In triangle \( ABC \), \( AB = AC \), and \( \angle A = 40^\circ \). If point \( D \) is on \( BC \) such that \( BD = 6 \) and \( DC = 8 \), verify if the triangle inequality theorem holds for triangle \( ABD \) and find the measure of \( \angle BDC \). Let's think step by step and output the final answer within \boxed{}.
user