I'm trying to find the dual cones for each of the following cones: $K=\left\{\left(x_{1}, x_{2}\right)\mid \left| x_{1}\right| \leq x_{2}\right\}$ and $K=\left\{\left(x_{1}, x_{2}\right) \mid x_{1} +x_{2} = 0\right\}$. My idea is to find all vectors $\mathbf{x}\in\mathbb{R}^2$ such that $\mathbf{x}^\top\mathbf{y}\geq 0, \forall \mathbf{y}\in K$. But it is too tedious to vertify each possible choices of $\mathbf{y}\in K$ and this does not give some insights of how the dual cones look like.
Could anyone give me some suggestions on how to find the dual cones for each of the cones: $K=\left\{\left(x_{1}, x_{2}\right)\mid \left| x_{1}\right| \leq x_{2}\right\}$ and $K=\left\{\left(x_{1}, x_{2}\right) \mid x_{1} +x_{2} = 0\right\}$? It is also appreciated if some geometry or intuition are provided.
Thanks.