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In the figure, M is the midpoint of side BC of a parallelogram

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In the figure, $M$ is the midpoint of side $BC$ of a parallelogram $A BC D$ such that $∠ B A M = ∠ D A M$ . Prove that $A D = 2 C D$

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In the figure,

M

is the midpoint of side

BC

of a parallelogram

A BC D

such that

∠ B A M = ∠ D A M

. Prove that

A D = 2 C D

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Text solutionVerified

It is given that

A BC D

is a parallelogram
so we know that

A D ∥ BC

from the figure we know that

∠ D A M

and

∠ A MB

are alternate angles

so we get

∠ D A M = ∠ A MB

we know that

∠ B A M = ∠ D A M

∠ B A M = ∠ A MB

From the figure we know that the sides opposite to equal angles are equal
so we get

BM = A B

we know the opposite sides of a parallelogram are equal

A B = C D

so we can write it as

MB = A B = C D .... (1)

we know that

M

is midpoint of the line

BC

so we get

BM = 1/2 BC

we know that

BC = A D

we get

MB = 1/2 A D

based on equation (1)

C D = 1/2 A D

by cross multiplication

A D = 2 C D

therefore it is proved that

A D = 2 C D

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Question Text	In the figure, $M$ is the midpoint of side $BC$ of a parallelogram $A BC D$ such that $∠ B A M = ∠ D A M$ . Prove that $A D = 2 C D$
Answer Type	Text solution:1
Upvotes	150

In the figure, M is the midpoint of side BC of a parallelogram ABCD such that ∠BAM=∠DAM. Prove that AD=2CD

Text solutionVerified

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In the figure, $M$ is the midpoint of side $BC$ of a parallelogram $A BC D$ such that $∠ B A M = ∠ D A M$ . Prove that $A D = 2 C D$