1. **State the problem:** We need to find the value of $x$ in a triangle where the three angles are given as $(63 + x)^\circ$, $(183 - x)^\circ$, and $2x^\circ$. 2. **Recall the triangle angle sum rule:** The sum of the interior angles of any triangle is always $180^\circ$. 3. **Set up the equation:** $$ (63 + x) + (183 - x) + 2x = 180 $$ 4. **Simplify the equation:** $$ 63 + x + 183 - x + 2x = 180 $$ $$ (63 + 183) + (x - x + 2x) = 180 $$ $$ 246 + 2x = 180 $$ 5. **Isolate $x$:** $$ 2x = 180 - 246 $$ $$ 2x = -66 $$ 6. **Solve for $x$:** $$ x = \frac{-66}{2} $$ $$ x = -33 $$ 7. **Interpretation:** The value of $x$ is $-33$. This means the angles are: - $63 + (-33) = 30^\circ$ - $183 - (-33) = 216^\circ$ - $2 \times (-33) = -66^\circ$ Since angles in a triangle cannot be negative or exceed $180^\circ$, this suggests the given angle expressions do not form a valid triangle for any real $x$. However, mathematically, the solution to the equation is $x = -33$. **Final answer:** $$ x = -33 $$