In a completely inelastic collision, two objects stick together after colliding. Momentum is conserved, but kinetic energy is not. Let the mass of each object be $m$ and initial speed be $v$. The final speed is $\frac{v}{2}$. Let the angle between their initial velocities be $\theta$.

Conserving momentum in the x and y directions and combining gives: ${(2m·\frac{v}{2})}^2={\left(mv\right)}^2+{\left(mv\right)}^2+2\left(mv\right)\left(mv\right)\cos \theta$. Simplifying, $v^2=2v^2+2v^2\cos \theta$, leading to $\cos \theta =-\frac{1}{2}$.

Final Answer: $120°$