@yangtheary I don't see what exactly you want to do with Maple for this question. But I follow my guess. Since you didn't specify which two edges of your triangle are equal, I check all three possibilities.
1- AC=BC: In this case D=C and we have ABD=ABC and since ABC=BAC, the answer is 20 degrees.
2- AB=BC: Since AD=BC, the triangle ADB also becomes isosceles. Then 2(?)+20=180 which implies ABD=?=80.
3- AB=AC: In this case, note that the triangle ABC can be uniquely determined by the assumptions of your question plus length of AD. And two cases for two different values of AD are geometrically similar (one is a scaled version of the other). Therefore the angles are the same and only size of edges are changed (by a multiplying to a fixed ratio). Now I fixed AD=1. Without loss of generality (at most with a geometric translation and rotation which none of them change the size of angles and edges) assume that A=(0,0) and AC is on the positive side of the x-axis. Then it's trivial that D=(1,0) and the AB edge is on the line y=tan(20)*x. The point C is (xC,0) which we yet don't know the value of xC. The line which BC is on it, is y=-tan(80)*(x-xC). The point B is the unique point on the intersection of the two lines and with the distance 1 from C. Therefore we need to solve the following system of equations. Here I use Maple. Just note that B is on these two lines, by considering one of them, say the first one, we have B=(xB,tan(20)*xB). Therefore we have two equations and two unknowns.
Now we have the coordinates of both B and C. But knowing B is enough for us. Again using Maple we compute the arctan of ABD. To this end, just choose an arbitrary point on one of the two edges of this angle, say D from BD, then project it on the other edge. Call this new point M. Then tan(ABD)=(distance of M and D)/(distance of M and AB). The wollowing Maple computations is related to all these steps. To find M, we first found distance of M from the line of AB. Then solved an equation determining the unique point on line of AB with the found distance from M.
So the answer in the case of AC=AB is (almost) 10 degrees.
I also put the plot of my above code below.