Almost Congruent Triangles: Geometry, Arithmetic, and Generalizations

We characterize pairs of noncongruent triangles having the same three angle measures and twoequal, noncorresponding side lengths, and study their primitive integer parametrizations, finite chains,and asymptotic counting laws. The golden ratio governs the admissible geometric range and the acute–obtuse classification. For an associated family of tetrahedra we determine exact volumes, a Pythagoreancriterion for integrality, and a sharp universal divisibility property. We also obtain the limiting distri-bution of the similarity ratios, explicit expressions for the geometry of opposite edges, the inradius andcircumradius, the distance between their centers, and the dihedral angles, together with Diophantinecriteria for the rationality of their sines. The right-triangle specialization yields a dissection of theKepler triangle and a Fibonacci substitution matrix. For rational similarity ratios we establish an ob-struction to monohedral face-to-face tilings of three-dimensional Euclidean space. Previously knownresults are identified and the scope of the tiling conclusions is stated precisely.