In an isosceles triangle, two angles are equal. Let's denote the angles as follows:

• Angle 1: $5x - 7$
• Angle 2: $8x - 55$

Step-by-Step Solution:

Since it's an isosceles triangle, we assume that either:

1. Angle 1 is equal to Angle 2, or
2. Both are equal to the third angle, which also means the sum of all angles in the triangle is $180^\circ$.

Case 1: Angle 1 = Angle 2

If Angle 1 and Angle 2 are the equal angles of the isosceles triangle, then: $5x - 7 = 8x - 55$ Solving for $x$:

1. Move terms involving $x$ to one side: $-7 + 55 = 8x - 5x$ $48 = 3x$
2. Divide by 3: $x = 16$

So, $x = 16$ if Angle 1 equals Angle 2.

Verification:

• Substituting $x = 16$ into Angle 1: $5(16) - 7 = 80 - 7 = 73^\circ$
• Substituting $x = 16$ into Angle 2: $8(16) - 55 = 128 - 55 = 73^\circ$

Thus, both angles are $73^\circ$, confirming that the triangle is indeed isosceles.

Final Answer:

The value of $x$ is $\boxed{16}$.

Would you like additional details, or do you have any questions?

---

Additional Questions for Further Exploration:

1. How would the solution change if the triangle were not isosceles?
2. What are other possible configurations of angles in an isosceles triangle?
3. How would you find the third angle in this triangle?
4. Can we solve for $x$ if given only one angle?
5. What if the expressions for the angles were different?

Tip: Always verify solutions by plugging values back into the original equations to ensure consistency.