505. In figure, D is any point on the base BC produced of an isosceles triangle ∆ABC. Prove that AD > AB.

Given: D is any point on the base BC produced of an isosceles triangle ABC.
To Prove: AD >  AB.
Proof: ∵ ABC is an isosceles triangle
∴ AB = AC    ...(1)
∴ ∠ABC = ∠ACB    ...(2)
| Angles opposite to equal sides of a triangle are equal
In ∆ACD,
Ext. ∠ACB >  ∠CDA
| An exterior angle of a triangle is greater than each of its interior opposite angles
⇒ ∠ABC >  ∠BDA    | From (2)
⇒ ∠ABD >  ∠BDA AD > AB
| Side opposite to greater angle is longer.