Question

9. In the adjoining figure, BD and CD are angle bisectors of exterior angle B and
exterior angle C of ∆ ABC respectively. If AB = AC, prove that AD bisects angle A
and angle D both?​

Answer 1

Answer:

Extend the line BC to E

BD and CD are angular bisectors,

∴∠ABD=∠DBC=x and ∠ACD=∠DCE=y

∠ABC=2x and ∠ACE=2y

Consider △ABC,

∠ACE=∠ABC+∠BAC ------exterior angle is equal to sum of interior opposite angle

2y=2x+∠A

y−x=

2

∠A

------(i)

Consider △BCD,

∠DCE=∠DBC+∠BDC ------exterior angle is equal to sum of interior opposite angle

y=x+∠D

y−x=∠D------(ii)

From(i) and (ii)

∠D=

2

1

∠A