  A graph is $k$-degenerate if every subgraph has minimum degree at most $k$.
We provide lower bounds on the size of a maximum induced 2-degenerate subgraph
in a triangle-free planar graph. We denote the size of a maximum induced
2-degenerate subgraph of a graph $G$ by $\alpha_2(G)$. We prove that if $G$ is
a connected triangle-free planar graph with $n$ vertices and $m$ edges, then
$\alpha_2(G) \geq \frac{6n - m - 1}{5}$. By Euler's Formula, this implies
$\alpha_2(G) \geq \frac{4}{5}n$. We also prove that if $G$ is a triangle-free
planar graph on $n$ vertices with at most $n_3$ vertices of degree at most
three, then $\alpha_2(G) \geq \frac{7}{8}n - 18 n_3$.
