Starting point. Sketch free body diagrams for where the strings intersect. There will be two of them..

Consider the intersection of the strings on the left. It will have three forces on it: A tension (say T1) acting to the left and up at an angle \(\displaystyle \theta\), a second tension (T2) acting to the right, and a tension acting downward due to the weight (T3 = mg). The mass is not accelerating (or even moving) so the net force on it will be 0 N.

You can set up the other intersection. The tension to the left will be T2 also, as it's the same string.

See where you can go from here and let us know.

-Dan