Use the intercept Theorem to prove that the converse of the Mid-point Theorem

Answer

To prove the converse of the Mid-point Theorem using the Intercept Theorem, we will follow these steps:

### Step-by-Step Solution

1. **Given Information**:
- Let \( P \) be the midpoint of segment \( XY \) in triangle \( XYZ \).
- It is given that line segment \( PQ \) is parallel to line segment \( YZ \).

2. **Construction**:
- We will draw a line \( MN \) through point \( X \) such that \( MN \) is parallel to \( YZ \).

3. **Establishing Parallel Lines**:
- Since \( PQ \) is parallel to \( YZ \) (given), and \( MN \) is also drawn parallel to \( YZ \), we can conclude that \( PQ \) is parallel to \( MN \).

4. **Using the Transversal**:
- The line segment \( XZ \) acts as a transversal to the parallel lines \( PQ \) and \( MN \).

5. **Applying the Intercept Theorem**:
- According to the Intercept Theorem, if a transversal intersects two parallel lines, then it divides the segments on the transversal proportionally.
- Therefore, the segments \( XQ \) and \( QZ \) on the transversal \( XZ \) will be equal, i.e., \( XQ = QZ \).

6. **Conclusion**:
- Since we have shown that \( XQ = QZ \), we have proved that if \( P \) is the midpoint of \( XY \) and \( PQ \) is parallel to \( YZ \), then \( XQ \) is equal to \( QZ \). This concludes the proof of the converse of the Mid-point Theorem.

### Summary of Steps
1. Identify given information and what needs to be proved.
2. Construct a parallel line through a point.
3. Establish relationships between parallel lines.
4. Use a transversal to apply the Intercept Theorem.
5. Conclude the proof based on equal segments.