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To solve this problem, we need to find the value of \(x\) in the circle diagram provided. The angle \(x\) is marked inside the circle, and there is an external angle marked as \(70^\circ\). Let's explore how these angles relate and how we can find \(x\).

### Understanding the Problem

The angle \(70^\circ\) is formed between two tangents drawn from a point outside the circle to the circle itself. Let's break down the key geometry principles involved:

1. **Tangent-Secant Angle Theorem**: The angle formed between a tangent and a chord through the point of contact (like angle \(x\)) is equal to half the difference of the intercepted arcs (the arc opposite the angle and the arc adjacent to it).
  
2. **External Angle Theorem**: For an angle formed outside a circle by two tangents, the angle is equal to half the difference of the intercepted arcs.

In this specific case, the external angle theorem applies, and we can use the following formula:

\[
\text{Angle} = \frac{1}{2} (\text{Difference of intercepted arcs})
\]

### Step-by-Step Solution

Here, we have:

- External angle = \(70^\circ\).
- Internal angle \(x\) (between a tangent and a chord).

The external angle is equal to half the difference of the measures of the intercepted arcs \(B\) and \(A\), where \(B\) is the larger arc opposite the external angle and \(A\) is the arc adjacent to the angle \(x\).

Thus:

\[
70^\circ = \frac{1}{2} (B - A)
\]

The angle \(x\) is the interior angle made by the tangent and the chord, so:

\[
x = \frac{1}{2} \times (\text{Arc A})
\]

Given that the external angle and the internal angle subtend the same arcs, the arc difference can be directly related:

\[
x = \frac{1}{2} \times (B - 70^\circ)
\]

Thus:

\[
x = \frac{1}{2} \times 70^\circ = 35^\circ
\]

This internal relationship simplifies our calculations:

\[
x = 70^\circ / 2 = 35^\circ
\]

### Final Answer

The angle \(x\) is:

\[
\boxed{35^\circ}
\]

### Explanation

- **Tangent-Secant Angle Theorem**: External angles in circles relate directly to the internal angles through arcs, making half the difference or sum of arcs, in this case, related to each tangent or chord.

- **External Angle Calculation**: Knowing the external angle and its impact helps identify the internal angle easily with simple division, which we confirmed through:

  - **External Angle**: \[70^\circ = \frac{1}{2} (\text{Difference of intercepted arcs})\]
  - **Internal Angle**: Thus, \(x\) is directly derived from the external relationship.

### Further Exploration

1. What is the measure of an angle formed by two chords intersecting inside the circle?
2. How do you find the angle between two secants intersecting outside the circle?
3. What is the Inscribed Angle Theorem, and how does it apply to circles?
4. How do tangent-secant theorems differ when applied to different circle positions?
5. What are some properties of cyclic quadrilaterals in circle geometry?
6. How do arc lengths relate to angles in circles?
7. What happens if two tangents are drawn from a single external point?
8. How does the Alternate Segment Theorem relate to tangent angles in circles?

### Tip

Remember, angles within and outside a circle follow specific rules like the Tangent-Secant and Alternate Segment Theorems. Knowing these can simplify complex circle problems.

Do you have any questions or need more details on this topic?

Learn how to find the value of x in a circle diagram using the Tangent-Secant Theorem and the External Angle Theorem. This problem involves understanding circle geometry, angles between tangents and chords, and the relationship between internal and external angles in circles. Suitable for high school students studying circle geometry.

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