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This is NOT homework. I remember solving this question many years ago (at least 10 years ago). I am trying to recall the solution again and am just not able to. The question is -

In a triangle ABC, AD is the angle bisector of angle BAC. AB = CD. Prove that angle BAC = 72 degrees.

A diagram is attached.

NOTE - I am actually not even sure the question has enough information as given. I am just recalling from memory. If you think more information is required give an appropriate reason as to why you think that is true.

Usually, the way I see if the question has enough information is that I try to construct a triangle given the above properties. If I can construct a unique triangle, then of course the information is enough. Else not.

Anyway, give it a try.

https://www.dropbox.com/s/wpvei01hw1nr25l/2012-06-06%2000.39.06.jpg