To find the values of P and θ, we resolve the forces vertically and horizontally, and use Lami's Theorem.

Vertical Resolution:
The vertical components of the forces must sum to zero for equilibrium:

\(3P \sin 55^{\circ} + P \sin θ + P \sin θ = 20\)

\(3P \sin 55^{\circ} = 20\)

Solving for P:

\(P = \frac{20}{3 \sin 55^{\circ}} \approx 8.14 \text{ N}\)

Horizontal Resolution:
The horizontal components of the forces must also sum to zero:

\(3P \cos 55^{\circ} = 2P \cos θ\)

\(\cos θ = \frac{3 \cos 55^{\circ}}{2}\)

Solving for θ:

\(\cos θ = 1.5 \cos 55^{\circ}\)

\(θ \approx 30.6^{\circ}\)

Alternatively, using Lami's Theorem:

\(\frac{3P}{\sin 90^{\circ}} = \frac{20}{\sin 125^{\circ}}\)

Solving gives:

\(P = 8.14 \text{ N}\)

And for the horizontal forces:

\(\frac{3P}{\sin 90^{\circ}} = \frac{2P \cos θ}{\sin 145^{\circ}}\)

\(\cos θ = 1.5 \sin 145^{\circ}\)

\(θ \approx 30.6^{\circ}\)