Since OB and OD are radii, OB=OD, as illustrated by the BLUE SEGMENTS in the figure above. Thus, ∠OBD = ∠BDO. Since it is given then BD=DA -- as illustrated by the PINK SEGMENTS in the figure above -- ∠BDA = ∠BAO. A radius drawn to a point of tangency forms a RIGHT ANGLE. Thus, radius OB and tangent BA form a right angle, implying that ∠OBA = 90. We can PLUG IN THE ANSWERS, which represent the value of ∠BAO. When we satisfy the constraint that ∠BDA = ∠BAO and that ∠OBD = ∠BDO, the result must be that ∠OBA = 90. D: 45 Since ∠BDA = ∠BAO and ∠OBD = ∠BDO -- and ∠BDA and ∠BDO must sum to 180 -- the following figure is yielded: Here, ∠OBA = 90+45 = 135. Eliminate D. B: 30 Since ∠BDA = ∠BAO and ∠OBD = ∠BDO -- and ∠BDA and ∠BDO must sum to 180 -- the following figure is yielded: Here, ∠OBA = 60+30 = 90. Success! The correct answer is B.

I wouldn't follow a messy string of calculations to solve this. How much time do we have for this anyway. Better to just observe the basics: 1. We all know isosceles triangles have equal base angles. There are two isosceles triangles in this picture. Label the angels. 2. A tangent to a circle makes a 90 degree angle with the radius. 3. Combine these two ideas and you get one equation in one variable. Regards, Mike Miagi www.google.com/+MikeMiagi

prachi18oct wrote: Please explain! The three angles of a triangle add up to 180 degrees. Looking at triangle 0BA we know that because AB is tangent to the circle, ∠OBA = 90. So we have one of the three angles of triangle OBA. So the other two angles of triangle OBA add up to 180 - 90 = 90, meaning ∠BAO + ∠BOA = 90. Now, since ∠OBA = 90, ∠ABD + ∠OBD = 90. Great. Why is this great? Because BD = DA, meaning triangle ABD is isosceles, and ∠ABD = ∠BAO So that means ∠BAO + ∠OBD = 90. We already know that ∠BAO + ∠BOA = 90. Therefore ∠OBD =∠BOA. Now, we're almost there because OB and OD are radii, forming an isosceles triangle. So we know that ∠OBD = ∠ODB. From the above we also know that ∠OBD =∠BOA. So all three angles of triangle OBD are the same and it's an equilateral triangle with all angles measuring 60 degrees. So ∠BOD = 60 and ∠BAO = 90 - 60 = 30 Choose B.