An induced matching of a graph G is a set M of edges of G such that every two vertices of G that are incident with distinct edges in M have distance at least 2 in G. The maximum number of edges in an induced matching of G is the induced matching number νs(G) of G. In contrast to ordinary matchings, induced matchings in graphs are algorithmically difficult. Next to hardness results and efficient algorithms for restricted graph classes, lower bounds are therefore of interest.We show that if G is a connected graph of order n(G), maximum degree at most 3, girth at least 6, and without a cycle of length 7, then νs(G)≥n(G)−15, and we characterize the graphs achieving equality in this lower bound.