ABCD is a cyclic quadrilateral in which AB is parallel to DC and AB is a diameter of the circle. Given ` angle BED = 65^(@) ` , Calculate
(i) ` angle DAB `
(ii) ` angle BDC `

Answer

To solve the problem step by step, we will analyze the cyclic quadrilateral ABCD with the given conditions.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- ABCD is a cyclic quadrilateral.
- AB is parallel to DC.
- AB is the diameter of the circle.
- Angle BED = 65°.

2. **Use the Property of Angles Subtended by the Same Chord:**
- Since AB is the diameter, angle DAB subtended by chord DB at point A is equal to angle BED (subtended at point E).
- Therefore, we have:
\[ \angle DAB = \angle BED = 65° \]

3. **Determine Angle ADB:**
- Since AB is the diameter of the circle, angle ADB (formed at point D) is a right angle (90°) because the angle in a semicircle is always 90°.
- Thus, we have:
\[ \angle ADB = 90° \]

4. **Find Angle ABD:**
- In triangle ADB, we can find angle ABD using the fact that the sum of angles in a triangle is 180°.
- Therefore:
\[ \angle ADB + \angle DAB + \angle ABD = 180° \]
- Substituting the known values:
\[ 90° + 65° + \angle ABD = 180° \]
- Simplifying gives:
\[ \angle ABD = 180° - 90° - 65° = 25° \]

5. **Use the Parallel Lines Property:**
- Since AB is parallel to DC, angle ABD is equal to angle BDC (alternate interior angles).
- Therefore:
\[ \angle BDC = \angle ABD = 25° \]

### Final Answers:
- (i) Angle DAB = 65°
- (ii) Angle BDC = 25°