In a cyclic quadrilateral ABCD, side AB is extended to a point X. If ∠XBC = 82° and the angle ∠ADB at the opposite vertex is 47°, what is the measure of angle ∠BDC in degrees?

Introduction / Context:
This question combines properties of cyclic quadrilaterals with exterior angles of triangles. It tests whether a student can correctly relate inscribed angles that subtend the same chord and use basic angle chasing in a cyclic figure.

Given Data / Assumptions:
- ABCD is a cyclic quadrilateral, so all four vertices lie on a circle.
- Side AB is extended beyond B to point X.
- ∠XBC, an exterior angle at vertex B of triangle ABC, is 82°.
- ∠ADB is given as 47°.
- We are asked to find ∠BDC in degrees.

Concept / Approach:
There are two important concepts:
- In a triangle, an exterior angle is equal to the sum of the two opposite interior angles.
- In a circle, inscribed angles that subtend the same chord are equal. Here both ∠ACB and ∠ADB subtend chord AB, so they are equal.
Once we use ∠ADB = 47° to determine ∠ACB, we can use the exterior angle at B to find ∠BAC, and then use the inscribed angle theorem again to connect ∠BAC with ∠BDC, since both subtend chord BC.

Step-by-Step Solution:
Step 1: Because ABCD is cyclic, and both ∠ACB and ∠ADB subtend chord AB, we have ∠ACB = ∠ADB = 47°. Step 2: Consider triangle ABC. The exterior angle at B, ∠XBC, equals the sum of the two interior opposite angles ∠BAC and ∠ACB. Step 3: Write the relation: ∠XBC = ∠BAC + ∠ACB. Step 4: Substitute the values: 82° = ∠BAC + 47°. Step 5: Solve for ∠BAC: ∠BAC = 82° − 47° = 35°. Step 6: Now note that ∠BAC and ∠BDC both subtend chord BC of the circle, so these two inscribed angles are equal. Step 7: Therefore ∠BDC = ∠BAC = 35°.

Verification / Alternative Check:
We can check consistency by confirming that no other angle condition is violated. The fact that ∠ACB = 47° and ∠BAC = 35° implies that ∠ABC = 180° − (35° + 47°) = 98°. Together with the exterior angle 82°, this is consistent since ∠XBC is outside the triangle and equal to 82°. There is no conflict among the angle relations, confirming that 35° for ∠BDC is correct.

Why Other Options Are Wrong:
Option a, 40°, would imply a different measure for ∠BAC and hence would break the equality of inscribed angles subtending chord BC.
Option c, 30°, and option d, 25°, also do not match the derived angle of 35° based on the external angle relationship and inscribed angle theorem, and they would make the sum of angles in triangle ABC inconsistent with the given exterior angle.

Common Pitfalls:
A common mistake is to forget that inscribed angles subtending the same chord are equal, or to misuse the cyclic quadrilateral property that opposite angles sum to 180°, even when the information given is better handled by the inscribed angle theorem. Another pitfall is misidentifying which angles subtend which chords. Drawing the circle carefully and marking chords AB and BC clearly can prevent such confusion.

Final Answer:
The measure of angle ∠BDC is 35°.