To find the measure of angle \( M \), we can set up an equation using the property that the sum of the angles in a quadrilateral equals 360 degrees. In this case, we have four angles, with measures \( 80^\circ \), \( 9x - 1 \), \( 9x - 5 \), and \( 16 + 9x \).
We can write the equation:
\[ 80 + (9x - 1) + (9x - 5) + (16 + 9x) = 360 \]
Next, simplify the equation by combining like terms. This yields:
\[ 80 - 1 - 5 + 16 + 27x = 360 \]
\[ 90 + 27x = 360 \]
Finally, isolate \( x \) by subtracting 90 from both sides and then dividing by 27:
\[ 27x = 270 \]
\[ x = 10 \]
Now, substitute \( x \) back into each angle expression to find their measures, including angle \( M \).