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To prove that triangles \( \triangle ADB \) and \( \triangle CDB \) are congruent, we'll use the information provided and fill in the correct statements:

### Given Information:
- \( AB = BC = CD = DA \) (All sides of the quadrilateral are equal, making it a rhombus)
- \( BD \) is the common side between both triangles.

### Proof Steps:

1. **\( AD = CD \)** - This is given directly in the problem (as both \( AD \) and \( CD \) are equal sides of the rhombus).
   
2. **\( \angle ABD = \angle CDB \)** - Since the quadrilateral is a rhombus, the diagonals bisect each other at right angles. Therefore, these angles are equal by the property of rhombus diagonals.

3. **\( BD \) is common** - \( BD \) is a shared side between both triangles \( \triangle ADB \) and \( \triangle CDB \).

### Conclusion:
By the **Side-Angle-Side (SAS)** congruence rule:
- Side \( AD = CD \) (Given)
- Angle \( \angle ABD = \angle CDB \) (Diagonals of a rhombus bisect each other at right angles)
- Side \( BD \) is common.

Thus, \( \triangle ADB \cong \triangle CDB \).

Would you like further details, or do you have any questions? Here are some additional related questions:

1. What is the definition of congruent triangles?
2. How do the properties of a rhombus help in proving triangle congruence?
3. What is the Side-Angle-Side (SAS) congruence rule?
4. Can you list the properties of a rhombus related to its diagonals?
5. How do you identify congruent parts of congruent triangles?

**Tip:** Remember that in a rhombus, diagonals always bisect each other at right angles and are a key element in proving triangle congruence within the rhombus.

Prove that triangle ADB is congruent to triangle CDB in a rhombus where all sides are equal.

Learn how to prove that two triangles in a rhombus are congruent using the Side-Angle-Side (SAS) theorem. This detailed solution breaks down the steps to show that triangle ADB is congruent to triangle CDB, highlighting key properties of rhombuses and congruent triangles.

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