https://file.aiquickdraw.com/mathbot/file/172711285096240iigltf.jpg

Let's solve the problem step by step.

### Given Information:
- The external angles in the figure are labeled.
- The angles given are:
  - \( 60^\circ \) (in green at the top)
  - \( 3x \) (in purple at the left)
  - \( x + 90^\circ \) (in purple on the bottom-right)
  - \( x \) (in purple on the bottom-left)

### Solution Approach:

1. **Sum of Exterior Angles of a Triangle**:
   For any polygon, the sum of the exterior angles is always \( 360^\circ \).

   Here, we have four exterior angles: \( 60^\circ \), \( 3x \), \( x + 90^\circ \), and \( x \).

2. **Setting up the Equation**:
   \[
   60 + 3x + (x + 90) + x = 360
   \]

3. **Simplifying the Equation**:
   Combine the terms:
   \[
   60 + 3x + x + 90 + x = 360
   \]
   \[
   60 + 90 + 5x = 360
   \]
   \[
   150 + 5x = 360
   \]

4. **Solving for \( x \)**:
   Subtract 150 from both sides:
   \[
   5x = 360 - 150
   \]
   \[
   5x = 210
   \]

   Now, divide by 5:
   \[
   x = \frac{210}{5} = 42
   \]

### Final Answer:
\[
x = 42^\circ
\]

Would you like more details, or do you have any further questions?

Here are some related questions:
1. How do we calculate the sum of interior angles in any polygon?
2. Can you explain how exterior and interior angles are related in polygons?
3. How would you solve this if the angles were labeled differently?
4. What is the difference between an interior and exterior angle?
5. Can you always use the exterior angle sum theorem in irregular polygons?

**Tip**: Always double-check your equation setup when dealing with multiple angles to ensure you haven't left any out.

Find the value of x in a polygon where the exterior angles are 60°, 3x, x+90°, and x.

Learn how to solve a geometry problem by finding the value of x using the sum of exterior angles theorem. The problem involves exterior angles of a polygon with values 60°, 3x, x+90°, and x. Suitable for students in Grades 8-10.

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