1. **State the problem:** We have three points A, B, and C on a circle with angles labeled as $x$, $2(2x + 20)$ degrees, and $5x + 40$ degrees. We need to find the value of $x$, then determine if segment AC is a diameter of the circle. 2. **Recall the circle angle rule:** The sum of angles around a point on a circle is $180^\circ$ if they form a straight line (like a diameter). Since AC is a candidate diameter, the angles around it should sum to $180^\circ$. 3. **Set up the equation:** The three angles given are adjacent around the circle at points A, B, and C. Their sum should be $180^\circ$ if AC is a diameter: $$x + 2(2x + 20) + (5x + 40) = 180$$ 4. **Simplify the equation:** $$x + 4x + 40 + 5x + 40 = 180$$ $$ (x + 4x + 5x) + (40 + 40) = 180$$ $$10x + 80 = 180$$ 5. **Isolate $x$:** $$10x + 80 = 180$$ $$10x = 180 - 80$$ $$10x = 100$$ $$x = \cancel{\frac{10x}{10}}{\frac{100}{10}} = 10$$ 6. **Calculate the sum of angles using $x=10$:** $$5x + 40 + x + 2(2x + 20) = 5(10) + 40 + 10 + 2(2(10) + 20)$$ $$= 50 + 40 + 10 + 2(20 + 20)$$ $$= 100 + 2(40) = 100 + 80 = 180$$ 7. **Conclusion:** Since the sum of the angles is $180^\circ$, AC is indeed a diameter of the circle. **Final answers:** - $x = 10^\circ$ - Sum of angles $= 180^\circ$ - AC is a diameter of the circle.