Rectangle

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 (55)

• 2 pairs of sides are parallel.

  AB is parallel to CD, BC is parallel to DA

• All four angles are right angles, 90°.

  ∠DAB = ∠ABC = ∠BCD = ∠CDA = 90°.

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

  ∠DAB + ∠ABC = 90° + 90° = 180°, ∠ABC + ∠BCD = 90° + 90° = 180°

• All 4 angles add up to 360°.

  True of every quadrilateral, however odd it looks.

• The diagonals are equal and cut each other in half.

  AC = BD = 109.8, and they cross at the midpoint of both.

  Why are the two diagonals exactly the same length?

  1. Draw AC. It cuts the rectangle into two triangles, and because all four corners are right angles, each one has the long side and the short side of the rectangle for its two shorter sides.

  2. Now draw the other diagonal, BD. It does exactly the same thing — its triangles are built from the same long side and the same short side, meeting at the same right angle.

  3. Two triangles built from the same two lengths meeting at the same angle are identical. So the sides opposite that angle — AC and BD — have to match.

  4. This is what a leaning parallelogram loses. Tip a rectangle over and one diagonal stretches while the other shortens, because the corners stop being right angles. The equal diagonals come from the 90°, not from the parallel sides.

02 Why it works

A rectangle is a parallelogram first, so it keeps everything a parallelogram has. The four right angles are the extra.

Because every angle is 90°, the neighbouring-angles rule stops being useful here — 90° + 90° = 180° is true but tells you nothing you did not already know. The property worth remembering is the last row: the diagonals are equal, which a leaning parallelogram's are not.

A rectangle does not have to be long and thin. Make its sides equal and it is a square, which is still a rectangle.

03 Where it sits

See the whole family on Quadrilaterals.