1. **State the problem:** Solve for $x$ given the angles $(8x - 3)^\circ$ and $(16x - 33)^\circ$ which are part of a triangle, and then find $m\angle CXY$ given angles $48^\circ$ and $89^\circ$ in the quadrilateral figure. 2. **Use the triangle angle sum rule:** The sum of angles in a triangle is $180^\circ$. 3. **Set up the equation for the triangle:** $$(8x - 3) + (16x - 33) + \text{third angle} = 180$$ Since the two angles are adjacent on the left side and the figure suggests these two angles sum to the top left angle, the third angle is the angle at the top right vertex, which is not given explicitly but can be found by subtracting the sum of the two given angles from $180^\circ$. 4. **Sum the two given angles:** $$8x - 3 + 16x - 33 = 24x - 36$$ 5. **Find the third angle:** $$180 - (24x - 36) = 180 - 24x + 36 = 216 - 24x$$ 6. **Since the two angles $(8x - 3)^\circ$ and $(16x - 33)^\circ$ are adjacent and form a straight line (linear pair), their sum is $180^\circ$:** $$ (8x - 3) + (16x - 33) = 180 $$ 7. **Solve for $x$:** $$ 24x - 36 = 180 $$ $$ 24x = 180 + 36 $$ $$ 24x = 216 $$ $$ x = \frac{216}{24} $$ $$ x = 9 $$ 8. **Find $m\angle CXY$ in the quadrilateral:** Given $\angle D = 48^\circ$ and $\angle E = 89^\circ$, and $X$ and $Y$ are midpoints creating segment $XY$ inside the figure. 9. **Use the fact that $CX = XE$ and $DY = YE$ (midpoints), so triangles $CXY$ and $DYE$ are isosceles or have special properties.** 10. **Calculate $m\angle CXY$ using the triangle sum rule in triangle $CXY$ or by angle subtraction:** Since $\angle D = 48^\circ$ and $\angle E = 89^\circ$, and $XY$ is inside the figure, $m\angle CXY$ is complementary to these angles in the polygon. 11. **Sum of angles around point $X$ or in the polygon is $360^\circ$, so:** $$ m\angle CXY = 180 - 48 - 89 = 43^\circ $$ **Final answers:** $$ x = 9 $$ $$ m\angle CXY = 43^\circ $$