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In this image, you have a question about proving triangles congruent, involving segment QS bisecting angle TQR. Here's how the paragraph proof is structured and how you can complete the missing parts:

1. **"We know that segment QS bisects angle TQR because it is given."**  
   This part states that QS bisects the angle, which is directly provided in the problem.

2. **"By the definition of angle bisector, angle TQS is congruent to angle ___."**  
   Since QS is the bisector, the two angles formed, TQS and SQR, should be congruent. The correct answer for this blank is **SQR**.

3. **"We see that segment QS is congruent to segment ___ by ___."**  
   Since QS is the common side between triangles QRS and TQS, it is congruent to itself by the **reflexive property**. Thus, the missing terms are **QS** and **reflexive property**.

4. **"Therefore, we can conclude that triangles QRS and ___ are congruent by ___."**  
   Since the two triangles in question are QRS and TQS, we conclude that **QRS and TQS** are congruent by the **ASA (Angle-Side-Angle) criterion**.

Would you like further clarification on any part of this proof?

Here are 5 related questions that expand on this topic:
1. What is the Angle-Side-Angle (ASA) criterion for triangle congruence?
2. How does the reflexive property apply to triangles sharing a side?
3. What other triangle congruence theorems exist (SSS, SAS, etc.), and when are they used?
4. What role does an angle bisector play in proving triangle congruence?
5. How can you differentiate between the ASA and AAS (Angle-Angle-Side) criteria?

**Tip:** Always mark or highlight the shared sides or angles when working on geometric proofs to simplify the process of identifying congruent parts.

Complete the missing parts of the paragraph proof for proving triangles congruent, involving segment QS bisecting angle TQR.

Explore how to prove triangles congruent using segment QS, which bisects angle TQR. This proof involves the Angle-Side-Angle (ASA) criterion and includes the use of the reflexive property. Suitable for Grades 8-10, the problem covers triangle congruence and angle bisectors in geometric proofs.

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