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This problem is related to the exterior angles of an \( n \)-sided polygon. Here's a breakdown of the question:

### Given:
- Five exterior angles of a polygon are: \( 20^\circ, 30^\circ, 34^\circ, 39^\circ, 57^\circ \).
- The remaining \( (n - 5) \) exterior angles are each \( 12^\circ \).
- We need to find the value of \( n \) (the number of sides of the polygon).

### Formula for exterior angles:
The sum of the exterior angles of any polygon is always \( 360^\circ \), regardless of the number of sides.

### Approach:
1. Let the total number of sides of the polygon be \( n \).
2. The sum of the five given exterior angles is:
   \[
   20^\circ + 30^\circ + 34^\circ + 39^\circ + 57^\circ = 180^\circ.
   \]
3. The remaining \( (n - 5) \) exterior angles are each \( 12^\circ \), so the sum of the remaining exterior angles is:
   \[
   (n - 5) \times 12^\circ.
   \]
4. The sum of all the exterior angles must equal \( 360^\circ \). Thus, the equation becomes:
   \[
   180^\circ + (n - 5) \times 12^\circ = 360^\circ.
   \]

### Solving the equation:
\[
180^\circ + (n - 5) \times 12^\circ = 360^\circ
\]
\[
(n - 5) \times 12^\circ = 360^\circ - 180^\circ = 180^\circ
\]
\[
n - 5 = \frac{180^\circ}{12^\circ} = 15
\]
\[
n = 15 + 5 = 20.
\]

Thus, the polygon has **20 sides**.

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Would you like more details or clarifications on any step?

### Related Questions:
1. How is the sum of exterior angles of a polygon always \( 360^\circ \)?
2. What is the relationship between exterior and interior angles of a polygon?
3. How would the solution change if the given angles were interior angles?
4. What is the formula for finding the number of sides given equal exterior angles?
5. How do we calculate interior angles of a regular polygon with \( n \) sides?

### Tip:
For any polygon, the sum of the exterior angles is always constant at \( 360^\circ \), regardless of the number of sides. This helps in solving problems related to polygons efficiently.

Five exterior angles of an n-sided polygon are 20°, 30°, 34°, 39°, and 57°. The remaining (n - 5) exterior angles are each equal to 12°. Find the value of n.

This problem helps students calculate the number of sides (n) of a polygon when given five exterior angles (20°, 30°, 34°, 39°, and 57°) and knowing that the remaining exterior angles are 12°. The sum of exterior angles of any polygon is 360°, making this an ideal geometry problem for Grades 9-10.

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