Let the value of the larger angle of the rhombus be X0. Angle DСВ = X0.
The diagonals of the rhombus divide the angle at the apex in half, then the angle is DCA = X / 2.
Since, by condition, CM is the bisector of the angle DCA, then the angle DCM = DCA / 2 = (X / 2) / 2 = X / 4.
The sum of the adjacent angles of the rhombus is 180, then the angle CDA = 180 – DCB = 180 – X.
Consider a triangle DСМ, the sum of the interior angles of which will be equal to:
(180 – X) + 90 + X / 4 = 180.
3 * X / 4 = 90.
X = 90 * 4/3 = 120.
Angle ВСD = 120.
Answer: The larger angle of the rhombus is 120.
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