ABC is an isosceles triangle. AB = CB. The angles BCD and BAK are equal. BK = 2 cm. Find BD.

Consider the newly formed triangles ABK and BCD. It is known from the condition that the angle ВСD is equal to the angle ВАК. In addition, the angle ABC is common. Since the triangle is isosceles, the side AB is equal to the side BC. Thus, in accordance with the second equality of the triangle, if side AB and two adjacent angles BAC and ABK of one triangle are equal to side BC and two adjacent angles BCD and DBK, then such triangles are equal. Accordingly, the VK side is equal to the BD side and is equal to 2 cm.