1. **Problem Statement:** We have two similar triangles, ABC and XYZ. We know that $\angle Y = 30^\circ$ in triangle XYZ, and we want to find the measure of $\angle B$ in triangle ABC. 2. **Key Concept:** In similar triangles, corresponding angles are congruent. This means $\angle A = \angle X$, $\angle B = \angle Y$, and $\angle C = \angle Z$. 3. **Apply the Concept:** Since $\angle B$ corresponds to $\angle Y$, and $\angle Y = 30^\circ$, it follows that: $$\angle B = \angle Y = 30^\circ$$ 4. **Answer:** The measure of $\angle B$ is 30.