Only thing which is sufficient to prove two triangle as similar is
"all the angles should be equal"
So does not matter whether they have a common base or not. If you can prove that all of their angles are equal then it's enough to say that they are similar triangles.
Angle ADB = Angle ADC
Angle ABD = Angle ACD
Angle BAD = Angle CAD
=> ABC and ABD are similar triangles....
Also there are other cases (but in these cases you will end up proving that both triangles are equivalent, (equivalent triangles are subset of similar triangles))
1) Common base BC and the two attached angles...I.e. <ACB =<DCB and <ABC = <DBC
2) two sides and included sides are equal... I.e. AB = DB, BC = BC and <ABC = <DBC
In both of the cases we have equivalent triangles which will obviously be similar
Hope it helps!
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