Let the masses and initial speeds be

Since both bodies have the same mass and same speed, let each mass be $m$ and each initial speed be $v$.

Use conservation of momentum

The collision is completely inelastic, so after collision they stick together.

Final combined mass $= 2m$.

Final speed is given as $\dfrac{v}{2}$.

Hence final momentum magnitude is $P_f = (2m)\left(\frac{v}{2}\right) = mv.$

Therefore, the magnitude of the vector sum of the two initial momenta must also be $mv$.

Write the initial momenta

Each body has momentum magnitude $p = mv.$

Let the angle between the two initial velocity vectors be $\theta$. Since momentum is along velocity, the angle between the momentum vectors is also $\theta$.

The magnitude of the resultant of two equal vectors of magnitude $mv$ at angle $\theta$ is $|\vec p_1 + \vec p_2| = \sqrt{(mv)^2 + (mv)^2 + 2(mv)^2\cos\theta}.$

So, $|\vec p_1 + \vec p_2| = mv\sqrt{2+2\cos\theta}.$

Equate with final momentum

We found final momentum magnitude to be $mv$, so $mv\sqrt{2+2\cos\theta} = mv.$

Cancelling $mv$, $\sqrt{2+2\cos\theta} = 1.$

Squaring both sides, $2+2\cos\theta = 1.$

$2\cos\theta = -1$

$\cos\theta = -\frac{1}{2}.$

Find the angle

$\theta = 120^\circ.$

Final Answer

The angle between the initial velocities is $\boxed{120}.$