1. **State the problem:** We need to find the measure of angle $m\angle XYW$ in the given triangle $XYZ$ with exterior point $W$ on segment $XY$. 2. **Given information:** - $\angle WXY = 64^\circ$ - $\angle WZY = 64^\circ$ - $\angle Y$ is split into two parts: $(5x - 9)^\circ$ and $(3x + 5)^\circ$ - Right angles at $WZ$ with $WX$ and at $Z$ with extension of $XY$ 3. **Use the triangle angle sum property:** The sum of interior angles in triangle $XYZ$ is $180^\circ$. 4. **Express the angle at $Y$:** $$m\angle Y = (5x - 9) + (3x + 5) = 8x - 4$$ 5. **Sum of angles in triangle $XYZ$:** $$64 + 64 + (8x - 4) = 180$$ 6. **Simplify the equation:** $$128 + 8x - 4 = 180$$ $$124 + 8x = 180$$ 7. **Isolate $x$:** $$8x = 180 - 124$$ $$8x = 56$$ $$x = \frac{56}{8}$$ $$x = 7$$ 8. **Find $m\angle Y$ using $x=7$:** $$m\angle Y = 8(7) - 4 = 56 - 4 = 52^\circ$$ 9. **Find $m\angle XYW$:** Since $W$ lies on $XY$ and $\angle WXY = 64^\circ$, and $m\angle XYW$ is the exterior angle adjacent to $\angle WXY$, then $$m\angle XYW = 180^\circ - 64^\circ = 116^\circ$$ **Final answer:** $$m\angle XYW = 116^\circ$$