Accurate predictions of orthology and paralogy relationships are necessary to infer human molecular function from experiments in model organisms. Previous genome-scale approaches to predicting these relationships have been limited by their use of protein similarity and their failure to take into account multiple splicing events and gene prediction errors. We have developed PhyOP, a new phylogenetic orthology prediction pipeline based on synonymous rate estimates, which accurately predicts orthology and paralogy relationships for transcripts, genes, exons, or genomic segments between closely related genomes. We were able to identify orthologue relationships to human genes for 93% of all dog genes from Ensembl. Among 1:1 orthologues, the alignments covered a median of 97.4% of protein sequences, and 92% of orthologues shared essentially identical gene structures. PhyOP accurately recapitulated genomic maps of conserved synteny. Benchmarking against predictions from Ensembl and Inparanoid showed that PhyOP is more accurate, especially in its predictions of paralogy. Nearly half (46%) of PhyOP paralogy predictions are unique. Using PhyOP to investigate orthologues and paralogues in the human and dog genomes, we found that the human assembly contains 3-fold more gene duplications than the dog. Species-specific duplicate genes, or "in-paralogues," are generally shorter and have fewer exons than 1:1 orthologues, which is consistent with selective constraints and mutation biases based on the sizes of duplicated genes. In-paralogues have experienced elevated amino acid and synonymous nucleotide substitution rates. Duplicates possess similar biological functions for either the dog or human lineages. Having accounted for 2,954 likely pseudogenes and gene fragments, and after separating 346 erroneously merged genes, we estimated that the human genome encodes a minimum of 19,700 protein-coding genes, similar to the gene count of nematode worms. PhyOP is a fast and robust approach to orthology prediction that will be applicable to whole genomes from multiple closely related species. PhyOP will be particularly useful in predicting orthology for mammalian genomes that have been incompletely sequenced, and for large families of rapidly duplicating genes.