1. **Problem Statement:** Estimate the measure of angle $\angle WXY$ formed at point $X$ by segments $WX$ and $XY$. 2. **Understanding the angle:** The angle $\angle WXY$ is the angle between the two line segments $WX$ and $XY$ meeting at point $X$. 3. **Visualizing directions:** Segment $WX$ points slightly downward to the left, which corresponds roughly to a direction of about $225^\circ$ (southwest) if we consider standard compass directions. 4. Segment $XY$ points upward to the right, roughly at about $45^\circ$ (northeast). 5. **Calculating the angle between two directions:** The angle between two directions $\theta_1$ and $\theta_2$ is given by the absolute difference: $$\text{Angle} = |\theta_2 - \theta_1|$$ 6. Substitute the approximate directions: $$|45^\circ - 225^\circ| = | -180^\circ| = 180^\circ$$ 7. However, the angle between two segments at a point is the smaller angle formed, so we take the smaller angle between $180^\circ$ and $360^\circ - 180^\circ = 180^\circ$. 8. Since both are $180^\circ$, the angle is a straight line, but the problem states the segments form an angle, so the directions must be slightly different. 9. Given the description "slightly downward to the left" and "upward to the right," a reasonable estimate is that the angle is about $90^\circ$. 10. **Final estimate:** The measure of $\angle WXY$ is approximately $90^\circ$, rounded to the nearest $10^\circ$. **Answer:** $90^\circ$