In a cyclic quadrilateral ABCD, the diagonal AC bisects the angle C. Which of the following statements is/are correct? I. ∠ABD = ∠ADB II. The diagonal BD is parallel to the tangent of the circle at A

Since AC bisects ∠C, we have ∠ACB = ∠ACD. Angles subtended by equal arcs are equal: ∠ABD subtends arc AD and ∠ACD subtends arc AD, so ∠ABD = ∠ACD. Similarly ∠ADB = ∠ACB. Since ∠ACB = ∠ACD, we get ∠ABD = ∠ADB, so I is correct. For II: the tangent at A makes an angle with chord AB equal to the angle in the alternate segment (∠ADB). Since ∠ABD = ∠ADB, BD subtends equal angles, and the tangent at A is parallel to BD. So II is also correct.