I've been trying stuff with angle sums, Sine law and Thales theorem, but either I'm making very bad mistakes or I'm just tired - either way I would love some outside input. Thank you.
I'll leave out the degree symbol from my equations.
Let'd denote $\gamma$ as the angle BDA. Then the angle BDC $=180 - \gamma$.
Let's also denote $\beta$ as the angle DBC. From symmetry, we get that the angle ACB $=\alpha + \beta$. Now, looking at the lower triangle, we obtain $$ (\beta) + (\alpha + \beta) + (180-\gamma) = 180 \qquad \Rightarrow \qquad \alpha + 2\beta - \gamma = 0 $$ From the upper triangle we get $$ 20 + \alpha + \gamma = 180 \qquad \Rightarrow \qquad \alpha + \gamma = 160 $$ Maybe you can continue from here?