Quadrilateral

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 = 92.2, BC = 50.6, CD = 62.8, DA = 46.2

• All 4 angles add up to 360°.

  True of every quadrilateral, however odd it looks.

  Why is it always 360°?

  1. Draw one diagonal, from A across to C. It cuts the quadrilateral into two triangles and adds nothing else.

  2. The angles of any triangle add up to 180°. There are two triangles here, so together their angles come to 180° + 180° = 360°.

  3. Every one of those angles is part of a corner of the quadrilateral, and no corner is left out — ∠DAB is split between the two triangles, ∠BCD likewise, and ∠ABC and ∠CDA each sit whole inside one of them.

  4. So the quadrilateral's four angles add to exactly what the two triangles add to: 360°. Nothing about the shape was assumed, which is why it holds for every quadrilateral, however lopsided.

02 Why it works

There is only one thing every quadrilateral has in common besides its four sides, and it is the row above: the angles always add up to 360°. No matter how lopsided the shape, that total never moves.

Everything else you may have learned — equal sides, parallel sides, right angles, equal diagonals — belongs to a particular kind of quadrilateral, not to all of them. The shape drawn above has none of them.

This is why a paper that says only “ABCD is a quadrilateral” is telling you almost nothing, and why the first move in such a question is nearly always the 360°.

03 Where it sits

See the whole family on Quadrilaterals.