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We are tasked with finding the number of sides of a regular polygon, where the measure of each interior angle is 24 degrees more than 38 times the measure of each exterior angle.

### Let’s break this down:
1. **Interior and Exterior Angle Relationship**:
   The sum of the interior and exterior angles of any polygon is always \(180^\circ\).
   - Interior angle \( = 180^\circ - \) Exterior angle.

2. **Exterior Angle Formula for Regular Polygons**:
   For a polygon with \(n\) sides, the measure of each exterior angle is:
   \[
   \text{Exterior angle} = \frac{360^\circ}{n}
   \]
   Therefore, the measure of the interior angle is:
   \[
   \text{Interior angle} = 180^\circ - \frac{360^\circ}{n}
   \]

3. **Given Equation**:
   According to the problem, the interior angle is 24 degrees more than 38 times the exterior angle. This gives the equation:
   \[
   180^\circ - \frac{360^\circ}{n} = 38 \times \frac{360^\circ}{n} + 24
   \]

### Step-by-step solution:

1. Combine terms to form the equation:
   \[
   180^\circ - \frac{360^\circ}{n} = 38 \times \frac{360^\circ}{n} + 24
   \]

2. Simplify and group like terms:
   \[
   180^\circ - 24 = 38 \times \frac{360^\circ}{n} + \frac{360^\circ}{n}
   \]
   \[
   156^\circ = 39 \times \frac{360^\circ}{n}
   \]

3. Solve for \(n\):
   \[
   \frac{156^\circ}{39} = \frac{360^\circ}{n}
   \]
   \[
   n = \frac{360^\circ \times 39}{156^\circ}
   \]
   \[
   n = 90
   \]

Thus, the polygon has **90 sides**.

---

Would you like a deeper explanation or have any other questions?

Here are five questions that can expand on this topic:
1. How do the interior and exterior angles of polygons relate to each other?
2. What is the formula for the sum of the interior angles of any polygon?
3. How does increasing the number of sides affect the measure of interior angles in regular polygons?
4. Can a polygon with fewer sides have an interior angle relation similar to the one in this problem?
5. What is the significance of a polygon's exterior angle sum always being \(360^\circ\)?

**Tip**: In problems involving regular polygons, always remember the key relationship between interior and exterior angles: their sum is always \(180^\circ\).

The measure of each interior angle of a regular polygon is 24 more than 38 times the measure of each exterior angle. Find the number of sides of the polygon.

Learn how to find the number of sides of a polygon where the measure of each interior angle is 24 degrees more than 38 times the measure of its exterior angle. This problem involves key geometry concepts and formulas related to polygons and angles, ideal for students in Grades 9-12.

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