 Solution

Using the decomposition from part (ii), $U_{\theta }^{-1}$ maps Alice's $|{\psi }_1⟩$ to $|0⟩$ and $|{\psi }_2⟩$ to $|1⟩$. Alice obtains $0$ or $1$, each with probability $1/2$. Conditional on outcome $0$, Bob has $|{\psi }_1⟩$; conditional on outcome $1$, Bob has $|{\psi }_2⟩$.