Find the value of $x$ .

A pentagon formed by five rays. The five exterior angles, in order, are labeled 2 x degrees, 71 degrees, 85 degrees, 44 degrees, and 3 x degrees respectively.

To find the value of $ x $ in this pentagon, we can use the property of the exterior angles of any polygon. The sum of the exterior angles of any polygon is always $ 360^\circ $.

Given:
- The five exterior angles are $ 2x $, $ 71^\circ $, $ 85^\circ $, $ 44^\circ $, and $ 3x $ degrees.

Let's set up the equation to sum these angles:

\[
2x + 71 + 85 + 44 + 3x = 360
\]

Now, combine like terms:

\[
5x + 200 = 360
\]

Next, isolate $ x $ by subtracting 200 from both sides:

\[
5x = 160
\]

Now, divide both sides by 5:

\[
x = 32
\]

So, the value of $ x $ is:

\[
\boxed{32}
\]

Would you like a more detailed explanation, or do you have any questions?

Here are some related questions to expand on this concept:
1. How would the solution change if it were a hexagon with different angle values?
2. What is the importance of exterior angle properties in polygons?
3. How does this apply to irregular polygons versus regular polygons?
4. Can interior angles be determined from exterior angles?
5. What happens if we increase or decrease one of the angles; how does it affect the equation?

**Tip:** Remember, the sum of exterior angles for any polygon always equals $ 360^\circ $, regardless of the number of sides.

Find the value of $x$. A pentagon formed by five rays. The five exterior angles, in order, are labeled 2x degrees, 71 degrees, 85 degrees, 44 degrees, and 3x degrees respectively.

Learn how to find the value of x in a pentagon formed by rays using the property that the sum of exterior angles of any polygon is always 360°. The solution involves algebraic equations and the exterior angle theorem. Suitable for students in Grades 7-9.

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