# In parallelogram ABCD, the length of the segment AB = 4. The bisector of angle A intersects the side BC

**In parallelogram ABCD, the length of the segment AB = 4. The bisector of angle A intersects the side BC at point K, and the continuation of the side CD at point E. Find the length of the segment KC if EC = 1.**

Since AK, by condition, is the bisector of the angle BAD, then the angle BAK = KAD. Angle DАК = ВКА as criss-crossing angles at the intersection of parallel straight lines ВС and АD secant АС, then angle ВАК = ВКА, and triangle ABK is isosceles, ВС = AB = 4 cm.

Let us prove that triangles ABK and EKC are similar. Angle BKA = EKC as vertical angles, angle BAK = KEC as criss-crossing angles at the intersection of parallel straight lines DE and AB secant AE. Then the triangles ВКА and ЕКС are similar in two angles.

Then AB / CE = BK / KC.

4/1 = 4 / KC.

COP = 4/4 = 1 cm.

Answer: The length of the CS segment is 1 cm.