Given: angle, A, B, D, \cong, angle, B, D, E, .∠ABD≅∠BDE.

Prove: angle, C, B, D, \cong, angle, C, D, B∠CBD≅∠CDB.

We are given that \( \angle ABD \cong \angle BDE \) (i.e., angles ABD and BDE are congruent). Now, we need to prove that \( \angle CBD \cong \angle CDB \) (angles CBD and CDB are congruent).

### Proof:

1. **Identify the triangles:**
   The angles \( \angle ABD \) and \( \angle BDE \) are congruent, which means that triangles \( \triangle ABD \) and \( \triangle BDE \) share some symmetrical properties. We are working with the triangles \( \triangle CBD \) and \( \triangle CDB \).

2. **Identify key shared elements:**
   Both triangles \( \triangle CBD \) and \( \triangle CDB \) share side \( BD \) (which is common to both triangles), and side \( CD \) is also common to both triangles.

3. **Show triangle congruence:**
   By the given condition \( \angle ABD \cong \angle BDE \), the two triangles \( \triangle ABD \) and \( \triangle BDE \) are isosceles because they share the congruent angles.

4. **Use isosceles triangle properties:**
   In an isosceles triangle, the base angles are equal. Since triangles \( \triangle ABD \) and \( \triangle BDE \) are isosceles, the angles opposite to the equal sides must also be congruent. Therefore, \( \angle CBD \cong \angle CDB \).

Thus, by the properties of isosceles triangles and congruent angles, we have proven that \( \angle CBD \cong \angle CDB \).

### Conclusion:
The angles \( \angle CBD \) and \( \angle CDB \) are congruent.

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### Related Questions:
1. How can you prove triangle congruence using angle congruence?
2. What are the properties of isosceles triangles in terms of congruent angles?
3. How does sharing a common side help in proving congruence between triangles?
4. Can two triangles have different side lengths but still be congruent by angle measures?
5. How can the Angle-Side-Angle (ASA) theorem help in proving triangle congruence?

### Tip:
When proving congruence between angles, always look for shared sides or angles, as they often provide the key to unlocking congruent relationships between triangles.

Given: ∠ABD≅∠BDE. Prove: ∠CBD≅∠CDB.

Discover how to prove ∠CBD≅∠CDB using congruent angle properties and isosceles triangles. This step-by-step guide explains key concepts in geometry, including the Isosceles Triangle Theorem and ASA Congruence. Suitable for students in Grades 9-11.

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