1. Let the circumcenter of $\triangle A B C$ be $O$, and
$$
3 \overrightarrow{O A}+4 \overrightarrow{O B}+5 \overrightarrow{O C}=\mathbf{0} \text {. }
$$

Then the size of $\angle C$ is $\qquad$
$-1.45^{\circ}$
Notice, $|\overrightarrow{O A}|=|\overrightarrow{O B}|=|\overrightarrow{O C}|$.
From the given information,
$$
\begin{array}{l}
-5 \overrightarrow{O C}=3 \overrightarrow{O A}+4 \overrightarrow{O B} \\
\Rightarrow 25 \overrightarrow{O C}^{2}=9 \overrightarrow{O A}^{2}+24 \overrightarrow{O A} \cdot \overrightarrow{O B}+16 \overrightarrow{O B}^{2} \\
\Rightarrow \overrightarrow{O A} \cdot \overrightarrow{O B}=0 \\
\Rightarrow \angle A O B=90^{\circ} .
\end{array}
$$

Since $O$ is the circumcenter, we have,
$$
\angle A C B=\frac{1}{2} \angle A O B=45^{\circ} \text {. }
$$