We study random graphs with latent geometric structure, where the probability of each edge depends on the underlying random positions corresponding to the two endpoints. We focus on the setting where this conditional probability is a general monotone increasing function of the inner product of two vectors; such a function can naturally be viewed as the cumulative distribution function of some independent random variable. We consider a one-parameter family of random graphs, characterized by the variance of this random variable, that smoothly interpolates between a random dot product graph and an Erdős--Rényi random graph. We prove phase transitions of detecting geometry in these graphs, in terms of the dimension of the underlying geometric space and the variance parameter of the conditional probability. When the dimension is high or the variance is large, the graph is similar to an Erdős--Rényi graph with the same edge density that does not possess geometry; in other parameter regimes, there is a computationally efficient signed triangle statistic that distinguishes them. The proofs make use of information-theoretic inequalities and concentration of measure phenomena.