## Geometry Diagram: Circle with Tangent and Inscribed Angle
### Overview
The image depicts a circle labeled **O** with center **O**. A tangent line **CD** touches the circle at point **C**, and point **D** lies outside the circle. Points **A** and **B** are on the circle, connected to **O** via radii **OA** and **OB**. The problem asks for the measure of **angle A**, with options provided. A labeled angle of **50°** is marked at point **D**.
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### Components/Axes
- **Circle**: Centered at **O**, labeled as "circle O."
- **Tangent Line**: **CD** touches the circle at **C** (tangent property: **OC ⊥ CD**).
- **Points**:
- **A**, **B**: On the circle, connected to **O** (radii **OA**, **OB**).
- **C**: Point of tangency.
- **D**: External point where **CD** extends.
- **Angles**:
- **∠D = 50°** (marked at point **D**).
- **∠A**: To be determined (options: 20°, 25°, 40°, 50°).
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### Detailed Analysis
1. **Tangent Property**: Since **CD** is tangent to the circle at **C**, **OC ⊥ CD** (angle **∠OCD = 90°**).
2. **Triangle OCD**:
- **∠OCD = 90°** (tangent-radius property).
- **∠D = 50°** (given).
- **∠COD = 180° - 90° - 50° = 40°** (sum of angles in a triangle).
3. **Central Angle ∠COD = 40°**: This angle subtends arc **CB**.
4. **Inscribed Angle Theorem**: The inscribed angle subtended by the same arc **CB** is half the central angle:
- **∠CAB = ½ × 40° = 20°**.
5. **Angle A**: If **angle A** refers to **∠CAB**, it equals **20°**.
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### Key Observations
- The tangent **CD** and radius **OC** form a right angle (**90°**).
- The central angle **∠COD** (40°) directly relates to the inscribed angle **∠CAB** via the inscribed angle theorem.
- The given **50°** at **D** is critical for calculating **∠COD**.
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### Interpretation
The problem leverages the **tangent-radius perpendicularity** and **inscribed angle theorem**. The **50°** at **D** allows calculation of the central angle **∠COD = 40°**, which is halved to find the inscribed angle **∠A = 20°**. This aligns with option **A: 20°**. The diagram emphasizes the relationship between external angles (e.g., **∠D**) and internal circle properties (central/inscribed angles).
