The rhombus

Are squares the only quadrilaterals with all four sides equal? Find out by drawing. Draw two equal sides AD and AB meeting at, say, 50° (on purpose not a right angle). Then set your compasses to that same length and draw arcs from B and from D; where they cross is C.

You now have a quadrilateral with four equal sides and an angle of 50°, and it's clearly not a square. Any angle less than 180° would have worked. A quadrilateral with all sides the same length is a rhombus.

Deduction 9: the angles. In rhombus GAME, draw the diagonal AE.

• In ∆GAE, GE = GA, so its base angles are equal: a = d.
• In ∆MAE, ME = MA, so b = c.
• And ∆GAE ≅ ∆MAE (SSS), so a = b, c = d and ∠G = ∠M.

Putting it together, a = b = c = d: the diagonal makes four equal angles with the sides.

Apply that to our rhombus ABCD with its 50° angle. In ∆ADB, a + a + 50 = 180, so a = 65°. The angles of ABCD are then 50°, 130°, 50°, 130°: opposite angles equal.

There's a neater route to the same place. Those four equal angles at the diagonal are alternate angles for two pairs of lines, and equal alternate angles mean parallel lines. So EM ∥ GA and GE ∥ AM.

So a rhombus has parallel opposite sides: every rhombus is a parallelogram, and every parallelogram property comes with it free. Neighbouring angles add to 180°; opposite angles are equal; the diagonals bisect each other.

Where does the square fit? A square is a rectangle; its opposite sides are parallel, so it's a parallelogram; and all its sides are equal, so it's a rhombus too. In the Venn diagram the squares are exactly the overlap of the rectangles and the rhombuses, with both inside the parallelograms.

Deduction 10: the angle between the diagonals. In rhombus GAME, ∆GEO ≅ ∆MEO, so ∠GOE = ∠MOE. They make a straight angle, so they add to 180°, and each must be 90°.

The six properties of a rhombus:

• all sides equal;
• opposite sides parallel;
• neighbouring angles add to 180°, opposite angles equal;
• diagonals bisect each other;
• diagonals bisect the angles;
• diagonals meet at 90° (but, unlike a square's, they need not be equal).