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This problem appears to involve angles related to a circle, with an exterior angle formed by two intersecting lines. In such cases, the exterior angle theorem for circles can be applied.

### The theorem states:
If two lines intersect outside a circle, the measure of the angle formed by these lines is equal to half the difference of the intercepted arcs.

Here, we have:
- An exterior angle of \(23^\circ\),
- An arc inside the circle with a measure of \(14x - 22^\circ\).

Using the exterior angle theorem, we can write the equation:

\[
23^\circ = \frac{1}{2} \times \left( \text{Arc measure} \right)
\]

Substituting the given arc:

\[
23^\circ = \frac{1}{2} \times (14x - 22)
\]

Now, solving for \(x\):

1. Multiply both sides of the equation by 2 to eliminate the fraction:

\[
46 = 14x - 22
\]

2. Add 22 to both sides:

\[
68 = 14x
\]

3. Divide both sides by 14:

\[
x = \frac{68}{14} = 4.857
\]

Therefore, the value of \(x\) is approximately \(4.857\).

Would you like a deeper explanation on how this formula is applied or additional details?

Here are 5 related questions for further practice:
1. What is the general form of the exterior angle theorem in circles?
2. How do you calculate the measure of an intercepted arc?
3. Can this method be applied when the lines intersect inside the circle instead of outside?
4. What are the different types of angles formed in a circle, and how do they relate to the arcs?
5. How would the solution change if the angle inside the circle was unknown instead of the exterior angle?

**Tip:** When dealing with geometry problems, always check for theorems related to specific figures like circles or triangles to simplify the solving process.

Solve for x using the exterior angle formed by the circle and the intersecting lines. The exterior angle is 23°, and the arc measure inside the circle is given by (14x - 22)°.

Learn how to solve for x using the Exterior Angle Theorem in a circle. Given an exterior angle of 23° and an arc measure of (14x - 22)°, this problem covers important concepts in geometry, including the Exterior Angle Theorem. Suitable for students in Grades 9-11.

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