Before revision:  We prove that the size of a maximum matching is at least \beta(n)/2 where \beta(n) is the minimum, over all \Theta_6-graphs with n vertices, of the minimum size of a blocking set.
After revision:  We prove that the size of a maximum matching is at least \beta(n)/2 where \beta(n) is the minimum, over all theta-six graphs with n vertices, of the minimum size of a blocking set.