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To prove that the line segment joining the midpoints of the two non-parallel sides of a trapezium bisects its diagonals, let’s proceed with the given trapezium \(ABCD\), where \(AB\) and \(CD\) are parallel sides, and \(P\) and \(Q\) are the midpoints of the non-parallel sides \(AD\) and \(BC\), respectively.

### Proof Using the Midpoint Theorem:

1. **Apply the Midpoint Theorem**:
   Since \(P\) and \(Q\) are midpoints of \(AD\) and \(BC\), respectively, the line segment \(PQ\) will be parallel to \(AB\) and \(CD\) (by the Midpoint Theorem). Furthermore, \(PQ\) will be half the sum of \(AB\) and \(CD\) because it is the line joining the midpoints of the two non-parallel sides of a trapezium.

2. **Diagonals and Intersection Point**:
   Let the diagonals \(AC\) and \(BD\) intersect at point \(O\). We need to show that \(O\) is the midpoint of both diagonals \(AC\) and \(BD\).

3. **Using Similar Triangles**:
   Since \(PQ\) is parallel to both \(AB\) and \(CD\), triangles \(APQ\) and \(CDB\) are similar by AA similarity (corresponding angles are equal because of parallel lines).
   
   - This similarity implies that the line \(PQ\), which joins the midpoints of \(AD\) and \(BC\), divides the diagonals \(AC\) and \(BD\) proportionally.
   - Consequently, \(O\) must be the midpoint of both diagonals because it lies on the line \(PQ\), which is exactly halfway between the parallel lines \(AB\) and \(CD\).

4. **Conclusion**:
   Hence, the line segment \(PQ\), which joins the midpoints of the non-parallel sides of trapezium \(ABCD\), bisects the diagonals \(AC\) and \(BD\) at point \(O\).

This completes the proof.

Would you like further details on any specific step, or do you have additional questions?

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### Relative Questions to Explore Further:
1. How does the Midpoint Theorem apply to other quadrilaterals?
2. Why is point \(O\) guaranteed to be the midpoint of the diagonals in this trapezium?
3. What are some real-world applications of properties of trapeziums?
4. How would this proof change if the trapezium was isosceles?
5. Can this result be generalized to other types of quadrilaterals?

### Tip:
In geometry, understanding the relationships between midpoints and parallel lines often simplifies complex proofs involving trapeziums and other quadrilaterals.

Prove that the line segment joining the midpoints of two non-parallel sides of a trapezium bisects its diagonals.

Learn how to prove that the line segment joining the midpoints of two non-parallel sides of a trapezium bisects its diagonals. This problem involves the midpoint theorem and properties of similar triangles, ideal for Grade 9 geometry students.

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