Will angle chasing help?

We have AB is external bisector of CAD and DB is internal bisector of CDA. Then, we have CB is external bisector of ACD. So ,we have ACB=(180-60):2=60

just draw the angular bisector of angle ACD. Let it meet BD in Z. then quad BAZC is cyclic. Then after some simple angle chasing the problem is over.

Let us call the intersection point of $AC$ and $BD$ $E$ and $\angle BCE=\theta$ and $\angle EBC=\alpha$ .Since $DE$ is the internal bisector of angle $D$ we have $\dfrac{AE}{EC}=\dfrac{AD}{DC}.$ Applying sine-rule in $\bigtriangleup ABE,\bigtriangleup BEC$ $2 AE=\dfrac{BE}{\sin 80^{\circ}}\\ \dfrac{EC}{\sin\alpha}=\dfrac{BE}{\sin \theta}\\ \Longrightarrow 2\dfrac{AD}{DC}\sin\alpha=\dfrac{\sin\theta}{\sin80^{\circ}}.$ Leave that aside for a second and apply sine-rule in $\bigtriangleup ABD,\bigtriangleup BDC$ ,we get, $2AD=\dfrac{BD}{\sin 100^{\circ}}\\ \dfrac{DC}{\sin\alpha}=\dfrac{BD}{\sin(60+\theta)}\\ \Longrightarrow 2\dfrac{AD}{DC}\sin\alpha=\dfrac{\sin(60+\theta)}{\sin 100^{\circ}}\\ \Longrightarrow \dfrac{\sin(60+\theta)}{\sin 100^{\circ}}=\dfrac{\sin\theta}{\sin 80^{\circ}}\\ \Longrightarrow \sin(60+\theta)=\sin\theta$ Expanding the left hand side and evaluating we obtain $\tan\theta=\sqrt{3}$ , hence $\theta=60^{\circ}$ .And done!

With out loss of generality, let AB=1, and angle BCA=X.
$\angle ACD=180 - 20 - 50 - 50 = 60 ~ \text{Applying Sin Law to triangles ABC, ABD, CBD, we get}\\ \dfrac 1 {BC}=\dfrac {SinX}{Sin80}, ~~~~~~~\dfrac {BD} 1 =\dfrac {Sin100}{Sin50}, ~~~~~~~~\dfrac{BC}{BD}=\dfrac {Sin50}{Sin(60+X)}, \\ \text{Equating the product of the left sides of the three equations with the right sides, we get,}\\ 1=\dfrac{Sin100}{Sin80}*\dfrac {SinX}{Sin(60+X)}, ~ But ~\dfrac {SinX}{Sin(60+X)}=\dfrac 1{Sin60CotX+Cos60} ~\\ \implies Sin60CotX+Cos60=\dfrac{Sin100}{Sin80}. Calculating, X= \Large ~\color{rted}{60}$

Most probably angle DBC will be shorter than 30° hence either DBC could be 10 or 20 (assumption) by substituting 10 you get answer 60 while you have 3 tries and you have two cases hence you will be able to get answer (I know its hilarious)