①当$\angle A$为顶角时，等腰三角形两底角的度数为：$\dfrac{180^{\circ}-80^{\circ}}{2}=50^{\circ}$

$\therefore$特征值$k=\dfrac{80^{\circ}}{50^{\circ}}=\dfrac{8}{5}$

②当$\angle A$为底角时，顶角的度数为：$180^{\circ}-80^{\circ}-80^{\circ}=20^{\circ}$

$\therefore$特征值$k=\dfrac{20^{\circ}}{80^{\circ}}=\dfrac{1}{4}$，

综上所述，特征值$k$为$\dfrac{8}{5}$或$\dfrac{1}{4}$，

故答案为：$A$、$C$.

①当$\angle A$为顶角时，等腰三角形两底角的度数为：$\dfrac{180^{\circ}-80^{\circ}}{2}=50^{\circ}$

$\therefore$特征值$k=\dfrac{80^{\circ}}{50^{\circ}}=\dfrac{8}{5}$

②当$\angle A$为底角时，顶角的度数为：$180^{\circ}-80^{\circ}-80^{\circ}=20^{\circ}$

$\therefore$特征值$k=\dfrac{20^{\circ}}{80^{\circ}}=\dfrac{1}{4}$，

综上所述，特征值$k$为$\dfrac{8}{5}$或$\dfrac{1}{4}$，

故答案为：$A$、$C$.