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To find the value of \( x \) in this problem, let’s analyze the angles in this shape.

The figure consists of a polygon with the following angles:

1. Interior angles given:
   - \( 71^\circ \)
   - \( 85^\circ \)
   - \( 44^\circ \)

2. Variable angles represented by:
   - \( 2x \) degrees
   - \( 3x \) degrees

In a closed polygon, the sum of the interior angles depends on the number of sides. Since we are not explicitly told the shape's number of sides, we can focus on treating the problem as a linear angle equation.

Given that this figure appears to be a pentagon, we can use the sum of the interior angles formula for an \( n \)-sided polygon:
\[
\text{Sum of angles} = (n - 2) \times 180^\circ
\]
For a pentagon (\( n = 5 \)):
\[
\text{Sum of angles} = (5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ
\]

Now we can set up an equation:
\[
71^\circ + 85^\circ + 44^\circ + 2x + 3x = 540^\circ
\]

Combine terms:
\[
200^\circ + 5x = 540^\circ
\]

Subtract \( 200^\circ \) from both sides:
\[
5x = 340^\circ
\]

Divide by 5:
\[
x = 68^\circ
\]

### Answer:
\[
x = 68^\circ
\]

Would you like a further breakdown of this solution, or do you have other questions? Here are some related questions to deepen your understanding:

1. How do we determine the sum of angles in a polygon with any given number of sides?
2. If we change one of the angles, how would that affect the value of \( x \)?
3. What are some common properties of pentagons and other polygons?
4. How would this approach change if the figure had six sides instead of five?
5. Can you identify cases where polygons have exterior angles?

**Tip:** In any polygon, the sum of interior angles increases by \(180^\circ\) for each additional side added to the shape.

Find the value of x in a polygon with given angles of 71°, 85°, 44°, and variable angles 2x° and 3x°.

Solve for x in a polygon with angles of 71°, 85°, 44°, 2x°, and 3x°. This problem applies the Polygon Interior Angle Sum Theorem and basic algebra to form and solve an equation. Suitable for students in Grades 8-10.

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