BD is the bisector of angle ABC. point N lies on the bisector. Angles ANB and BNC are equal to AB = 8cm, find BC.

Consider triangles ABN and BCN. In it, side BN is common, angle ABN = angle CBN and angle ANB = angle BNC. So, we have one side in common and two equal corners adjacent to it. Hence, according to the second criterion of equality of triangles, triangle ABC is equal to triangle BCN, which means that the corresponding sides of these triangles are equal. It follows that side AB is equal to side BC. It turns out that BC = 8 cm.
Answer: BC is 8 centimeters long.