Square

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.

• All four sides are the same length.

  AB = BC = CD = DA, each 62 units here.

• 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, cross at right angles, and cut each other in half.

  AC = 87.7 and BD = 87.7, meeting at 90° at the midpoint of both.

  Why do a square's diagonals do two things at once?

  1. Because a square inherits from both parents, and each gives it one of the two.

  2. From the rectangle: four right angles make the two diagonals the same length.

  3. From the rhombus: four equal sides make the two diagonals cross at right angles.

  4. A square has four equal sides and four right angles, so it gets both results at once — diagonals that are equal in length and meet at 90°. No other quadrilateral manages both.

02 Why it works

A square is the end of the family. It passes the rhombus test (four equal sides) and the rectangle test (four right angles), so it is both — not something separate that resembles them.

That is worth saying plainly because papers ask it directly. Is a square a rectangle? Yes. Is a rectangle a square? Only sometimes. The relationship runs one way.

Turning it changes nothing. A square balanced on one corner is still a square, even though it looks like the diamond a rhombus is usually drawn as.

03 Where it sits

See the whole family on Quadrilaterals.