In the given problem, we have an isosceles triangle $\triangle ABC$, where $AB \cong BC$. This means that sides $AB$ and $BC$ are equal in length.

Given information:

• $AB = 4x - 3$
• $BC = 2x + 5$
• $CA = x + 2$

Steps to solve:

1. Set $AB$ equal to $BC$ since $AB \cong BC$:
   $4x - 3 = 2x + 5$

2. Solve for $x$: $4x - 2x = 5 + 3$ $2x = 8$ $x = 4$

3. Find the lengths of each side:

   • AB: $AB = 4x - 3 = 4(4) - 3 = 16 - 3 = 13$

   • BC: $BC = 2x + 5 = 2(4) + 5 = 8 + 5 = 13$

   • CA: $CA = x + 2 = 4 + 2 = 6$

Final answers:

• $x = 4$
• $AB = 13$
• $BC = 13$
• $CA = 6$

Would you like a further breakdown of any steps or explanations?

Here are some related questions you might find useful:

1. What are the properties of an isosceles triangle?
2. How do we determine the angles in an isosceles triangle?
3. Can you solve a similar problem if all side lengths involve variables?
4. How do the angles in a triangle relate to the side lengths?
5. What if $AB$ and $BC$ were not congruent—how would the solution change?

Tip: Always check if any additional conditions are given, such as angle congruency, as they may provide further insights into the problem.