In the circle below, DOB is a diameter.




                ∠ADO =  blank                                    ∠ABC =  blank

                ∠OCA =  blank                                    ∠ACB =  blank

                ∠OBC =  blank                                    ∠CAB =  blank

24° 28° 34° 38° 52° 58° 114° 120°



Let's break down the problem using the following rules of circles and angles:

### Given:
- \( DOB \) is a diameter of the circle.
- Various angles need to be found using the provided angle values and geometric properties.

### Key Properties to Consider:
1. **Angle in a semicircle (Thales' Theorem)**: Any angle subtended by a diameter on the circumference is a right angle (i.e., \( \angle ADO = 90^\circ \) or \( \angle ABC = 90^\circ \)).
2. **Central Angle Theorem**: The angle subtended by an arc at the center of a circle is twice the angle subtended at any point on the remaining part of the circle.

Let’s use this information to determine each angle.

### Assigning the angles:
- \( \angle ADO = 90^\circ \) (by Thales' Theorem).
- From the options given, we select the appropriate angles for the other positions using properties of circles and angle relationships:

#### Trying the angles:
Based on the given information (24°, 28°, 34°, 38°, 52°, 58°, 114°, 120°), let's assume the following:

1. \( \angle ADO = 90^\circ \) is already established by Thales' Theorem.
2. \( \angle ABC = 90^\circ \) (from Thales' theorem as well, since \( DOB \) is the diameter).
3. \( \angle OCA = 34^\circ \).
4. \( \angle ACB = 58^\circ \) (could be a supplementary angle).
5. \( \angle OBC = 38^\circ \).
6. \( \angle CAB = 28^\circ \).

Let me know if you'd like to discuss further details or clarifications on any part of the solution!

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**Relative questions:**
1. How is Thales' theorem applied to a circle with a diameter?
2. Can you explain how the central angle theorem affects angles in a circle?
3. What is the relationship between angles on a semicircle and the diameter?
4. How do you determine if two angles are supplementary in a circle?
5. How would the sum of angles in a triangle within a circle relate to the diameter?

**Tip:** Always remember that the angle subtended by a diameter of a circle is a right angle. This fact is key to solving many circle-related angle problems.

In the circle below, DOB is a diameter.

                ∠ADO = blank                               ∠ABC = blank

                ∠OCA = blank                               ∠ACB = blank

                ∠OBC = blank                               ∠CAB = blank

24° 28° 34° 38° 52° 58° 114° 120°

Learn how to solve a circle geometry problem involving a diameter and angles, using the Thales' theorem and central angle theorem. This problem covers key geometric concepts related to circle theorems and is suitable for students in Grades 8-10.

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