Trapezium

The idea

A trapezium is a quadrilateral with at least one pair of parallel opposite sides; the two angles at the ends of each of the other sides add up to 180°.

Only one pair of parallel sides

A parallelogram has two pairs of parallel sides. Now take a quadrilateral PQRS in which only PQ ∥ SR. The other two sides, PS and QR, lean in any way they like.

What can you say about ∠P and ∠S, the angles at the two ends of the side PS?

• They are equal
• They add up to 180°
• Nothing: they can be anything

Between two rails

PQ and SR lie on two parallel rails. Drag the purple dots to make different shapes. The numbers add up the angles in pairs. Which pairs always add up to the same total?

The leaning sides

A trapezium is a quadrilateral with at least one pair of parallel opposite sides; the two angles at the ends of each of the other sides add up to 180°.

In every shape, ∠P + ∠S and ∠Q + ∠R came to 180°. The pairs at the ends of the parallel sides, ∠P + ∠Q and ∠S + ∠R, changed from shape to shape.

Why? The side PS cuts across the two parallel lines PQ and SR. ∠P and ∠S are interior angles on the same side of PS, and those add up to 180°. The side QR does the same for ∠Q and ∠R.

Two pairs of 180° make 360°, the total of the angles of any quadrilateral.

If PS and QR are parallel as well, the shape is a parallelogram. It has at least one pair of parallel sides, so it is a trapezium too.

Notes

A trapezium is a quadrilateral with at least one pair of parallel opposite sides; the two angles at the ends of each of the other sides add up to 180°.

Check yourself

In trapezium PQRS, PQ ∥ SR and ∠P = 70°. How large is ∠S?

Answer: 110 °

PQ ∥ SR, so the angles at the ends of PS add up to 180°: ∠S = 180° − 70° = 110°.

A parallelogram has two pairs of parallel sides. A trapezium needs at least one pair. Is a parallelogram also a trapezium?

In trapezium PQRS, PQ ∥ SR, ∠P = 65° and ∠Q = 80°. How large is ∠S?

Answer: 115 °

∠S is at the other end of PS from ∠P, so ∠S = 180° − 65° = 115°.

In trapezium PQRS with PQ ∥ SR, which pair of angles must add up to 180°?

• Yes, it has at least one pair of parallel sides — correct. Yes! Two pairs is more than "at least one", so every parallelogram is a trapezium too.
• No, a trapezium has exactly one pair. The rule says "at least one pair", not "exactly one". A shape with two pairs also has one.
• Only a rectangle is a trapezium. Any quadrilateral with at least one pair of parallel sides is a trapezium, not only rectangles.
• ∠P and ∠S — correct. Yes! They are at the two ends of the side PS, and PQ ∥ SR.
• ∠P and ∠Q. They are at the two ends of the parallel side PQ. Their total changes from shape to shape.
• ∠P and ∠R. They are opposite corners. There is no rule that makes them add up to 180°.