Quadrilaterals

The angle sum of a quadrilateral, and the sides, angles and diagonals of parallelograms, rectangles, rhombuses, squares, trapeziums and kites.

Key ideas

The angles of any quadrilateral add up to 360°.
Parallelogram: opposite sides parallel and equal, opposite angles equal, neighbouring angles add to 180°, diagonals bisect each other.
Rectangle: a parallelogram with right angles; its diagonals are equal.
Rhombus: all sides equal; its diagonals bisect each other at right angles and bisect its angles.
Square: a rectangle and a rhombus at once; its diagonals are equal, perpendicular and make 45° with the sides.
Trapezium: exactly one pair of parallel sides. Kite: two pairs of equal neighbouring sides, diagonals at right angles.

Angle sum	∠A + ∠B + ∠C + ∠D = 360°
Parallelogram	∠A = ∠C, ∠B = ∠D, ∠A + ∠B = 180°
Rhombus diagonals	half-diagonals and a side form a right triangle

Saying a rhombus has equal diagonals (only a square does).
Taking all angles of a parallelogram as 90°.
Calling every quadrilateral with perpendicular diagonals a rhombus (a kite has them too).

In square PQRS of side 10 cm, the diagonals meet at O. Find ∠POQ, ∠OPQ and the perimeter.

Working: the diagonals of a square are perpendicular and bisect its 90° angles

Answer: 90°, 45°, 40 cm