1. $\frac{1}{\sqrt{2}}$ is irrational. Assume $\frac{1}{\sqrt{2}} = \frac{p}{q}$, where $p$ and $q$ are integers. Then $\sqrt{2} = \frac{q}{p}$, which implies $\sqrt{2}$ is rational. But $\sqrt{2}$ is irrational, so the assumption is false, proving that $\frac{1}{\sqrt{2}}$ is irrational.
2. $1 + \sqrt{5}$ is irrational. Assume $1 + \sqrt{5} = \frac{p}{q}$, then $\sqrt{5} = \frac{p - q}{q}$, which implies $\sqrt{5}$ is rational. But $\sqrt{5}$ is irrational, so the assumption is false.
3. $6 + \sqrt{2}$ is irrational. Assume $6 + \sqrt{2} = \frac{p}{q}$, then $\sqrt{2} = \frac{p - 6q}{q}$, which implies $\sqrt{2}$ is rational. But $\sqrt{2}$ is irrational, so the assumption is false.