Derive and apply the properties and definitions of: special types of…

The idea

This topic requires you to derive and apply the defining properties of special quadrilaterals, triangles, and other plane figures using correct geometric language. A square has four equal sides and four right angles; a rectangle has four right angles with opposite sides equal; a parallelogram has two pairs of parallel sides; a trapezium has one pair of parallel sides; a kite has two pairs of adjacent equal sides; a rhombus has four equal sides. For triangles, classify by sides and angles. 'Derive' means reasoning from facts, like showing a rhombus has opposite angles equal using parallel lines. 'Apply' means using properties to find missing angles. 'Other plane figures' includes polygons. Appropriate language includes terms like parallel, perpendicular, bisect, and adjacent.

What earns the marks

• A square is a quadrilateral with four equal sides and four right angles, so it is also a rectangle, a rhombus and a parallelogram.
• A rectangle has four right angles and opposite sides equal and parallel; its diagonals are equal in length.
• A parallelogram has two pairs of parallel sides, opposite sides equal, opposite angles equal, and diagonals that bisect each other.
• A trapezium has exactly one pair of parallel sides; an isosceles trapezium has equal base angles and equal diagonals.
• A kite has two pairs of adjacent equal sides; one pair of opposite angles is equal, and one diagonal is a line of symmetry.
• A rhombus has four equal sides, opposite sides parallel, opposite angles equal, and diagonals that bisect each other at right angles.
• Triangles are classified by sides as equilateral, isosceles or scalene, and by angles as acute, right or obtuse; the angle sum is 180°.
• Other plane figures include regular and irregular polygons; a regular polygon has equal sides and equal interior angles.

Common mistakes and how to put them right

• Thinking a rhombus must have right angles.

  Put it right: A rhombus has four equal sides but its angles need not be 90°; a square is the special rhombus with right angles.

• Confusing a trapezium with a parallelogram.

  Put it right: A trapezium has exactly one pair of parallel sides, while a parallelogram has two pairs.

• Believing a kite always has all four sides equal.

  Put it right: A kite has two pairs of adjacent equal sides, which may be of different lengths.

• Using 'diamond' instead of 'rhombus' or 'kite'.

  Put it right: Use the precise terms rhombus and kite as defined in the specification.

Exam tips

• When a question asks you to derive a property, show each step: state the given shape, quote the relevant definition, then use angle or side facts to reach the result.
• Learn the minimum defining properties for each shape, then deduce the rest; for example, from a square's four equal sides and right angles you can deduce parallel sides.
• In multiple-choice questions, eliminate options that describe a different shape or use imprecise language such as 'slanted rectangle' instead of parallelogram.
• Check whether the question says 'exactly one pair of parallel sides' for a trapezium; if it says 'at least one pair', that would include parallelograms, so read carefully.