From the diagram itself, we know that the circle is inscribed in the triangle, which is to say, the three sides of the triangle are tangent to the circle. Side BC is tangent to the circle at point E, so angle AEC must be 90 degrees. This is because a radius is always perpendicular to a tangent line at the point of tangency.
We want to know: Is angle CBD a right angle?
Statement #1: AE is parallel to BD
Well, AE is perpendicular to BC, and if AE is parallel to BD, that means BD must also be perpendicular to BC. Two parallel lines have to be perpendicular to the same line. If BD is perpendicular to BC, that means the angle where they meet, angle CBD, is a right angle. This statement is sufficient.
Statement #2: BC=BD
Well, that would tell us that triangle BCD is an isosceles triangle, and angle BCD = angle BDC. Angle CBD would be the vertex angle of an isosceles triangle. The vertex angle of an isosceles triangle can be acute, right, or obtuse, and in any of those cases, it's possible to inscribe a circle into the triangle. This statement gives us absolutely no information about whether angle CBD is right or not. This statement is insufficient.
Answer = A.
Please let me know if you have any questions.
Mike
