Tertius is an Inductive Logic Programming system that performs confirmatory induction, i.e., it looks for the n clauses that have the highest value of a confirmation evaluation function. In this setting, background knowledge is very useful because it can improve the reliability of the evaluation function, assigning minimal confirmation to clauses that are implied by the background knowledge and increasing the confirmation of the remaining clauses. We propose the algorithms Background1 and Background2 that look for clauses in the background that imply the clause under evaluation by Tertius. Both are based on a simplified implication test that is correct with respect to theta-subsumption but not complete. The implication test is not complete because we want to keep the run time inside acceptable bounds. We compare Background1 with Background2 on two datasets. The results show that Background2 is more efficient than Background1. Moreover, we also present the algorithm Preprocess that infers new clauses from the background knowledge in order to exploit it as much as possible. The algorithm modifies the consequence finding algorithm proposed by Inoue by reducing its execution time while giving up completeness.