8. Given that $E$ is a point on the extension of side $C D$ of the cyclic quadrilateral $A B C D$, and $I$ is the incenter of $\triangle A B C$. If $\angle A B C=70^{\circ}, \angle A C B=60^{\circ}, D E=D A$, then the degree measure of $\angle D E I$ is $\qquad$
8. $25^{\circ}$.

From the given information, we have
$$
\begin{array}{l}
\angle B A C=50^{\circ}, \angle E D A=\angle A B C=70^{\circ}, \\
\angle D E A=\angle D A E=55^{\circ}, \\
\angle C I A=90^{\circ}+\frac{1}{2} \angle A B C=180^{\circ}-\angle C E A .
\end{array}
$$

Therefore, points $C$, $I$, $A$, and $E$ are concyclic.
Thus, $\angle D E I=\angle C A I=\frac{1}{2} \times 50^{\circ}=25^{\circ}$.