Question

In a parallelogram ABCD ,the angle bisector of angle A bisects BC.Will angle bisector of B also bisect AD??Give reason.​

Answer 1

Answer:

We have a parallelogram ABCD, where the angle bisector of A meets BC at its mid-point P.

Let us draw a line that is parallel to the lines AB and CD and passes through P and meets the line AD at Q.

Due to such a construction, it can be said that Q is the mid-point of AD.

Now, angle BAP = angle APQ (alternate angles) … (i)

and, angle BAP = angle PAQ (because AP is the bisector of angle A) … (ii)

So, from (i) and (ii), we have angle APQ = angle PAQ.

Therefore, the triangle APQ is an isosceles triangle and AQ = PQ … (iii)

Now, we have AQ = (1/2)AD (by construction) … (iv)

and, PQ = AB (by construction) … (v)

Substituting equations (iv) and (v) into (iii) we have

(1/2)AD = AB

=> AD = 2AB … Q.E.D

Answer 2

In parallelogram ABCD ,

Bisector of ∠A bisects BC at X

∵ AD││BC and AX cuts them so

∠DAX = ∠AXB (alternate angles)

∠DAX = ∠XAB (AX is bisector of ∠A)

∴∠AXB = ∠XAB

AB= BX (sides opposite of equal angles)

Now,AB/AD= BX/BC

=AB/AD=BX/2BX

(since,X is mid point of BC)

=AB/AD=1/2

Hence,

AB:AD=1:2.

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