Segments AB and CD intersect at point O so that CO = DO, angle ACO = angle BDO

Segments AB and CD intersect at point O so that CO = DO, angle ACO = angle BDO, AO = 4cm. What is the segment BО?

Consider triangles ВDО and АСО:

DО = СО (by condition), angle АСО = angle ВDО (by condition), angle BOD = angle СОА (vertical angles). This means that the triangles BDO and ACO are equal according to the second sign of equality of triangles (along the side and the corners adjacent to it).

So AO = BO = 4 cm.

Answer: the segment BO is 4 cm.