To find the value of \( z \), we start by calculating the measure of each interior angle of a regular polygon. The formula to find the measure of each interior angle for a regular polygon with \( n \) sides is:
\[ \text{Interior angle} = \frac{(n-2) \times 180}{n} \]
For a 9-sided polygon (nonagon), we substitute \( n = 9 \) into the formula:
\[ \text{Interior angle} = \frac{(9-2) \times 180}{9} = \frac{7 \times 180}{9} = \frac{1260}{9} = 140 \text{ degrees} \]
We are given that:
\[ m\angle a = (4z + 68)^\circ \]
Setting this equal to the measure of each interior angle for the polygon gives us:
\[ 4z + 68 = 140 \]
Next, we solve for \( z \):
- Subtract 68 from both sides:
\[ 4z = 140 - 68 \]
\[ 4z = 72 \]
- Divide both sides by 4:
\[ z = \frac{72}{4} = 18 \]
Thus, the value of \( z \) is
\[ \boxed{18} \]