1. **State the problem:** We need to find the measure of angle $C$ in rhombus $BCDE$ where $m\angle C = x + 56^\circ$ and $m\angle D = x$.\n\n2. **Recall properties of a rhombus:** Opposite angles in a rhombus are equal, and adjacent angles are supplementary (sum to $180^\circ$).\n\n3. **Set up the equation:** Since $\angle C$ and $\angle D$ are adjacent, their measures add up to $180^\circ$. So,\n$$ (x + 56) + x = 180 $$\n\n4. **Simplify the equation:**\n$$ 2x + 56 = 180 $$\n\n5. **Isolate $x$:**\n$$ 2x = 180 - 56 $$\n$$ 2x = 124 $$\n$$ x = \frac{124}{2} $$\n$$ x = 62 $$\n\n6. **Find $m\angle C$:**\n$$ m\angle C = x + 56 = 62 + 56 = 118^\circ $$\n\n**Final answer:** $m\angle C = 118^\circ$