1. **State the problem:** We are given a triangle with angles at vertices A, B, and C labeled as $21x - 1^\circ$, $26x + 6^\circ$, and $11x + 1^\circ$ respectively. We need to find the measure of angle $C$. 2. **Recall the triangle angle sum rule:** The sum of the interior angles of any triangle is always $180^\circ$. So, $$ (21x - 1) + (26x + 6) + (11x + 1) = 180 $$ 3. **Combine like terms:** $$ 21x - 1 + 26x + 6 + 11x + 1 = 180 $$ $$ (21x + 26x + 11x) + (-1 + 6 + 1) = 180 $$ $$ 58x + 6 = 180 $$ 4. **Solve for $x$:** $$ 58x = 180 - 6 $$ $$ 58x = 174 $$ $$ x = \frac{174}{58} = 3 $$ 5. **Find $m\angle C$:** Substitute $x=3$ into $11x + 1$: $$ 11(3) + 1 = 33 + 1 = 34^\circ $$ **Final answer:** $m\angle C = 34^\circ$