Parallelogram

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.

• Both pairs of opposite sides are equal.

  AB = CD (95), BC = DA (57.7)

• 2 pairs of sides are parallel.

  AB is parallel to CD, BC is parallel to DA

• Opposite angles are equal.

  ∠DAB = ∠BCD (64.3°), ∠ABC = ∠CDA (115.7°)

• Any two angles next to each other add up to 180°.

  ∠DAB + ∠ABC = 64.3° + 115.7° = 180°, ∠ABC + ∠BCD = 115.7° + 64.3° = 180°

  Why do they always add up to 180°?

  1. Take the two corners next to each other, ∠DAB and ∠ABC.

  2. Cut them off and put them down side by side. They fill a straight line exactly — and a straight line is 180°.

  3. That is what happens. Here is why it has to. Opposite angles are equal, so going round the shape the four angles are a, b, a, b.

  4. Every quadrilateral's angles add to 360°, so a + b + a + b = 360°. That is 2a + 2b = 360°, and halving both sides gives a + b = 180°.

  5. Both facts it rests on are in the table above. Nothing new was needed — the 180° is what those two together force.

• All 4 angles add up to 360°.

  True of every quadrilateral, however odd it looks.

• The diagonals cut each other in half.

  AC = 130.8 and BD = 87.2 — different lengths, but they cross at the midpoint of both.

02 Why it works

Everything in the table follows from the one condition. Once both pairs of sides are parallel, the opposite sides have to be equal, the opposite angles have to match, and each pair of neighbours has to add to 180° — none of it is a separate rule to memorise.

That last property is the one papers lean on hardest: give a child one angle of a parallelogram and every other angle follows. If ∠DAB is 64°, then ∠ABC is 180° − 64° = 116°, ∠BCD matches ∠DAB, and ∠CDA matches ∠ABC.

A parallelogram is not required to have equal sides or right angles. Add four equal sides and it becomes a rhombus; add four right angles and it becomes a rectangle; add both and it is a square. All three are still parallelograms.

03 Where it sits

See the whole family on Quadrilaterals.