In triangle ABC the point D lies on side AC, and angle ABD is equal to angle ACB. If AB = 6 cm and AD = 4 cm, what is the length of AC?

Triangles $ABD$ and $ACB$ share the angle at A, and the given condition angle $ABD$ = angle $ACB$ supplies a second equal angle, so they are similar with $B$ matching $C$ and $D$ matching $B$. Corresponding sides give $AB/AC=AD/AB$, so $A{B}^2=AD\times AC$, that is $36=4\times AC$ and $AC=9$ cm. The value 5 cm is $DC$, the leftover piece of $AC$.