1. **State the problem:** We have a triangle with three angles: $b + 18^\circ$, $111^\circ$, and $2b$. We need to find the value of $b$. 2. **Recall the triangle angle sum rule:** The sum of the interior angles of any triangle is always $180^\circ$. So, $$ (b + 18^\circ) + 111^\circ + 2b = 180^\circ $$ 3. **Set up the equation:** Combine like terms: $$ b + 18 + 111 + 2b = 180 $$ $$ (b + 2b) + (18 + 111) = 180 $$ $$ 3b + 129 = 180 $$ 4. **Solve for $b$:** Subtract 129 from both sides: $$ 3b = 180 - 129 $$ $$ 3b = 51 $$ Divide both sides by 3: $$ b = \frac{51}{3} = 17 $$ 5. **Conclusion:** The value of $b$ is $17^\circ$. This means the angles are $b + 18 = 17 + 18 = 35^\circ$, $111^\circ$, and $2b = 34^\circ$, which sum to $35 + 111 + 34 = 180^\circ$, confirming our solution.