Proof: Take BC as the side and make an equilateral triangle BCE to connect AE on the same side.

So BE=BC=CE

Angle BEC= angle CBE=60 degrees.

Because ABC= angle ACB=50 degrees

So AB=AC

Because AE=AE

So the triangle is congruent triangles ACE (SAS).

So angle AEB= angle AEC= 1/2 angle BEC=30 degrees.

Because angle BAC+ angle ABC+ angle ACB= 180 degrees.

So I also put BAC=80 degrees.

Because angle BAD= angle BAC- angle DAC

Angle DAC= 10 degree

So the angle BAD=70 degrees

Because angle ABD= angle ABC- angle DBC

Angle DBC= 10 degree

So the angle ABD=40 degrees

Because angle ABD+ angle BAD+ angle ADB= 180 degrees.

So the angle ADB=70 degrees

So angle BAD= angle ADB=70 degrees.

So AB=DB

Because angle ABE= angle CBE- angle ABC=60-50= 10 degrees.

So angle ABE= angle DBC= 10 degrees.

So the triangle is congruent triangles DBC (SAS).

So angle AEB= angle DCB

So the angle DCB=30 degrees