Properties of 2D Shapes — AQA GCSE Maths Revision Notes
What you'll learn
This topic covers the named triangles and quadrilaterals and the properties that define them. By the end of this guide you should be able to classify a triangle or quadrilateral, state its properties, and use them to find missing angles.
You should also be able to describe the diagonals of each quadrilateral, count lines of symmetry and rotational symmetry, recognise how the quadrilaterals relate to one another, and give proper reasons in an angle problem.
The organising idea is that each shape is defined by its properties, not by how it looks. A square is not "a shape that looks square" but any quadrilateral with four equal sides and four right angles — which is why a square is also a rectangle, also a rhombus, and also a parallelogram, all at once. Working from the defining properties rather than the picture is what lets you answer "is this always, sometimes or never true?" questions, and it is also what stops you assuming a triangle is isosceles just because it looks it on a diagram that is not to scale.
Key terms and definitions
Polygon — a closed shape with straight sides.
Vertex — a corner. The plural is vertices.
Diagonal — a line joining two vertices that are not next to each other.
Bisect — cut exactly in half.
Perpendicular — at right angles.
Line of symmetry — a mirror line where one half of the shape reflects onto the other.
Rotational symmetry — the number of positions in a full turn where the shape looks unchanged. Order 1 means no rotational symmetry.
Congruent — identical in shape and size.
Core concepts
Types of triangle
Triangles are classified by their sides, and each classification brings angle properties with it.
An equilateral triangle has three equal sides and three equal angles of 60°. It has 3 lines of symmetry and rotational symmetry of order 3.
An isosceles triangle has two equal sides and two equal base angles, which sit opposite the equal sides. It has 1 line of symmetry and rotational symmetry of order 1.
A scalene triangle has three different sides and three different angles, with no symmetry at all.
A right-angled triangle contains one angle of 90°. It may also be isosceles, in which case the other two angles are 45° each.
The angles of any triangle add to 180°, whatever its type.
Types of quadrilateral
A square has four equal sides, four right angles, 4 lines of symmetry and rotational symmetry of order 4.
A rectangle has opposite sides equal and four right angles, with 2 lines of symmetry and rotational symmetry of order 2.
A parallelogram has two pairs of parallel and equal sides, with opposite angles equal. It has no lines of symmetry but rotational symmetry of order 2, which surprises people.
A rhombus has four equal sides with opposite sides parallel and opposite angles equal. It has 2 lines of symmetry — along its diagonals — and rotational symmetry of order 2.
A trapezium has exactly one pair of parallel sides. An isosceles trapezium, where the two non-parallel sides are equal, has 1 line of symmetry.
A kite has two pairs of adjacent equal sides and one pair of equal angles. It has 1 line of symmetry and rotational symmetry of order 1.
The angles of any quadrilateral add to 360°, because it splits into two triangles.
Diagonals
The diagonals distinguish shapes that otherwise look similar, and questions test them directly.
In a square, the diagonals are equal, bisect each other, and meet at right angles.
In a rectangle, they are equal and bisect each other, but do not meet at right angles.
In a rhombus, they bisect each other at right angles, but are not equal.
In a parallelogram, they bisect each other but are neither equal nor perpendicular.
In a kite, one diagonal bisects the other at right angles, but they do not bisect each other.
Two useful summaries: the diagonals are equal only in the square and rectangle, and perpendicular only in the square, rhombus and kite.
How the quadrilaterals relate
The definitions overlap, and the overlaps matter.
Every square is a rectangle, a rhombus and a parallelogram, because it satisfies all their conditions.
Every rectangle and every rhombus is a parallelogram.
A parallelogram is not generally a rectangle, since its angles need not be 90°.
This hierarchy is what "always, sometimes, never" questions probe. "A rhombus is a square" is sometimes true — true when its angles happen to be right angles. "A square is a rhombus" is always true.
Symmetry
A line of symmetry is a mirror line. Folding along it puts one half exactly onto the other.
Rotational symmetry counts the positions in a full turn where the shape looks identical to how it started. Every shape returns to itself after a full turn, so the minimum order is 1, which means no rotational symmetry.
A regular polygon with n sides has n lines of symmetry and rotational symmetry of order n.
The parallelogram is the one to watch: order 2 rotational symmetry but no lines of symmetry at all.
Angle notation
Exam questions name angles with three letters, and reading the notation correctly matters.
Angle ABC means the angle at B — the middle letter is always the vertex, with the outer letters naming the two arms.
So in a quadrilateral PQRS, angle PQR sits at Q and angle QRS sits at R. Reading the first letter as the vertex is a common misstep that sends the whole answer to the wrong corner.
A single letter may be used where there is no ambiguity, and the notation is sometimes written with a small angle symbol in front.
Which quadrilaterals tessellate
Every triangle and every quadrilateral tessellates, which surprises people who expect only the regular shapes to work.
For a quadrilateral, the four angles total 360°, so placing four copies round a point — one of each angle — fills the space exactly with no gaps.
For a triangle, the three angles total 180°, so six copies fit round a point.
This is why floor and wall patterns so often use irregular four-sided tiles: the angle sum guarantees they will fit, whatever their shape.
Angle problems and giving reasons
Missing-angle questions combine these properties with the standard angle facts, and each step needs a named reason.
Useful reasons include "base angles of an isosceles triangle are equal", "angles in a triangle add to 180°", "opposite angles of a parallelogram are equal", and "co-interior angles between parallel lines add to 180°".
Never assume a triangle is isosceles or a pair of lines parallel because the diagram suggests it. Look for the markings — matching dashes for equal sides, matching arrows for parallel lines — since diagrams are not drawn to scale.
Where a shape has parallel sides, the parallel-line angle facts become available, which is why parallelograms and trapeziums appear so often in these questions.
Worked examples
Example 1: An isosceles triangle
An isosceles triangle has an apex angle of 36°. Find the base angles, giving reasons.
The angles of a triangle add to 180°, so the two base angles together measure 180 − 36 = 144°.
Base angles of an isosceles triangle are equal, so each is 144 ÷ 2 = 72°.
Both reasons are needed for full marks: the angle sum, and the equality of the base angles.
Example 2: Identifying a shape from its diagonals
A quadrilateral has diagonals that bisect each other at right angles but are not equal in length. What shape is it?
Diagonals bisecting each other rules out the kite, where only one bisects the other.
Meeting at right angles rules out the rectangle and the general parallelogram.
Being unequal rules out the square, whose diagonals are equal.
The shape is a rhombus.
Example 3: An "always, sometimes, never" question
Is the statement "a parallelogram has a line of symmetry" always, sometimes or never true?
A general parallelogram has no lines of symmetry, only rotational symmetry of order 2.
But a rectangle is a parallelogram and has 2 lines of symmetry, and a rhombus is a parallelogram and also has 2.
So the statement is sometimes true — false for a general parallelogram, true when it is also a rectangle or a rhombus.
Answering "never" is the trap, and it comes from picturing only the slanted parallelogram rather than working from the definition.
Common mistakes and how to avoid them
Assuming a triangle is isosceles from the picture. Look for the matching dashes; diagrams are not to scale.
Giving a parallelogram lines of symmetry. It has none, though it has rotational symmetry of order 2.
Saying the diagonals of a rectangle are perpendicular. They are equal and bisect each other, but do not meet at right angles.
Thinking rotational symmetry of order 1 means some symmetry. Order 1 means none.
Treating the shape names as mutually exclusive. A square is also a rectangle, a rhombus and a parallelogram.
Giving an angle with no reason. Half the marks in these questions are for the reasons.
Confusing a kite with a rhombus. A kite's equal sides are adjacent; a rhombus has all four equal.
Exam technique for "Properties of 2D Shapes"
Check the diagram markings before anything else — dashes for equal sides, arrows for parallel lines, squares for right angles. Only use a property the markings justify.
Name a reason beside every angle you calculate, using the standard wording.
For "always, sometimes, never" questions, test the statement against a general example and a special case before answering.
When identifying a shape from its properties, work through the candidates eliminating each one, and say what ruled it out.
Learn the diagonal properties as a small table. They are the fastest way to tell the similar-looking quadrilaterals apart.
Remember the two angle sums — 180° for a triangle, 360° for a quadrilateral — as the starting point for most missing-angle problems.
Quick revision summary
Each shape is defined by its properties, not by how it looks — which is why a square is simultaneously a rectangle, a rhombus and a parallelogram.
Triangles: equilateral (3 equal sides, 60° each), isosceles (2 equal sides, 2 equal base angles), scalene (all different), right-angled (one 90°). Angles total 180°.
Quadrilaterals: square, rectangle, parallelogram (opposite sides parallel and equal, opposite angles equal), rhombus (4 equal sides), trapezium (one pair of parallel sides), kite (two pairs of adjacent equal sides). Angles total 360°.
Diagonals: equal only in the square and rectangle; perpendicular only in the square, rhombus and kite; bisecting each other in the square, rectangle, rhombus and parallelogram.
A parallelogram has no lines of symmetry but rotational symmetry of order 2. Order 1 means no rotational symmetry.
A regular polygon with n sides has n lines of symmetry and rotational symmetry of order n.
In angle problems, use only properties the markings justify, and give a named reason for every step.