The question is based on the geometry of a circle and the properties of angles subtended by the diameter and chords in a circle. Let’s analyze and solve each part systematically.

Given:

• $AB$ is the diameter of the circle.
• $AB \parallel CD$.
• Angle relationships in the circle are to be evaluated for the table provided.

We proceed by explaining and solving each part of the table:

a) $\angle ADB = 90^\circ$

• Reason: According to the property of a circle, the angle subtended by the diameter of a circle on the circumference is always a right angle. Hence, $\angle ADB = 90^\circ$.

b) $\angle ACD = \angle ABD$

• Reason: Since $AB \parallel CD$, the angles subtended by these parallel chords at the circumference on the same side of the circle are equal. This is due to the property of equal arcs subtending equal angles at the circumference.

c) $\angle BAC = \angle ACD$

• Reason: Both $\angle BAC$ and $\angle ACD$ are angles subtended by the same chord $AC$ on the circumference. According to the angle subtended by the same chord theorem, they are equal.

d) $\angle BAC + \angle BAD = 180^\circ$

• Reason: The total angle subtended by any two adjacent arcs of the circle is $180^\circ$, as the points lie on a semicircle (formed by the diameter $AB$).

Final Table:

Would you like further clarification on any part?
Here are 5 related questions to explore:

1. What are other properties of angles subtended by a chord in a circle?
2. How do parallel chords affect angle relationships in a circle?
3. Can you derive the angle subtended by a diameter mathematically?
4. How does the cyclic quadrilateral property apply in this diagram?
5. What happens if the diameter is not parallel to the chord $CD$?

Tip: Always start solving circle geometry problems by identifying key elements like the diameter, chords, and the center, as they often simplify the reasoning.