1. **State the problem:** Calculate the size of the largest angle in the triangle with angles $x + 72^\circ$, $4x - 18^\circ$, and $x$. 2. **Use the triangle angle sum rule:** The sum of the interior angles of a triangle is always $180^\circ$. 3. **Set up the equation:** $$x + 72 + 4x - 18 + x = 180$$ 4. **Simplify the equation:** $$x + 4x + x + 72 - 18 = 180$$ $$6x + 54 = 180$$ 5. **Isolate $x$:** $$6x = 180 - 54$$ $$6x = 126$$ 6. **Divide both sides by 6:** $$\cancel{6}x = \frac{126}{\cancel{6}}$$ $$x = 21$$ 7. **Calculate each angle:** - $x = 21^\circ$ - $x + 72 = 21 + 72 = 93^\circ$ - $4x - 18 = 4(21) - 18 = 84 - 18 = 66^\circ$ 8. **Identify the largest angle:** The largest angle is $93^\circ$. **Final answer:** The largest angle measures $93^\circ$.