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This problem involves triangles and the property of equal angles between multiple triangles, suggesting that similar triangles are involved. Let's break it down:

### Given:
- The angles \( \angle ABC = \angle CAB = \angle DEB = \angle BDE \), implying similarity between triangles \( \triangle ABC \) and \( \triangle DEB \).
- \( AE = 21 \) and \( ED = 27 \).
- We need to find \( CA \).

### Step-by-Step Solution:

1. **Similarity of triangles**: 
   Since the angles are equal, the triangles \( \triangle ABC \) and \( \triangle DEB \) are similar by the AA (Angle-Angle) criterion.

   Therefore, we can write a ratio of corresponding sides:
   \[
   \frac{CA}{AB} = \frac{AE}{ED}.
   \]
   
2. **Given lengths**:
   - \( AE = 21 \),
   - \( ED = 27 \).
   
   Using the ratio of corresponding sides from the similar triangles, we substitute the values:
   \[
   \frac{CA}{AB} = \frac{21}{27} = \frac{7}{9}.
   \]
   
3. **Relation between \( CA \) and \( AB \)**:
   From this, we have:
   \[
   CA = \frac{7}{9} \times AB.
   \]

4. **Conclusion**: 
   To find \( CA \), we need the length of \( AB \), which isn't provided directly in the problem. However, the relationship between \( CA \) and \( AB \) is clear: \( CA \) is \( \frac{7}{9} \) of \( AB \).

### Final Answer:
   \( CA = \frac{7}{9} \times AB \), where \( AB \) is yet to be determined.

Would you like further details or clarifications?

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Here are 5 relative questions to expand on the solution:
1. What is the property of similar triangles in terms of their corresponding sides and angles?
2. How do you derive the ratio of corresponding sides in similar triangles?
3. Could the triangle's height or any additional length help find \( AB \)?
4. What are some applications of the AA criterion in triangle similarity?
5. How would the solution change if a different pair of angles were equal?

**Tip**: Always look for angle or side relationships when you see triangles with equal angles, as it usually hints at similarity.

In the diagram, we have \(\angle ABC = \angle CAB = \angle DEB = \angle BDE\). Given that \(AE = 21\) and \(ED = 27\), find \(CA\).

Solve this geometry problem involving triangle similarity using the AA criterion. With angles \(\angle ABC = \angle CAB = \angle DEB = \angle BDE\), and side lengths \(AE = 21\) and \(ED = 27\), learn how to find the length of side \(CA\). Suitable for students in Grades 9-12.

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