Let $L=L_1 \cup ... \cup L_n$ be the union of $n$ distinct lines through the origin in $\mathbb{R}^{3}$. I'd like a convincing argument that $\mathbb{R}^{3} \setminus L$ is homotopy equivalent to a wedge of $n$ circles (if that is true). In fact, I especially care about the case $n=2$.

I know this sounds like a homework problem, but I have other purposes in mind (this space naturally showed up as the fibre in a certain fibration) and I don't find typical text-book explanations of such problems very convincing, so I would appreciate a clear answer.