In $\Delta \mathrm{WXY}, \overline{W Y}$ is extended through point $\mathrm{Y}$ to point $\mathrm{Z}$, $\mathrm{m} \angle Y W X=(2 x+9)^{\circ}$, $\mathrm{m} \angle W X Y=(2 x+16)^{\circ}$, and $\mathrm{m} \angle X Y Z=(8 x+1)^{\circ}$. Find $\mathrm{m} \angle W X Y$.

Step 1 ：The problem involves a triangle and an extended line, forming an exterior angle. The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles. In this case, \( \angle YWZ = \angle YWX + \angle WXY \).

Step 2 ：We can set up an equation using this relationship: \( (2x + 9) + (2x + 16) = (8x + 1) \).

Step 3 ：Solving this equation gives us the value of \( x = 6 \).

Step 4 ：Substituting \( x = 6 \) back into the expression for \( \angle WXY \), we get \( \angle WXY = 2x + 16 = 28 \).

Step 5 ：Final Answer: The measure of \( \angle WXY \) is \( \boxed{28^{\circ}} \).