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$A B C D$ is a parallelogram, $M$ is a parallelogram, is the mid-point of BD and BM bisects $∠ B$ is the mid-point of BD and BM bisects . Then, from the following choices, the value of $∠ A M B$ . Then, from the following choices, the value of is

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45°

60°

90°

75°

answer is C.

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Detailed Solution

A B C D

is a parallelogram, with

M

as the mid-point of

B D a n d B M

bisects

∠ B

, as shown in the figure given below.
AM is joined.
Question Image

BM bisects

∠ B

∴ ∠ 1 = ∠ 2

……(1)

A D ‖ B C

and BD is the transversal.
So,

∠ 2 = ∠ 3

. [Alternate interior angles are equal] ……(2)
From (1) and (2),

∠ 1 = ∠ 2 = ∠ 3

So,

∠ 1 = ∠ 3

.
Then,
AD = AB [Sides opposite to equal angles are equal] …….(3)
AB = CD [Opposite side of a parallelogram] ……(4)
From (3) and (4),

A B = C D = B C = A D

Thus, ABCD is a rhombus and the diagonals bisect each other at

90 °

.
So,

∠ A M B = 90 °

.
Hence, option 3) is correct.

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