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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?

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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.

Find x and the length of each side if ΔABC is an isosceles triangle with AB ≅ BC.

Learn how to find the value of x and the lengths of sides AB, BC, and CA in an isosceles triangle where AB ≅ BC. This problem covers algebraic solving techniques and properties of isosceles triangles. Suitable for students in Grades 7-9.

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