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Find the value of x.


A. 65
B. 110
C. 180
D. 55

To find the value of \( x \) in the given diagram, we need to use the properties of angles around a circle and the external angles.

1. **Identify the given angles:**
   - \( \angle 70^\circ \)
   - \( \angle 110^\circ \)

2. **Understand the geometry:**
   - The angle \( 110^\circ \) is an external angle formed by a tangent and a chord (or secant).
   - The angle \( x \) is an external angle at the point of tangency.

3. **Use the properties of external angles:**
   - The external angle \( \angle 110^\circ \) is equal to the sum of the internal opposite angles (interior opposite angles).
   - Therefore, \( \angle x = 180^\circ - \angle 70^\circ - \angle 110^\circ \) because the sum of angles in a triangle is \( 180^\circ \).

4. **Calculate the value of \( x \):**
   - Let's add the angles \( 70^\circ \) and \( 110^\circ \):
   \[
   70^\circ + 110^\circ = 180^\circ
   \]
   - Since the sum of these angles is \( 180^\circ \), \( \angle x = 180^\circ - 180^\circ = 0^\circ \).

However, this approach leads to a contradiction, so we must reconsider:

5. **Revisiting the angle relationships:**
   - Since \( 70^\circ \) and \( x \) are angles outside the circle at the tangents, and \( 110^\circ \) is the external angle directly:
   \[
   \angle x = 180^\circ - \angle 70^\circ = 180^\circ - 70^\circ = 110^\circ
   \]

Therefore, the correct value of \( x \) is \( 110^\circ \).

**Correct Answer:**
\[ \boxed{110} \]

Do you want more details or have any questions?

Here are some related questions you can ask:
1. How do you find angles formed by tangents and chords?
2. What are the properties of external angles in a triangle?
3. How can you use the tangent-secant theorem in circle geometry?
4. What is the relationship between interior and exterior angles in polygons?
5. How do you apply angle sum properties in various geometric shapes?

**Tip:** Remember, the exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles.

Learn how to find the value of x in circle geometry by applying the properties of external angles. This problem involves identifying angles formed by tangents and chords, and using the angle sum property of triangles. Suitable for high school students.

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