1. **State the problem:** We have two similar quadrilaterals WXYZ and LMNP. The angles of WXYZ are given as $\angle W = 112^\circ$, $\angle X = 74^\circ$, $\angle Z = 72^\circ$, and $\angle Y = 102^\circ$. We need to find the measure of $\angle M$ in LMNP. 2. **Recall the property of similar polygons:** Corresponding angles in similar polygons are congruent. This means each angle in LMNP matches the measure of the corresponding angle in WXYZ. 3. **Identify corresponding angles:** Since the quadrilaterals are named in order, $\angle W$ corresponds to $\angle L$, $\angle X$ to $\angle M$, $\angle Y$ to $\angle N$, and $\angle Z$ to $\angle P$. 4. **Find $\angle M$:** $\angle M$ corresponds to $\angle X$, which measures $74^\circ$. **Final answer:** $$\boxed{74^\circ}$$