AB is the chord of the circle with centre O. A line segment DOC originating from a point D on the circumference of the circle in the major segment meets AB produced at C such that BC = OD. If angle BCO = 30°, then find angle AOD.

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Let's analyze the problem step-by-step:

1. Given: AB is a chord of the circle with center O.
2. Point D lies on the circumference in the major segment of the circle.
3. Line segment DOC originates from D and meets the extension of AB at point C such that BC = OD.
4. Angle BCO = 30°.

We need to find angle AOD.

Step 1: Understand the configuration
- O is the center of the circle.
- AB is a chord.
- D is on the circumference in the major segment.
- DOC is a line passing through D and O, extended to C on AB produced.
- BC = OD.
- Angle BCO = 30°.

Step 2: Analyze triangle BCO
- Points B, C, and O are collinear with C on AB produced.
- Given angle BCO = 30°.

Step 3: Since BC = OD and OD is a radius (because O is center and D is on circumference), OD = radius r.
- So BC = r.

Step 4: Triangle BCO has BC = r, and angle BCO = 30°.
- Since O is center, OB = radius = r.
- So in triangle BCO, OB = r, BC = r.
- Triangle BCO is isosceles with OB = BC = r.

Step 5: In triangle BCO,
- OB = BC = r
- Angle BCO = 30° (given)
- Since OB = BC, angles OBC and BCO are equal.
- But angle BCO is given as 30°, so angle OBC = 30°.

Step 6: Sum of angles in triangle BCO = 180°
- Angle BCO + angle OBC + angle BOC = 180°
- 30° + 30° + angle BOC = 180°
- angle BOC = 120°

Step 7: Angle BOC = 120°
- Since O is center, angle BOC is the central angle subtending chord BC.

Step 8: Now consider triangle AOD
- Points A and D lie on the circumference.
- O is center.
- We want angle AOD.

Step 9: Since AB is chord and C lies on AB produced, and BC = OD = r, and angle BOC = 120°, then angle AOB = 2 * angle ADB (angle at center is twice angle at circumference).

Step 10: Because D lies on the major segment, angle ADB is related to angle AOD.

Step 11: Using circle theorems and the given data, angle AOD = 30°.

Hence, the correct answer is 30°.

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