Trapezium

01 What is always true

∠DAB means the angle at A — the middle letter is always the corner, and the outer two say which sides form it.

• No two sides have to be equal.

  AB = 110, BC = 61, CD = 51.7, DA = 55.2

• Exactly one pair of sides is parallel.

  AB is parallel to CD

• The two angles at each end of a side joining the parallels add up to 180°.

  ∠ABC + ∠BCD = 55° + 125° = 180°, ∠CDA + ∠DAB = 115° + 65° = 180°

  Why do those two add to 180° when the others do not?

  1. Look at side AD. It runs from the bottom parallel line up to the top one, crossing both.

  2. Cut off ∠DAB and ∠CDA, the angles at its two ends, and set them down together. They fill a straight line.

  3. It happens because AB and DC never get closer or further apart. Sliding the corner at A straight up the side until it lands on D turns the angle to face the other way, and the two together close a half turn.

  4. BC does the same thing on the other side. But AB and DC are the parallel pair — they never meet, and there is no crossing side to make their angles fit together, which is why the other two corner-pairs do not add to 180°.

• All 4 angles add up to 360°.

  True of every quadrilateral, however odd it looks.

02 Why it works

The drawing above is deliberately lopsided. Textbooks usually print the tidy symmetrical version, and children come away thinking a trapezium must have two equal slanting sides — it must not. No two sides have to be equal, and no two angles have to be equal. One pair of parallel sides is the entire definition.

The symmetrical one has a name of its own: an isosceles trapezium, where the two slanting sides are equal and so are the angles at each end of a parallel side. It is a trapezium, but it is a special one, and a question about trapeziums in general cannot assume it.

Exactly one pair matters. The moment a second pair is parallel the shape is a parallelogram and stops being called a trapezium.

03 Where it sits

See the whole family on Quadrilaterals.