MHB AB=AC,∠ABD=60°,∠ADB=70°,∠BDC=40°,find∠DBC

[Albert1]

$Quadrilateral \,\,ABCD,\overline{AB}=\overline{AC} ,\angle ABD=60^o,\angle ADB=70^o,
\angle BDC=40^o,find \,\, \angle DBC=?$

 

[Albert1]

$Quadrilateral \,\,ABCD,\overline{AB}=\overline{AC} ,\angle ABD=60^o,\angle ADB=70^o,
\angle BDC=40^o,find \,\, \angle DBC=?$

hint:

Spoiler

using the diagram:
construct a point E between $\overline{CD},and \,\, \angle ABE=70 ^o$
prove points C and E overlap
View attachment 6993

 

[Albert1]

Spoiler

from previous diagram we have:
Points $A,B,E,D$ cocyclic, so $\overset{\frown}{AB}=\overset{\frown}{AE}=140^o$
we get $\overline{AB}=\overline{AE}=\overline{AC}$
and points C and E overlap (since points C,E,D colinear)
$\therefore \angle DBC=\angle DBE=10^o$