In the figure, AB = BC, BD is the bisector of angle ABC, DC = 35, BC = 80. Find AD.

Given a triangle ABC with equal sides AB = BC, then triangle ABC is isosceles. In an isosceles triangle, the bisector drawn from the vertex is also the height of the triangle (forms a right angle with the base) and the median (divides the base in half). Therefore:

AD = DC = 35.

Answer: 35

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