1. **State the problem:** We need to find the measure of angle $\angle X$ in rhombus VWXY given that $m\angle W = 2p + 38^\circ$ and $m\angle V = 4p + 82^\circ$. 2. **Recall properties of a rhombus:** Opposite angles in a rhombus are equal, and adjacent angles are supplementary (sum to $180^\circ$). 3. **Set up the equation:** Since $\angle W$ and $\angle V$ are adjacent angles, we have: $$ (2p + 38) + (4p + 82) = 180 $$ 4. **Simplify the equation:** $$ 2p + 38 + 4p + 82 = 180 $$ $$ 6p + 120 = 180 $$ 5. **Solve for $p$:** $$ 6p = 180 - 120 $$ $$ 6p = 60 $$ $$ p = \frac{60}{6} $$ $$ p = 10 $$ 6. **Find $m\angle W$:** $$ m\angle W = 2p + 38 = 2(10) + 38 = 20 + 38 = 58^\circ $$ 7. **Find $m\angle X$:** Since $\angle X$ is opposite $\angle V$, and opposite angles are equal in a rhombus, $$ m\angle X = m\angle V = 4p + 82 = 4(10) + 82 = 40 + 82 = 122^\circ $$ **Final answer:** $$ m\angle X = 122^\circ $$