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We are tasked with finding the arc \( CD \) in the given circle, with the following information:

- Angle \( \angle DEC = 226.15^\circ \)
- Angle \( x = 46.15^\circ \)
- Segment \( DE \) is tangent to the circle at point \( E \), and similarly, segments that appear tangent should be considered tangent.

### Step-by-Step Approach:

1. **Identify key geometric relationships:**
   - Arc \( CD \) can be related to the tangent and the angles formed by the tangents. In this case, the angle \( \angle FCD \) will help determine the measure of the arc.

2. **Arc-Tangent Theorem:**
   - If we are given the external angle between two tangents (like \( \angle FCD \) or its complementary part \( x \)), the measure of the arc opposite the angle can be calculated using:
   \[
   \text{Arc } CD = 360^\circ - 2 \times x
   \]
   Here, \( x = 46.15^\circ \).

3. **Calculation:**
   \[
   \text{Arc } CD = 360^\circ - 2 \times 46.15^\circ = 360^\circ - 92.30^\circ = 267.70^\circ
   \]

Thus, the measure of arc \( CD \) is approximately **267.70°** (rounded to two decimal places).

Would you like more details or have any further questions?

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Here are some relative questions to expand your understanding:

1. How does the tangent-secant angle theorem apply in this case?
2. What are the properties of a tangent to a circle?
3. How is the angle formed between two tangents related to the intercepted arc?
4. Can the formula \( \text{Arc} = 360^\circ - 2 \times x \) be used in other situations? If yes, when?
5. What is the importance of the external angle in circle geometry?

**Tip**: Always double-check whether angles are internal or external to the circle when applying geometric theorems.

Find arc CD. Assume segments that look tangent to the circle are tangent. Round to 2 decimal places.

Learn how to find the measure of arc CD using the Arc-Tangent Theorem. Given angle x = 46.15° and external angle DEC = 226.15°, this problem involves circle geometry and tangent properties. Suitable for students in Grades 9-12.

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