Quadrilaterals: Thinking Puzzles

Logic puzzles on quadrilaterals: which shapes always have a property, colouring puzzles with clues, counting rectangles and mirror-image boards.

Key ideas

A square is also a rectangle, a rhombus and a parallelogram, so it has all their properties.
Rectangles and squares have four right angles and equal diagonals; rhombuses and squares have four equal sides.
The diagonals of every parallelogram (square, rectangle, rhombus) bisect each other; those of a square, rhombus and kite cross at right angles.
In a clue puzzle, use the definite clues first and cross out what each clue rules out.
To make one board the mirror image of another, flip the first board left to right and count the cells that differ.

Remember

Lines of symmetry	square 4, rectangle 2, rhombus 2, kite 1, parallelogram 0
Angle sum of a quadrilateral	360°
Angles in ratio 1 : 2 : 3 : 4	360° ÷ 10 = 36°; largest 144°
Parallelogram	opposite angles equal; neighbouring angles add to 180°
Rectangles in an m × n grid	(m + 1)m ÷ 2 × (n + 1)n ÷ 2
Squares in a 3 × 4 grid	12 + 6 + 2 = 20

Watch out

Forgetting that a square counts as a rectangle and as a rhombus.
Giving a parallelogram a line of symmetry: it has none (unless it is a rectangle or rhombus).
Counting only the 1 × 1 squares in a grid.

Worked example

Which shapes always have both at least one line of symmetry and diagonals that cross at right angles? Choose from square, rectangle, rhombus, parallelogram, kite, and write them in that order. (A kite here is not a rhombus.)

Working: at least one line of symmetry: square, rectangle, rhombus, kite; diagonals that cross at right angles: square, rhombus, kite

Answer: square, rhombus, kite