已知 $A(1,2), B(2,5), \overrightarrow{B C}=(-2,-4)$, 则 $\cos \langle\overrightarrow{A B}, \overrightarrow{A C}\rangle=$ ( )
$\overrightarrow{A B}=(1,3), \overrightarrow{A C}=\overrightarrow{A B}+\overrightarrow{B C}=(1,3)+(-2,-4)=(-1,-1)$,

所以 $\cos \langle\overrightarrow{A B}, \overrightarrow{A C}\rangle=\frac{\overrightarrow{A B} \cdot \overrightarrow{A C}}{|\overrightarrow{A B}| \cdot|\overrightarrow{A C}|}=\frac{-4}{\sqrt{10} \times \sqrt{2}}=-\frac{2 \sqrt{5}}{5}$, 故选 A.