To solve this problem, we need to determine the value of $x$ using the given information about the triangle's angles.

Step 1: Set up the equation

In a triangle, the sum of the interior angles is always $180^\circ$. Here, the three angles are:

• $89^\circ$
• $57^\circ$
• $(5x - 6)^\circ$

So, we can write the equation: $89 + 57 + (5x - 6) = 180$

Step 2: Simplify the equation

Combine the constants on the left side: $146 + 5x - 6 = 180$ $140 + 5x = 180$

Step 3: Solve for $x$

Isolate $x$ by first subtracting 140 from both sides: $5x = 40$ Then, divide by 5: $x = 8$

Answer

The value of $x$ is $8$, which corresponds to option A.

Would you like a further breakdown of the steps?

Here are five related questions to deepen your understanding:

1. How would the equation change if one of the given angles was different?
2. What if the angle expression was $(5x + 6)^\circ$ instead of $(5x - 6)^\circ$?
3. How do you find $x$ if a triangle’s angle sum is different (like in non-Euclidean geometry)?
4. What would happen if this was a right triangle with known side lengths instead of angles?
5. How would you solve if this triangle’s angles were expressed in terms of two different variables?

Tip: Remember, in any triangle, the sum of the angles will always be $180^\circ$ in Euclidean geometry.