Option : (C)
ABCD is a parallelogram.
BD is the diagonal and M is the mid-point of BD.
BD is a bisector of ∠B
We know that,
Diagonals of the parallelogram bisect each other
Therefore,
M is the mid-point of AC
AB || CD and BD is the transversal,
Therefore,
∠ABD = ∠BDC ...(i)
(Alternate interior angle)
∠ABD = ∠DBC ...(ii) (Given)
From (i) and (ii), we get
∠BDC = ∠DBC
In ΔBCD,
∠BDC = ∠DBC
BC = CD ...(iii)
(In a triangle, equal angles have equal sides opposite to them)
AB = CD and BC = AD ...(iv)
(Opposite sides of the parallelogram are equal)
From (iii) and (iv), we get
AB = BC = CD = DA
Therefore,
ABCD is a rhombus
∠AMB = 90°
(Diagonals of rhombus are perpendicular to each other)
