Two line segments PQ and RS intersect at X in such a way that `XP = XR`. If `/_PSX = /_RQX`, then one must have

Options

A. `PR = QS `
B. `PS = RQ`
C. `/_XSQ =/_XRP`
D. `ar(DeltaPXR) = ar(DeltaQXS)`

Answer

To solve the problem, we need to analyze the given information about the line segments PQ and RS that intersect at point X, with the conditions that \( XP = XR \) and \( \angle PSX = \angle RQX \).

### Step-by-Step Solution:

1. **Identify the Given Information:**
- We have two line segments PQ and RS intersecting at point X.
- The lengths of the segments are equal: \( XP = XR \).
- The angles formed at point X are equal: \( \angle PSX = \angle RQX \).

2. **Draw the Diagram:**
- Draw two intersecting lines, one representing line segment PQ and the other representing line segment RS.
- Mark the intersection point as X.
- Label the points as follows: P, Q on line PQ and R, S on line RS.

3. **Apply the Isosceles Triangle Property:**
- Since \( XP = XR \), triangle XPR is isosceles with \( XP = XR \).
- This means that the angles opposite to these equal sides are also equal. Therefore, \( \angle XPR = \angle XRP \).

4. **Use the Given Angle Condition:**
- We know that \( \angle PSX = \angle RQX \).
- By the property of angles in a triangle, we can say that if two angles are equal, the sides opposite those angles are also equal.

5. **Conclude the Relationship:**
- Since \( \angle PSX = \angle RQX \) and \( XP = XR \), we can conclude that triangle XPS is similar to triangle XQR.
- This leads us to the conclusion that the lengths of the segments PS and QR must also be equal.

6. **Final Statement:**
- Therefore, we conclude that \( PS = QR \).

### Conclusion:
The relationship derived from the given conditions is that \( PS = QR \).