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Let's go through each question on the worksheet:

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### **Question 6:**
**Given:**  
- \( PT \cong RS \),  
- \( \angle PTR \cong \angle RSP \).

**Prove:**  
- \( \triangle PQT \cong \triangle RQS \).

**Solution:**
1. **Statement:** \( PT \cong RS \)  
   **Reason:** Given

2. **Statement:** \( \angle PTR \cong \angle RSP \)  
   **Reason:** Given

3. **Statement:** \( \triangle PQT \cong \triangle RQS \)  
   **Reason:** AAS (Angle-Angle-Side)

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### **Question 7:**
**Given:**  
- \( BD \) is the angle bisector of \( \angle ABC \) and \( \angle ADC \).

**Prove:**  
- \( \triangle ABD \cong \triangle CBD \).

**Solution:**
1. **Statement:** \( BD \) is the angle bisector of \( \angle ABC \) and \( \angle ADC \)  
   **Reason:** Given

2. **Statement:** \( \angle ABD \cong \angle BCD \)  
   **Reason:** Definition of angle bisector

3. **Statement:** \( \angle BAD \cong \angle BCD \)  
   **Reason:** Vertical angles

4. **Statement:** \( BD \cong BD \)  
   **Reason:** Reflexive property

5. **Statement:** \( \triangle ABD \cong \triangle CBD \)  
   **Reason:** AAS (Angle-Angle-Side)

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### **Question 8: Reasoning**
**Question:** A student tells you that he can prove the AAS Theorem using the SAS Postulate and the Third Angles Theorem. Do you agree with him? Explain.

**Hint:** The SAS Postulate uses two pairs of sides, while AAS only needs one pair of sides and two angles. The Third Angles Theorem is used to find the third angle after two angles are congruent.

**Answer:** The student cannot prove the AAS Theorem using the SAS Postulate directly. The SAS Postulate requires two pairs of sides and the included angle, whereas AAS requires two pairs of angles and a non-included side. The Third Angles Theorem can be used to find the third angle, but this alone doesn’t fulfill the conditions needed for SAS.

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### **Question 9: Reasoning**
**Question:** Can you prove the triangles congruent? Justify your answer.

**Answer:**  
Yes, the triangles are congruent by the **SSS (Side-Side-Side)** theorem. All three sides of the triangles are marked as congruent, which satisfies the condition for congruence by SSS.

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### Questions to expand understanding:

1. How does the AAS theorem differ from the ASA theorem in terms of angle and side placement?
2. Can you give an example where SSS would not work but AAS or ASA would?
3. What is the importance of the reflexive property in triangle congruence proofs?
4. How can you use the Third Angles Theorem to solve congruence problems?
5. Why is the SAS postulate crucial in many congruence proofs?

**Tip:** Always double-check the conditions (angles or sides) that are given to ensure you're using the correct congruence theorem (e.g., SSS, SAS, ASA, AAS).

The image shows a practice worksheet on triangle congruence using AAS, ASA, and SSS theorems, with reasoning questions on congruence conditions.

Explore how to prove triangle congruence through geometric reasoning using the AAS, ASA, and SSS theorems. Learn key concepts like angle bisectors, reflexive property, and the Third Angles Theorem in this detailed practice worksheet. Suitable for Grades 9-11.

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