$\begin{aligned} & \vec{a}+\vec{b}+\vec{c}=0 \\ & \Rightarrow \vec{a}+\vec{b}=-\vec{c} \\ & \Rightarrow|\vec{a}+\vec{b}|^2=|\vec{c}|^2 \\ & \Rightarrow|\vec{a}|^2+|\vec{b}|^2+2|\vec{a}||\vec{b}| \cos \theta=|\vec{c}|^2 \\ & \Rightarrow 3^2+5^2+2 \times 3 \times 5 \cos \theta=7^2 \\ & \Rightarrow \cos \theta=\frac{15}{30}=\frac{1}{2}=\cos \frac{\pi}{3} \\ & \Rightarrow \theta=\frac{\pi}{3}\end{aligned}$