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In the given figure, we are tasked with calculating two angles: \(\angle ABC\) and \(\angle ADC\), given that \(\angle AOC = 130^\circ\).

### Solution:

1. **Angle at the center and angle at the circumference subtended by the same arc**:
   - In a circle, the angle subtended at the center is twice the angle subtended at the circumference by the same arc. This is known as the *angle at the center theorem*.
   - Arc \(AC\) subtends the central angle \(\angle AOC = 130^\circ\).
   - Therefore, the angle subtended by the same arc \(AC\) at point \(B\) on the circumference (i.e., \(\angle ABC\)) is:
   \[
   \angle ABC = \frac{1}{2} \times \angle AOC = \frac{1}{2} \times 130^\circ = 65^\circ
   \]

2. **Opposite angles in a cyclic quadrilateral**:
   - Quadrilateral \(ABCD\) is a cyclic quadrilateral (all vertices lie on the circumference of the circle).
   - The opposite angles in a cyclic quadrilateral sum to \(180^\circ\).
   - Therefore, \(\angle ABC + \angle ADC = 180^\circ\).
   - We already found that \(\angle ABC = 65^\circ\), so:
   \[
   \angle ADC = 180^\circ - 65^\circ = 115^\circ
   \]

### Final Answer:
- \(\angle ABC = 65^\circ\)
- \(\angle ADC = 115^\circ\)

These results follow directly from the properties of circles and cyclic quadrilaterals.

Would you like further clarification on any of the steps or explanations? 

Here are 5 related questions to further explore this topic:
1. What is the angle subtended at the circumference by a semicircle?
2. Can all quadrilaterals be cyclic? Why or why not?
3. How do you prove that opposite angles in a cyclic quadrilateral are supplementary?
4. What happens if the central angle is a right angle? What can we say about the arc?
5. How can the inscribed angle theorem be applied to real-world problems involving circles?

**Tip:** Always remember that in a cyclic quadrilateral, the sum of opposite angles is 180°, a crucial property that simplifies solving many geometric problems in circles.

In Fig. 22.13, if ∠AOC = 130°, calculate ∠ABC and ∠ADC, giving reasons for your answers.

Learn how to calculate angles in a circle using the inscribed angle theorem and properties of cyclic quadrilaterals. Given ∠AOC = 130°, we find ∠ABC and ∠ADC step-by-step with clear reasoning. Suitable for students in Grades 8-10.

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