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This triangle problem shows an equilateral triangle (as indicated by the three \(60^\circ\) angles), with two sides labeled algebraically as \(7x + 19\) and \(11x - 89\). Since all sides of an equilateral triangle are equal, we can set these two expressions equal to each other and solve for \(x\).

### Step-by-Step Solution:

1. Set up the equation based on the equality of the sides:
   \[
   7x + 19 = 11x - 89
   \]

2. Rearrange to bring terms involving \(x\) to one side and constants to the other:
   \[
   19 + 89 = 11x - 7x
   \]
   \[
   108 = 4x
   \]

3. Solve for \(x\):
   \[
   x = \frac{108}{4} = 27
   \]

Thus, \(x = 27\).

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

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Here are 5 related questions that build on this problem:
1. What would happen if the angles were not all \(60^\circ\)?
2. How would you check if this triangle is indeed equilateral without given angles?
3. What if another side were given as a different expression? How would that affect solving for \(x\)?
4. How can we use these methods for other types of polygons?
5. How could we use this equation to solve for other unknowns if more sides were labeled?

**Tip:** In geometry, equilateral triangles are powerful tools, as knowing that all sides and angles are equal can simplify complex problems.

Solve for x given that two sides of an equilateral triangle are labeled as 7x + 19 and 11x - 89, with each angle equal to 60°.

Learn how to solve for x in an equilateral triangle where two sides are given as algebraic expressions, 7x + 19 and 11x - 89. This problem explores algebraic manipulation and properties of equilateral triangles, making it suitable for students in Grades 7-10.

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