To solve the problem, we analyze the given information and geometry of the circle:

Key Observations:

1. AB is the diameter: This means ∠ACB is 90° because an angle subtended by the diameter of a circle at the circumference is a right angle (by the semicircle property).

2. Given angles:

   • ∠ABD = 25°
   • ∠BAC = 30°

3. Required: ∠DCB.

Step-by-step Solution:

1. In triangle $\triangle ABD$:

   • The angle sum property of a triangle gives: $\angle BAD + \angle ABD + \angle ADB = 180^\circ$ Substituting the known values: $30^\circ + 25^\circ + \angle ADB = 180^\circ$ Solving for $\angle ADB$: $\angle ADB = 125^\circ$

2. Now, in the quadrilateral $ABCD$ inscribed in the circle:

   • Opposite angles of a cyclic quadrilateral are supplementary, so: $\angle ADB + \angle DCB = 180^\circ$ Substituting $\angle ADB = 125^\circ$: $\angle DCB = 180^\circ - 125^\circ = 55^\circ$

Final Answer:

The measure of $\angle DCB$ is 55°.

Answer: a. 55°

Would you like further clarification or a breakdown of any specific part?

Related Questions:

1. What property of circles ensures ∠ACB = 90°?
2. How does the cyclic quadrilateral property relate to this problem?
3. Why is the diameter subtending a right angle?
4. Can you find ∠ABC based on the given data?
5. How would this solution change if AB were not the diameter?

Tip: When solving problems involving circles, always identify whether any angles or lines