Question

If the bisector of the exterior angle C of a angle ABC is parellel to the sideAB, then prove that the triangle ABC is an isoceles triangle.

Answer 1

Let ∠ACD be the exterior angle of △ABC.Let CE is the bisector of ∠ACD and suppose CE ∥ BA.
Since, BA∥CE and BD is a transversal, then
,∠ABC = ∠ECD [Corresponding angles] ..............(1)
Since, BA∥CE and AC is a transversal, then,∠BAC = ∠ACE [Alternate interior angles] ..............(2)
Since, CE bisects ∠ACD, then,∠ACE = ∠ECD ...............(3)
∴ From (2), we get,∠BAC = ∠ECD [Using (3)] ..............(4)
From (1) and (2), we get,∠ABC = ∠BACIn △ABC, we have,∠ABC = ∠BAC [Proved Above]
⇒ AC = BC [Side opposite to equal angles are equal]⇒ △ ABC is isosceles.