The bisectors of angles A and D of parallelogram ABCD meet at a point lying on side BC. Find BC if AB = 13.

Let the bisectors intersect at point K;
Since ABCD is a parallelogram, AB = DC, AD = BC, AB || DC, AD || BC, therefore DC = 13cm;
From the condition, the angle (further ^) DAK = ^ KAС, ^ ADK = ^ KDС;
Based on the properties of two parallel straight lines intersected by a straight line ^ ADK = ^ DKС, ^ DAK = ^ AKB as internal versatile angles
From clause 4: Triangles AVK and DSC are isosceles, namely AB = BK, DC = KС, therefore BK = 13 cm, KС = 13 cm.
Since the point K lies on the BC, then BC = ВK + KС, therefore BC = 13 + 13, BC = 26 cm.

Answer: 26 cm.