Abstract: The classic Mattila-Sj\"olin theorem shows that if a compact subset of $\mathbb{R}^d$ has Hausdorff dimension at least $\frac{(d+1)}{2}$ then its set of distances has nonempty interior. In this talk, we present a similar result, namely that if the Hausdorff dimension of a compact subset $E \subset \mathbb{R}^d$, $d \geq 3$, is large enough then the set of congruence classes of triangles formed by triples of points of $E$ has nonempty interior. These types of results on point configurations with nonempty interior can be categorized as extensions and refinements of the statement in the well-known Falconer distance problem which establishes a positive Lebesgue measure for the set of distances instead of it having nonempty interior.