1. **State the problem:** We have a triangle ABC with angles at vertices A, B, and C given as $2x + 20$, $x + 20$, and $\frac{1}{2}x$ degrees respectively. We need to find the measure of angle ACB (angle at C). 2. **Recall the triangle angle sum rule:** The sum of the interior angles of any triangle is always 180 degrees. 3. **Set up the equation:** $$ (2x + 20) + (x + 20) + \frac{1}{2}x = 180 $$ 4. **Combine like terms:** $$ 2x + 20 + x + 20 + \frac{1}{2}x = 180 $$ $$ (2x + x + \frac{1}{2}x) + (20 + 20) = 180 $$ $$ \left(3 + \frac{1}{2}\right)x + 40 = 180 $$ $$ \frac{7}{2}x + 40 = 180 $$ 5. **Isolate the term with $x$:** $$ \frac{7}{2}x = 180 - 40 $$ $$ \frac{7}{2}x = 140 $$ 6. **Solve for $x$ by dividing both sides:** $$ x = \frac{140}{\cancel{1}} \times \frac{2}{\cancel{7}} = 40 $$ 7. **Find angle ACB:** $$ \text{Angle ACB} = \frac{1}{2}x = \frac{1}{2} \times 40 = 20^6 $$ **Final answer:** Angle ACB is 20 degrees, which corresponds to option A.