Answer
$120^{\circ}$
Work Step by Step
We know that the two initial objects have same mass and speed and that they collide to have half of the speed, meaning that they both have the same angle between each other. Thus, we must find this angle and multiply by two. Using conservation of momentum, we find:
$cos \theta = \frac{1}{2}\\ \theta = 60^{\circ}$
This means that the overall angle is:
$60^{\circ} \times 2 = 120^{\circ}$
