  $\Theta_6$-Graphs graphs are important geometric graphs that have many
applications especially in wireless sensor networks. They are equivalent to
Delaunay graphs where empty equilateral triangles take the place of empty
circles. We investigate lower bounds on the size of maximum matchings in these
graphs. The best known lower bound is $n/3$, where $n$ is the number of
vertices of the graph. Babu et al. (2014) conjectured that any $\Theta_6$-graph
has a (near-)perfect matching (as is true for standard Delaunay graphs).
Although this conjecture remains open, we improve the lower bound to
$(3n-8)/7$.
  We also relate the size of maximum matchings in $\Theta_6$-graphs to the
minimum size of a blocking set. Every edge of a $\Theta_6$-graph on point set
$P$ corresponds to an empty triangle that contains the endpoints of the edge
but no other point of $P$. A blocking set has at least one point in each such
triangle. 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. In the other direction, lower bounds on
matchings can be used to prove bounds on $\beta$, allowing us to show that
$\beta(n)\geq 3n/4-2$.
