https://file.aiquickdraw.com/mathbot/file/17266183151660fmqvqs7.jpg

Let's analyze each of the statements provided in the image for a **regular pentadecagon** (a polygon with 15 sides):

1. **Statement I**: "Desde cada vértice se pueden trazar 90 diagonales."
   - The formula for calculating the number of diagonals from a vertex in a polygon is:
     \[
     \text{Diagonals per vertex} = n - 3
     \]
     where \( n \) is the number of sides. For a pentadecagon, \( n = 15 \), so:
     \[
     15 - 3 = 12
     \]
     Thus, from each vertex, 12 diagonals can be drawn. The claim of 90 diagonals from each vertex is **false**.

2. **Statement II**: "Cada ángulo externo mide 24°."
   - The formula for calculating the external angle of a regular polygon is:
     \[
     \text{External angle} = \frac{360^\circ}{n}
     \]
     For a pentadecagon, \( n = 15 \), so:
     \[
     \frac{360^\circ}{15} = 24^\circ
     \]
     This statement is **true**.

3. **Statement III**: "La suma de los ángulos internos es 2340°."
   - The formula for the sum of the internal angles of a polygon is:
     \[
     \text{Sum of internal angles} = 180^\circ (n - 2)
     \]
     For a pentadecagon, \( n = 15 \), so:
     \[
     180^\circ \times (15 - 2) = 180^\circ \times 13 = 2340^\circ
     \]
     This statement is **true**.

### Conclusion:
- **True statements**: II and III.
- **False statement**: I.

Would you like more details or explanations on any part of this analysis?

Here are 5 related questions to expand your understanding:
1. How many total diagonals can be drawn in a regular pentadecagon?
2. How do you calculate the internal angle of a regular polygon?
3. How is the number of sides related to the measure of external angles?
4. What is the geometric significance of diagonals in a polygon?
5. How do the internal and external angles of polygons complement each other?

**Tip**: The internal and external angles of any polygon are supplementary, meaning they always add up to 180°.

Analice las siguientes afirmaciones para un pentadecágono regular:
I. Desde cada vértice se pueden trazar 90 diagonales.
II. Cada ángulo externo mide 24°.
III. La suma de los ángulos internos es 2340°.
¿Cuáles de ellas son verdaderas?

This question analyzes three statements about a regular pentadecagon, focusing on the number of diagonals, the measure of each external angle, and the sum of internal angles. Explore how to calculate diagonals, external angles, and the total internal angles using polygon formulas. Suitable for students in grades 9-12.

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