https://r2.erweima.ai/mathbot/image/1726765881485k348unm1.jpg

In the image, you are given a circle with five points labeled \( J, O, R, G, E \), and a tangent line at point \( J \). The angles \( \alpha \) are marked, and we are asked to calculate the value of angle \( x \), formed between points \( G \) and \( J \).

### Step-by-step Analysis:

1. **Given information**: There are five congruent angles labeled \( \alpha \) at point \( J \), implying that the angle subtended by the tangent line at \( J \) and any chord is the same.

2. **Inscribed angles theorem**: All angles inscribed on the circle are half of the central angle subtended by the same arc. Therefore, the five angles \( \alpha \) are subtended by the same arcs \( OJ, RJ, GJ, EJ \).

3. **Sum of the angles**:
   Since there are five equal angles \( \alpha \) at point \( J \), the total sum of these angles is \( 180^\circ \), because the angles at a tangent point form a semicircle.

4. **Finding angle \( x \)**:
   The angles \( \alpha \) are spread across the full circle (360°), and thus, \( x \) represents the angle subtended by one arc segment formed by two of these points on the circle.

   - Since there are five congruent segments, the total circle's angle \( 360^\circ \) is divided by 5.

\[
x = \frac{360^\circ}{5} = 72^\circ
\]

Thus, the value of \( x \) is \( \boxed{72^\circ} \).

Would you like more details or explanations? Here are some follow-up questions:

1. How does the tangent-secant angle theorem apply in this problem?
2. Why is the sum of the angles at point \( J \) equal to 180°?
3. Can you explain how the inscribed angle theorem helps in this situation?
4. What happens if the number of congruent angles increases to six? How would \( x \) change?
5. How do tangent lines relate to the circle's geometry in this context?

**Tip**: Remember that tangent lines are always perpendicular to the radius at the point of tangency!

Jorge resalta cinco puntos J, O, R, G, y E en una circunferencia; así como la recta tangente a la circunferencia en J, de tal manera que los cinco ángulos de medida α son congruentes. Calcule el valor de x.

This geometry problem involves five congruent angles formed in a circle with a tangent line at point J. Using the inscribed angle theorem and properties of a circle, find the value of x, the angle between G and J. Ideal for students in Grades 10-12.

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