triangle ABC is right angled at B. on the side AC, a point D is taken such that AD = DC and AB = BD. find the measure of angle CAB.

Given:
AD = DC and ABC is a right triangle at vertex B.
Since ABC is a right angle and angle in a semi circle is a right angle we can conclude that D (the midpoint of AC) is the center of circle passing through A, B and C.
Hence AD, DC and BD are all equal to the radius of that circle.
Hence, AB = BD = AD.
So ABD is an equilateral triangle.
Hence, Ð y = Ð A = 60^o o r Ð CAB = 60^o