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

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Solution

ANSWER:
Given: parallelogram ABCD, M is the midpoint of side BC and ∠BAM = ∠DAM.
To prove: AD = 2CD
Proof:
Since, AD∥BC and AM is the transversal.
So, ∠DAM=∠AMB
(Alternate interior angles)
But, ∠DAM=∠BAM (Given)
Therefore, ∠AMB=∠BAM
⇒AB=BM
(Angles opposite to equal sides are equal.) ...(1)
Now, AB = CD
(Opposite sides of a parallelogram are equal.)
⇒2AB=2CD⇒(AB+AB)=2CD
( AB = BM and MC = BM)
⇒BM+MC=2CD
⇒BC=2CD∴AD=2CD
AD=BC, Opposite sides of a parallelogram are equal.

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

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