Let $AD$ and $BC$ be the parallel sides of a trapezium $ABCD$. Let $P$ and $Q$ be the midpoints of the diagonals $AC$ and $BD$. If $AD = 16$ and $BC = 20$, what is the length of $PQ$?
1. **Identify the given elements and draw the necessary lines:**
   - We are given a trapezium \(ABCD\) with \(AD\) and \(BC\) as the parallel sides.
   - Let \(P\) and \(Q\) be the midpoints of the diagonals \(AC\) and \(BD\) respectively.
   - Given \(AD = 16\) and \(BC = 20\).

2. **Draw line \(EF\) parallel to \(AD\) and \(BC\):**
   - Draw line \(EF\) such that it is parallel to \(AD\) and \(BC\) and intersects \(P\) and \(Q\) (where \(E\) is on \(AB\) and \(F\) is on \(DC\)).

3. **Use the midpoint theorem:**
   - By the midpoint theorem, if a line segment joins the midpoints of two sides of a triangle, it is parallel to the third side and half as long.
   - Since \(P\) and \(Q\) are midpoints of \(AC\) and \(BD\) respectively, \(PQ\) is parallel to \(AD\) and \(BC\).

4. **Calculate the lengths of \(EP\) and \(EQ\):**
   - Since \(E\) and \(F\) are midpoints, \(EP\) and \(EQ\) are half the lengths of \(AD\) and \(BC\) respectively.
   - Therefore, \(EP = \frac{AD}{2} = \frac{16}{2} = 8\).
   - Similarly, \(EQ = \frac{BC}{2} = \frac{20}{2} = 10\).

5. **Determine the length of \(PQ\):**
   - Since \(PQ\) is the difference between \(EQ\) and \(EP\), we have:
     \[
     PQ = EQ - EP = 10 - 8 = 2
     \]

The final answer is \(\boxed{2}\).