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Preliminaries: SAS triangle congruence is an axiom. (1) implies one direction of the Isosceles Triangle Theorem, namely: If two sides of a triangle are congruent, then the angles opposite those sides are congruent. [⋆] [⋆] (2) implies that A point equidistant from distinct points P P and Q Q lies on the perpendicular bisector of the ¯ PQ P ...
The SAS criterion for congruence is generally taken as an axiom. From this, and using other postulates of Euclid, we can derive the ASA and SSS criterion. The proof proceeds generally by contariction. For ASA criterion, we cut one of the sides so as to make it equal to corresponding part of the other triangle, and then derive contradiction.
From what I learned, SSS belongs to Euclid's Book 1 and is the eighth proposition. It is not a postulate and can be proven from the previous propositions, the four postulates and the axioms. The fifth postulate is only applicable from proposition 29 onwards, where the discrepancies came in.:)
1. Two triangles are congruent if their corresponding sides are equal in length and their corresponding angles are equal in size. If you are given that corresponding sides are equal in length, you can easily apply the Cosine Rule and obtain that each of the corresponding angles are also equal. Hence the two triangles are congruent.
Sep 15, 2013 at 18:53. @Mayank The formula for the area of a triangle is 1 2fh 1 2 f h where f f is the base and h h is the height. It is standard to name the vertices of a triangle A, B, C A, B, C and to use the same letters for the angles, while the sides opposite are respectively a, b, c a, b, c so the vertex A A is where sides b b and c c ...
I have seen a proof of SAS congruence rule using rigid transformations. So, why isn't it a theorem. Also, are there any other proofs of it using the existing postulates? I am very confused regarding this matter.
If your definition of a triangle is written solely in terms of the three sides (A, B, C) and three angles accompanying them (a, b, c), basic rules like the sine rule for area can easily prove SSS by contradiction, etc, as a criterion for congruence is that all basic properties like area, volume, etc are the same. – egglog.
2. SAA is valid even in "neutral/absolute geometry", which takes no stand on the Parallel Postulate—or, equivalently, the Angle-Sum Theorem. (Importantly, Side-Angle-Side itself is a postulate in neutral geometry, because you have to start somewhere when comparing triangles.) See, for instance, this 1977 The Mathematics Teacher article ...
Geometry questions and answers. Theorem 11.1.9. SSS Congruence Theorem If the vertices of two triangles are in one-to-one correspon- dence such that all three sides of one triangle are congruent, respectively, to all three sides of the second triangle, then the triangles are congruent. hint for proof. (Use the Hinge Theorem)
Geometry questions and answers. In taxicab geometry, show that the SSS congruence theorem is false by finding triangles delta ABC and delta A'B'C' with AB = A' B', AC = A'C', and BC = B'C' but such that delta ABC deltaA'B'C'.