What you'll learn
This topic covers the properties of 2D shapes — triangles, quadrilaterals and other polygons. In this guide you will learn how to classify triangles and quadrilaterals, the properties of their sides, angles and diagonals, the meaning of symmetry, and how to use these properties to solve problems. This is core geometry knowledge that supports angle and construction work.
Key terms and definitions
Polygon — a closed shape with straight sides.
Quadrilateral — a four-sided polygon.
Parallel — lines that stay the same distance apart and never meet.
Line of symmetry — a line that divides a shape into two mirror-image halves.
Rotational symmetry — when a shape looks the same after a turn of less than 360°.
Core concepts
Types of triangle
Triangles are classified by sides and angles: equilateral (all sides and angles equal, 60° each), isosceles (two equal sides and two equal base angles), scalene (all different), and right-angled (one 90° angle). The angles of any triangle add to 180°.
Types of quadrilateral
Key quadrilaterals: the square (all sides equal, all 90°), rectangle (opposite sides equal, all 90°), parallelogram (opposite sides parallel and equal), rhombus (all sides equal, opposite sides parallel), trapezium (one pair of parallel sides) and kite (two pairs of adjacent equal sides). The angles of any quadrilateral add to 360°.
Diagonals and their properties
Diagonals help identify shapes: a square's diagonals are equal and bisect at right angles; a rectangle's are equal but not perpendicular; a rhombus's bisect at right angles but are unequal; a kite's cross at right angles with one bisecting the other. These facts often unlock problems.
Symmetry
A line of symmetry divides a shape into mirror-image halves — a square has 4, a rectangle 2, an equilateral triangle 3. Rotational symmetry order is how many times a shape matches itself in a full turn — a square has order 4, a parallelogram order 2.
Using properties to solve problems
To find missing angles or lengths, use the known properties — equal sides, equal angles, parallel lines and angle sums. For example, in an isosceles triangle the two base angles are equal, so knowing the apex angle lets you find them.
Worked examples
Example 1: Isosceles triangle
An isosceles triangle has an apex angle of 40°. Find each base angle.
Base angles are equal: (180 − 40) ÷ 2 = 70° each.
Example 2: Quadrilateral angles
Three angles of a quadrilateral are 90°, 100° and 80°. Find the fourth.
360 − (90 + 100 + 80) = 90°.
Example 3: Symmetry
State the order of rotational symmetry of a rectangle.
A rectangle maps onto itself twice in a turn, so order 2.
Common mistakes and how to avoid them
Wrong angle sum. Triangles add to 180°, quadrilaterals to 360°.
Confusing rhombus and square. A rhombus has equal sides but angles need not be 90°.
Forgetting isosceles base angles are equal. Use this to find missing angles.
Miscounting symmetry. Check lines and rotations carefully.
Assuming a trapezium has two pairs of parallel sides. It has only one.
Exam technique for Properties of 2D Shapes
Classify the shape by its sides and angles first.
Recall the angle sum — 180° for triangles, 360° for quadrilaterals.
Use diagonal properties to identify or solve shapes.
Apply symmetry when describing or completing shapes.
Use equal sides and angles to find missing values.
Quick revision summary
Triangles are classified as equilateral (60° each), isosceles (two equal sides and equal base angles), scalene or right-angled, and their angles add to 180°. Quadrilaterals include the square, rectangle, parallelogram, rhombus, trapezium (one parallel pair) and kite, with angles adding to 360°. Diagonals distinguish shapes — a square's are equal and perpendicular, a rhombus's are perpendicular but unequal, a kite's cross at right angles. Symmetry is measured by lines of symmetry (a square has 4) and rotational symmetry order (a square has order 4, a parallelogram order 2). Solve problems by using the properties — equal sides, equal angles, parallel lines and angle sums (isosceles base angles are equal). The common errors are using the wrong angle sum, confusing a rhombus with a square, forgetting equal base angles, and miscounting symmetry. Classify the shape, recall the angle sum, use diagonal and symmetry properties, and apply equal sides and angles to find missing values.