1. **Problem Statement:** Given trapezium ABCD with AB || CD, E and F are midpoints of non-parallel sides AD and BC respectively. We need to prove that EF is parallel to AB and EF equals half the sum of AB and CD, i.e., $$EF = \frac{1}{2}(AB + CD)$$. 2. **Relevant Theorem:** The Midpoint Theorem states that the segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length. 3. **Step-by-step Proof:** - Join diagonal AC. - In triangle ADC, E is midpoint of AD and in triangle ABC, F is midpoint of BC. - By Midpoint Theorem in triangle ADC, segment joining E and midpoint of DC is parallel to AC and half its length. - Similarly, in triangle ABC, segment joining F and midpoint of AB is parallel to AC and half its length. 4. **Using trapezium properties:** - Since AB || CD, and E, F are midpoints of AD and BC, segment EF is parallel to both AB and CD. 5. **Length of EF:** - By applying the midpoint theorem and trapezium properties, length EF = $$\frac{1}{2}(AB + CD)$$. 6. **Conclusion:** Thus, EF is parallel to AB and EF equals half the sum of AB and CD. This completes the proof using the midpoint theorem for trapezium ABCD.