If in $$\triangle PQR, \angle P = 120^\circ, PS \perp QR$$ at $$S$$ and $$PQ + QS = SR$$. then the measure of $$\angle Q$$ is:

Let the PQ = x and QS = y then SR = PQ + QS = x + y.

Take a point T on the SR so that QS = ST = y.

TR = SR - ST =  x + y - y = x

PT = TR = x so,

$$\angle TPR = \angle TRP = \theta$$

In triangle PTR - 

$$\angle TPR + \angle TRP +  \angle PTR = 180\degree$$

$$\angle PTR = 180\degree - 2\theta$$

$$\angle PTS = 180\degree - (180\degree - 2\theta) = 2\theta$$

$$\angle PTS = \angle PQS = 2\theta$$

($$\because$$ QP = PT)

In triangle PQR -

$$\angle PQR + \angle QRP +\angle RPQ = 180\degree$$

3$$\theta = 180\degree - 120 = 60\degree$$

$$\theta = 20\degree$$

$$\angle Q = 2\theta = 2 \times 20\degree = 40\degree$$