1. **State the problem:** We are given a triangle with three angles: $(4x + 19)^\circ$, $111^\circ$, and $26^\circ$. We need to find the value of $x$. 2. **Recall the triangle angle sum rule:** The sum of the interior angles of any triangle is always $180^\circ$. So, $$ (4x + 19) + 111 + 26 = 180 $$ 3. **Set up the equation:** $$ 4x + 19 + 111 + 26 = 180 $$ 4. **Combine like terms:** $$ 4x + (19 + 111 + 26) = 180 $$ $$ 4x + 156 = 180 $$ 5. **Isolate $x$:** $$ 4x = 180 - 156 $$ $$ 4x = 24 $$ 6. **Solve for $x$:** $$ x = \frac{24}{4} $$ $$ x = \cancel{\frac{24}{4}}6 $$ 7. **Final answer:** $$ \boxed{6} $$ This means the value of $x$ that satisfies the triangle's angle measures is 6.