1. **Stating the problem:** We need to find the measure of angle $\angle xyb$ given that $\angle xyb$ is formed by a line segment from point $y$ to $x$ making a $15^\circ$ angle with the horizontal line, and the horizontal line extends left to point $a$ and right to point $b$. 2. **Understanding the geometry:** The points $a$, $y$, and $b$ lie on a horizontal line. The angle $\angle xyb$ is the angle at point $y$ between the segment $yx$ and the segment $yb$. 3. Since $yb$ is along the horizontal line pointing right, and $yx$ forms a $15^\circ$ angle with the horizontal line, the measure of $\angle xyb$ is simply $15^\circ$. 4. **Final answer:** $$\angle xyb = 15^\circ$$