A Δ-groupoid is an algebraic structure which axiomatizes the combinatorics of a truncated tetrahedron. It is shown that there are relations of Δ-groupoids to rings, group pairs, and (ideal) triangulations of three-manifolds. In particular, we describe a class of representations of group pairs H⊂G into the group of upper triangular two-by-two matrices over an arbitrary ring R, and associate to that group pair a universal ring so that any representation of that class factorizes through a respective ring homomorphism. These constructions are illustrated by two examples coming from knot theory, namely the trefoil and the figure-eight knots. It is also shown that one can associate a Δ-groupoid to ideal triangulations of knot complements, and a homology of Δ-groupoids is defined.