Edexcel · GCSE Maths · 1MA1 · Foundation and Higher
M27 · Angles, polygons and shape properties
Revision notes, worked examples and methods for angles, polygons and shape properties.
Notes and quizzes ready · 50 questions · Video coming soon.
Foundation shows shared notes and questions. Higher includes the shared content and labelled Higher extensions. Changing tier starts the notes and a new quiz set.
Revise the key ideas
Lines, angles and polygons
Parallel lines never meet in a plane; perpendicular lines meet at 90°. A vertex is a corner, an edge joins vertices, and a plane is a flat surface. Use conventional labels, for example ∠ABC has its vertex at B.
Angles on a straight line sum to 180°, angles around a point sum to 360°, and vertically opposite angles are equal. A right angle is 90°. Parallel lines and a transversal
With parallel lines crossed by a transversal, corresponding angles are equal, alternate angles are equal, and co-interior angles sum to 180°. State the parallel-line reason when finding an unknown angle.
A triangle's interior angles sum to 180°. An isosceles triangle has two equal sides and two equal base angles; an equilateral triangle has three equal sides and three 60° angles.
A triangle's exterior angle equals the sum of its two opposite interior angles. If those are 43° and 61°, the exterior angle is 104°.
An n-sided polygon has interior-angle sum (n − 2) × 180°. Divide a polygon into triangles from one vertex to see why this works.
The exterior angles of a convex polygon, one at each vertex in the same turning direction, sum to 360°. For a regular n-gon, each exterior angle is 360°n and each interior angle is 180° minus that value.
Worked example: A regular pentagon has interior-angle sum 3 × 180° = 540°. Each interior angle is 108° and each exterior angle is 72°.
Quadrilaterals and symmetry
Any quadrilateral has interior-angle sum 360°. A parallelogram has parallel opposite sides, equal opposite sides and equal opposite angles; adjacent angles sum to 180°.
A rectangle is a parallelogram with four right angles and equal diagonals. A rhombus is a parallelogram with all four sides equal; its diagonals meet at right angles and bisect each other.
The diagonals of every parallelogram bisect each other. A square's diagonals are equal, perpendicular and angle-bisecting; a rectangle's diagonals are not generally perpendicular. State properties that actually follow from the named shape.
A square is both a rectangle and a rhombus. Its four sides are equal and its four angles are right angles. A kite has two pairs of adjacent equal sides and perpendicular diagonals, with one diagonal bisecting the other.
A trapezium has a pair of parallel sides. An isosceles trapezium has equal non-parallel sides and equal base angles. Use the definition stated in a question if conventions differ.
A line of symmetry reflects a shape onto itself. Rotational symmetry order counts how many times it fits itself during a full turn; a non-square rectangle has order 2.
Worked reasoning
Worked example: In an isosceles triangle with vertex angle 40°, the base angles are equal and total 140°. Each is 70°. Name the equal sides or angle property that justifies equality.
When proving a geometric statement, link each equality to a reason: parallel lines → alternate angles equal → triangle angle sum → required angle. A diagram is not assumed to scale unless the question says so.
Test yourself
50 questions · Sets of 10 from the selected tier. For fractions, use / when typing; for powers, use superscripts or ^. Follow each question's answer format. These quick checks support revision; practise full written solutions and proofs too.
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