Question: Select all of the statements that are always true for a trapezium.
Exactly one diagonal bisects a pair of opposite angles.
The diagonals bisect each other at right angles.
The opposite sides are equal.
The opposite sides are parallel.
None of these.
1. **State the problem:** We need to determine which statements are always true for a trapezium.
2. **Recall the definition and properties of a trapezium:**
- A trapezium (in American English, trapezoid) is a quadrilateral with exactly one pair of opposite sides parallel.
- The other pair of sides is not parallel.
- The diagonals generally do not bisect each other.
- Opposite sides are not necessarily equal.
- Diagonals do not necessarily bisect opposite angles or intersect at right angles.
3. **Analyze each statement:**
- "Exactly one diagonal bisects a pair of opposite angles": This is not always true for trapeziums. It can be true in special cases (like isosceles trapeziums) but not always.
- "The diagonals bisect each other at right angles": This is true for rhombuses and squares but not for trapeziums in general.
- "The opposite sides are equal": This is true for parallelograms but not for trapeziums.
- "The opposite sides are parallel": In trapeziums, only one pair of opposite sides is parallel, not both.
- "None of these": Since none of the above statements are always true for trapeziums, this is the correct choice.
**Final answer:** None of these.