1.

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

Answer»

It is given that ABCD is a parallelogram

So we know that AD || BC

From the figure we know that ∠ DAM and ∠ AMB are alternate angles

So we get

∠ DAM = ∠ AMB

We know that ∠ BAM = ∠ DAM

It can be written as

∠ BAM = ∠ AMB

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

So we get

BM = AB

We know that the opposite sides of a parallelogram are equal

AB = CD

So we can write it as

BM = AB = CD ……. (1)

We know that M is the midpoint of the line BC

So we get

BM = ½ BC

We know that BC = AD

We get

BM = ½ AD

Based on equation (1)

CD = ½ AD

By cross multiplication

AD = 2CD.

Therefore, it is proved that AD = 2CD.


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