In trapezium ABCD, AB is parallel to DC. P and Q are the mid-points of AD and BC respectively. BP produced meets CD produced at point E. Prove that :
PQ is parallel to AB.

Answer

To prove that line segment PQ is parallel to line segment AB in trapezium ABCD, where AB is parallel to DC, and P and Q are the midpoints of sides AD and BC respectively, we can follow these steps:

### Step-by-Step Solution

1. **Draw the trapezium ABCD**: Start by sketching trapezium ABCD with AB parallel to DC. Mark points P and Q as the midpoints of sides AD and BC respectively.

2. **Extend lines BP and CD**: Extend line segment BP and line segment CD until they meet at point E.

3. **Identify triangles**: We will focus on triangles PED and PAB.

4. **Use the midpoint property**: Since P is the midpoint of AD, we have:
\[ PD = AP \]

5. **Identify angles**: Note that angle EPD is equal to angle APB because they are vertically opposite angles.

6. **Use parallel lines**: Since AB is parallel to CD, the angle PED is equal to angle PBA (alternate interior angles).

7. **Apply the Angle-Side-Angle (ASA) criterion**: With PD = AP, angle EPD = angle APB, and angle PED = angle PBA, we can conclude that:
\[ \triangle PED \cong \triangle PAB \]
by the ASA criterion.

8. **Conclude equal sides**: From the congruence of triangles, we have:
\[ PE = PB \]
This means E is the midpoint of segment BE.

9. **Consider triangle ECB**: In triangle ECB, since P is the midpoint of EB and Q is the midpoint of CB, we can conclude that:
\[ PQ \text{ is the line segment joining the midpoints of sides EB and CB.} \]

10. **Use the midpoint theorem**: According to the midpoint theorem, the line segment joining the midpoints of two sides of a triangle is parallel to the third side. Therefore:
\[ PQ \parallel CE \]

11. **Relate CE to AB**: Since AB is parallel to CD, and CE is an extension of CD, we have:
\[ AB \parallel CE \]

12. **Conclude that PQ is parallel to AB**: Since both PQ and AB are parallel to CE, we can conclude:
\[ PQ \parallel AB \]

### Final Conclusion
Thus, we have proven that line segment PQ is parallel to line segment AB.