The side of the rhombus, with an angle of 120 degrees, is 4 cm, find its smaller diagonal.

Let ABCD be the given rhombus. The ABC angle is 120 °. BD is the smaller diagonal (the required one).

The diagonal of the rhombus is the bisector of the corner of the rhombus. This means that the angle ABD is equal to the angle CBD and is equal to 60 °.

Consider a triangle ABD: the angle ABD is 60 °, the angle ADB is also 60 ° (the opposite angles of the rhombus are equal and half of the angles are also equal).

This means that the angle BAD is also 60 ° (180 – (60 + 60) = 60).

Therefore, triangle ABD is equilateral, BD = AB = 4 cm.

Answer: the smaller diagonal of the rhombus is 4 cm.