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The image shows a problem involving a cyclic quadrilateral, where one needs to calculate a specific angle, labeled as \(\alpha\). The diagram represents a quadrilateral inscribed in a circle, with angles at the vertices and a question asking for the value of angle \(\alpha\).

Given the choices:
- A) \(75^\circ\)
- B) \(78^\circ\)
- C) \(45^\circ\)
- D) \(10^\circ\)
- E) \(72^\circ\)

### **Steps to Solve:**

1. **Understand the Properties of Cyclic Quadrilaterals:**
   - In a cyclic quadrilateral (a quadrilateral inscribed in a circle), the opposite angles are supplementary. This means the sum of the opposite angles equals \(180^\circ\).

2. **Identify the Known Angles:**
   - From the diagram, assume that we know three angles and need to calculate the fourth one, which is \(\alpha\).

3. **Calculate \(\alpha\):**
   - Let's denote the known angles as \( \angle A, \angle B, \angle C, \) and \( \angle D \), with \( \alpha = \angle D \). Since \( \angle B \) and \( \angle D \) are opposite angles, and the sum of opposite angles must be \(180^\circ\), you can find \( \alpha \) by subtracting the known opposite angle from \(180^\circ\).

### **Solution Calculation:**

Given that the exact values of the angles are not visible in the image, here's the general approach:
\[ \alpha = 180^\circ - \text{(Sum of the opposite angle)} \]

If, for instance, \(\angle B\) is \(105^\circ\), then:
\[ \alpha = 180^\circ - 105^\circ = 75^\circ \]

This example would make option A) \(75^\circ\) the correct answer.

### **Conclusion:**
If the known angle \( \angle B \) (opposite to \(\alpha\)) is given, you can apply this calculation directly.

Do you want further details on this solution or have any questions?

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**Relative Questions:**

1. What is the general formula for calculating angles in a cyclic quadrilateral?
2. How does the Inscribed Angle Theorem apply to this problem?
3. What are the conditions for a quadrilateral to be cyclic?
4. Can you explain the significance of supplementary angles in cyclic quadrilaterals?
5. How does one derive the angles if the circle's radius is known?

**Tip:** In any geometric problem involving circles, always consider the properties of cyclic figures, such as opposite angles summing to \(180^\circ\), which can simplify your calculations.

Learn how to calculate angle α in a cyclic quadrilateral problem with a detailed step-by-step solution. This problem involves applying the angle sum property of cyclic quadrilaterals and the Inscribed Angle Theorem. Suitable for high school students.

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