1. **Problem Statement:** Find the value of $x$ in rhombus $RSTU$ where the angle at vertex $R$ is $(x + 18)^\circ$ and the angle at vertex $S$ is $2x^\circ$. 2. **Important Properties of a Rhombus:** - Opposite angles in a rhombus are equal. - Adjacent angles are supplementary, meaning their sum is $180^\circ$. 3. **Set up the equation:** Since $R$ and $S$ are adjacent angles, their sum is $180^\circ$: $$ (x + 18) + 2x = 180 $$ 4. **Simplify the equation:** $$ x + 18 + 2x = 180 $$ $$ 3x + 18 = 180 $$ 5. **Isolate $x$:** $$ 3x + \cancel{18} - \cancel{18} = 180 - 18 $$ $$ 3x = 162 $$ 6. **Divide both sides by 3:** $$ \frac{3x}{\cancel{3}} = \frac{162}{\cancel{3}} $$ $$ x = 54 $$ 7. **Final answer:** The value of $x$ is $54^\circ$.