Question

in a parallelogram ABCD the bisector of angle A also bisects BC at X prove that a b equals to 2 AD​

Answer 1

Answer:

2Ab

Step-by-step explanation:

bisector of ∠A bisects the side BC at X.

Given, ABCD is a parallelogram.

∴ AD||BC (Opposite sides of the parallelogram are parallel)

Now, AD||BC and  AX is the transversal  ,

∴ ∠2 = ∠3 (Alternate angles)  ............(1)

and ∠1 = ∠2 (AX is the bisector of ∠A) ................(2)

From (1) and (2), we obtain

∠1 = ∠3

Now, in ΔABX,

∠1 = ∠3

⇒ AB = BX  ( If two angles of a triangle are equal, then sides opposite to them are equal)

⇒ 2AB = 2BX = BX + BX = BX + XC  ( X is the mid point of BC)

⇒ 2AB = BC

⇒ 2AB = AD (Opposite sides of a parallelogram are equal)

∴ AD = 2AB.

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

=AB = 2AD