https://file.aiquickdraw.com/mathbot/file/1727447572243b1omk1sa.jpg

Let's analyze the problem step by step.

We are given a quadrilateral **PQRS** with points **T** on **PS** and **U** on **RS**. The conditions are as follows:
- **PQ = RQ** (which implies that **PQRS** is symmetric along **QS**),
- \(\angle PQT = \angle RQU\),
- \(\angle TQS = \angle UQS\).

We need to **prove that \( QT = QU \)**.

### Step-by-step solution:

1. **Congruent triangles**: 
   We begin by focusing on the triangles **PQT** and **RQU**. For these triangles, we are given:
   - **PQ = RQ** (from the problem),
   - \(\angle PQT = \angle RQU\) (from the problem),
   - \(\angle TQS = \angle UQS\) (from the problem).
   
   This suggests that the two triangles might be congruent, as they have two pairs of equal angles and one side equal (**PQ = RQ**).

2. **Angle-Side-Angle (ASA) congruence**:
   By the **ASA (Angle-Side-Angle)** criterion for congruence:
   - The angle \(\angle PQT\) is equal to \(\angle RQU\),
   - The side **PQ** is equal to **RQ**,
   - The angle \(\angle TQS\) is equal to \(\angle UQS\).
   
   Hence, by ASA, the triangles **PQT** and **RQU** are **congruent**.

3. **Conclusion**:
   Since the triangles are congruent, their corresponding sides must be equal. Therefore, **QT = QU**.

Thus, we have proven that **QT = QU** as required.

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Would you like further details on any of the steps? Here are 5 related questions to expand on this topic:

1. What is the ASA criterion for congruence, and how does it work in geometry?
2. How can we apply symmetry in quadrilaterals to prove properties like equal lengths?
3. Can the SAS (Side-Angle-Side) congruence rule be applied here, and if so, how?
4. What other properties can we deduce about quadrilateral PQRS based on the given conditions?
5. How can this problem be generalized for other polygons or geometric figures?

### Tip:
When proving congruence between triangles, remember to check if one of the common congruence criteria (SSS, SAS, ASA, AAS, or RHS) applies directly.

In the figure PQRS is a quadrilateral and T and U are respectively points on PS and RS such that PQ = RQ, ∠PQT = ∠RQU, and ∠TQS = ∠UQS. Prove that QT = QU.

Learn how to prove that QT = QU in a quadrilateral PQRS using the ASA congruence theorem. This problem involves understanding congruent triangles and geometric reasoning. Suitable for high school students in Grades 9-12.

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