangle
• Congruent
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[zshekhtm]

Yes. Euclid's geometry was a great achievement, but from a contemporary standpoint not always rigorous enough. Hilbert put it on a solid foundation, which seems to be quite complex for a seemingly simple subject, but absolutely necessary to be really rigorous. Surprisingly, the fundamentals of almost any subject, if built absolutely rigorously, are quite complex. That's why I hesitated about how deep I should immerse my students into it.

[zshekhtm]

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[bluetwinky]

@zshekhtm

You are referring to "Foundations of Geometry" by David Hilbert?