We study the influence of clustering, more specifically triangles, on cascading failures in interdependent networks or systems, in which we model the dependence between comprising systems using a dependence graph. First, we propose a new model that captures how the presence of triangles in the dependence graph alters the manner in which failures transmit from affected systems to others. Unlike existing models, the new model allows us to approximate the failure propagation dynamics using a multi-type branching process, even with triangles. Second, making use of the model, we provide a simple condition that indicates how increasing clustering will affect the likelihood that a random failure triggers a cascade of failures, which we call the probability of cascading failures (PoCF). In particular, our condition reveals an intriguing observation that the influence of clustering on PoCF depends on the vulnerability of comprising systems to an increasing number of failed neighboring systems and the current PoCF, starting with different types of failed systems. Our numerical studies hint that increasing clustering impedes cascading failures under both (truncated) power law and Poisson degree distributions. Furthermore, our finding suggests that, as the degree distribution becomes more concentrated around the mean degree with smaller variance, increasing clustering will have greater impact on the PoCF. A numerical investigation of networks with Poisson and power law degree distributions reflects this finding and demonstrates that increasing clustering reduces the PoCF much faster under Poisson degree distributions in comparison to power law degree distributions.