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The image you've uploaded contains four geometric diagrams, each of which seems to ask for a proof of congruency between triangles based on given information. I will go through each one and provide the steps required to prove the congruency of the triangles.

### Diagram 1.1:
Given: \( \angle DAB = \angle BCD \), \( \overline{AB} \) is common, and \( \overline{DA} = \overline{CB} \).

**Proof:**  
1. \( \overline{AB} \) is common (Shared side).  
2. \( \overline{DA} = \overline{CB} \) (Given).  
3. \( \angle DAB = \angle BCD \) (Given).  
4. By the **SAS** (Side-Angle-Side) congruence rule, \( \triangle DAB \cong \triangle BCD \).

### Diagram 1.2:
Given: \( \overline{AB} = \overline{CD} \), \( \angle BAE = \angle CED \), and \( \overline{AE} = \overline{ED} \).

**Proof:**  
1. \( \overline{AB} = \overline{CD} \) (Given).  
2. \( \angle BAE = \angle CED \) (Given).  
3. \( \overline{AE} = \overline{ED} \) (Given).  
4. By the **ASA** (Angle-Side-Angle) congruence rule, \( \triangle BAE \cong \triangle CED \).

### Diagram 1.3:
Given: \( \overline{AB} = \overline{CD} \), \( \overline{AC} = \overline{BD} \), and \( \angle BAC = \angle DAB \).

**Proof:**  
1. \( \overline{AB} = \overline{CD} \) (Given).  
2. \( \overline{AC} = \overline{BD} \) (Given).  
3. \( \angle BAC = \angle DAB \) (Given).  
4. By the **SAS** (Side-Angle-Side) congruence rule, \( \triangle ABC \cong \triangle BDA \).

### Diagram 1.4:
Given: \( \overline{AB} = \overline{CD} \), \( \angle DAB = \angle BCD \), and \( \overline{AD} = \overline{BC} \).

**Proof:**  
1. \( \overline{AB} = \overline{CD} \) (Given).  
2. \( \angle DAB = \angle BCD \) (Given).  
3. \( \overline{AD} = \overline{BC} \) (Given).  
4. By the **ASA** (Angle-Side-Angle) congruence rule, \( \triangle DAB \cong \triangle BCD \).

These steps establish the congruency of the triangles in each of the diagrams provided. Let me know if you need further clarification or a more detailed explanation.

Would you like any additional details or have any questions?

Here are five questions you might find useful:
1. How can the ASA and SAS congruence rules be applied in other geometric proofs?
2. What are other congruence rules for triangles besides ASA and SAS?
3. How can you determine when to use which congruence rule?
4. What is the significance of a common side in proving congruence?
5. How does angle bisector theorem apply in triangle congruency?

**Tip:** When working on geometric proofs, always start by identifying the given information and the goal, then match them with the appropriate congruence rules (like SSS, SAS, ASA, or AAS).

Explore how to prove triangle congruency in each diagram using detailed step-by-step solutions. This problem covers key geometric concepts including ASA and SAS congruence rules. Suitable for high school students learning geometry.

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