Let $D$ be a strongly minimal set in the language $L$, and $D' \supset D$ an elementary extension with infinite dimension over $D$. Add to $L$ a unary predicate symbol $\mathbf{D}$ and let $T'$ be the theory of the structure $(D', D)$, where $D$ interprets the predicate $\mathbf{D}$. It is known that $T'$ is $\omega$-stable. We prove Theorem A. If $D$ is not locally modular, then $T'$ has Morley rank $\omega$. We say that a strongly minimal set $D$ is pseudoprojective if it is nontrivial and there is a $k < \omega$ such that, for all $a, b \in D$ and closed $X \subset D, a \in \mathrm{cl}(Xb) \Rightarrow$ there is a $Y \subset X$ with $a \in \mathrm{cl}(Yb)$ and $|Y| \leq k$. Using Theorem A, we prove Theorem B. If a strongly minimal set $D$ is pseudoprojective, then $D$ is locally projective. The following result of Hrushovski's (proved in $\S4$) plays a part in the proof of Theorem B. Theorem C. Suppose that $D$ is strongly minimal, and there is some proper elementary extension $D_1$ of $D$ such that the theory of the pair $(D_1, D)$ is $\omega_1$-categorical. Then $D$ is locally modular.