Answer: 45 degrees.
Step-by-step explanation: We know that the area of a sector of a circle with radius $r$ and central angle $\theta$ is given by:
$$\text{Sector area} = \frac{\theta}{360^\circ} \pi r^2$$
Let $x$ be the angle measure of the arc we are looking for. Then the area of the corresponding sector is:
$$8\pi = \frac{x}{360^\circ} \pi (8)^2 = \frac{x}{360^\circ} \cdot 64\pi$$
Simplifying, we get:
$$\frac{x}{360^\circ} = \frac{8}{64} = \frac{1}{8}$$
Multiplying both sides by $360^\circ$, we get:
$$x = \frac{360^\circ}{8} = 45^\circ$$
Therefore, the angle measure of the arc bounding the sector with area 8π square meters is 45 degrees.