HookBritain was mapped by drawing triangles across it
On 18 April 1936 an Ordnance Survey team set the first concrete trig pillar of the Retriangulation of Great Britain into the ground at Cold Ashby in Northamptonshire. Over the next twenty-odd years, under Brigadier Martin Hotine, roughly 6,500 of these squat white pillars were planted on hilltops across the country. From each pillar surveyors could see several others, and the whole map of Britain was rebuilt not by measuring thousands of distances — that would have been impossible over mountains and estuaries — but by measuring angles between pillars and letting geometry do the rest. Measure one baseline accurately, then measure the angles of the triangles hanging off it, and every triangle is pinned down. That single idea — that a triangle is fixed once you know enough of its angles and sides — is the engine of this whole section.
Everything in G4.1 is about knowing a shape well enough to be certain of it. You will name parts precisely, chase angles using a short list of rules, sort the special quadrilaterals by their properties, prove two shapes are identical (congruent) or scaled copies (similar), slide, spin, flip and enlarge them, learn the vocabulary of the circle, construct exact figures with only a ruler and a pair of compasses, and read 3D solids from their flat views. Foundation papers reward this section heavily — and, crucially, they reward reasons. A right angle with no justification often scores nothing; the same angle with 'alternate angles are equal' beside it scores full marks.
ModelThe angle rules — a short list that solves almost everything
There are only a handful of facts, and every angle chase is a chain of them. Angles at a point add to 360° (a full turn). Angles on a straight line add to 180° (a half turn). Vertically opposite angles are equal — where two lines cross, the angles facing each other match. When a straight line (a transversal) cuts a pair of parallel lines, three more facts appear: corresponding angles are equal (the 'F' shape), alternate angles are equal (the 'Z' shape), and co-interior angles add to 180° (the 'C' or 'U' shape).
Why do the parallel-line rules hold? Because the two parallel lines never converge, the transversal meets each at exactly the same tilt, so the pattern of angles at one crossing is copied at the other. That copying is what 'corresponding' means.
The two triangle facts fall straight out: the angles in a triangle add to 180°, and the exterior angle of a triangle equals the sum of the two interior angles it is not next to. You can see the 180° by tearing the three corners off a paper triangle and fitting them along a straight line — they always make a half turn. AQA does not accept 'it looks like 180', but it does accept the rule, quoted.
A transversal crosses two parallel lines. At the top crossing one angle is 118°. Find the angle marked x at the bottom crossing, sitting on the same side, between the two parallel lines. First, the 118° and the angle directly below it at the bottom are corresponding, so that lower angle is also 118°. The angle x is on a straight line with it, so x = 180° − 118° = 62°. Write the reason at every step — 'corresponding angles equal, so 118°; angles on a straight line sum to 180°, so x = 62°' — because AQA awards a mark for the reason, not just the 62.
MechanismPolygons — the (n − 2) × 180 machine
Split any polygon into triangles from one corner and you have the whole theory. A quadrilateral splits into 2 triangles, a pentagon into 3, a hexagon into 4 — always two fewer triangles than the number of sides. Since each triangle carries 180°, the interior angles of an n-sided polygon add to (n − 2) × 180°. A hexagon: (6 − 2) × 180 = 720°.
The exterior angles tell an even cleaner story. Walk once around the outside of any polygon and you turn through one full circle, so the exterior angles always add to 360°, no matter how many sides. That is the fast route into regular polygons, where every angle is identical: one exterior angle of a regular polygon is simply 360° ÷ n, and each interior angle is 180° minus that.
The 50p coin in your pocket is a regular-ish seven-sided shape (a heptagon of constant width, introduced in 1969); a football's black patches are regular pentagons; a honeycomb is a tessellation of regular hexagons, which fit with no gaps precisely because their 120° interior angles divide 360° exactly. Those are the three questions AQA recycles: find an angle, find the number of sides, or explain why a shape does or does not tessellate.
A regular polygon has an interior angle of 156°. How many sides? Work with the exterior angle, because those add to a clean 360°. The exterior angle is 180° − 156° = 24°. The number of sides is 360° ÷ 24° = 15 sides. Check it back through the interior formula: (15 − 2) × 180 = 2,340°, and 2,340 ÷ 15 = 156°. Going via the exterior angle turns a messy division into a one-line answer — that is the method examiners want to see.
ModelTriangles and quadrilaterals — sorted by property, not by look
Shapes are defined by what is true of them, not by how they are drawn. An isosceles triangle has two equal sides and — this is the marked fact — two equal base angles beneath them; an equilateral triangle has three equal sides and three 60° angles; a scalene triangle has all three different. A right-angled triangle simply contains a 90° angle.
The six special quadrilaterals form a family. A square has four equal sides and four right angles. A rectangle has four right angles and opposite sides equal. A parallelogram has two pairs of parallel sides and opposite angles equal. A rhombus is a parallelogram with four equal sides (a 'pushed-over square'), and its diagonals cross at right angles. A trapezium has just one pair of parallel sides. A kite has two pairs of adjacent equal sides and one pair of equal angles.
The key that unlocks the hardest of these questions is that the family is nested: a square is a special rectangle (it obeys every rectangle rule and adds equal sides), and a rectangle is a special parallelogram. So any true statement about a rectangle is automatically true about a square — a fact worth a mark on the 'always, sometimes, never' questions AQA loves.
In an isosceles triangle the apex angle at the top is 40°. Find the base angles. The three angles add to 180°, and the two base angles are equal, so together they are 180° − 40° = 140°. Each base angle is 140° ÷ 2 = 70°. The reason the two base angles are equal is the isosceles property — state it, because a later block shows how AQA asks you to prove exactly that fact.
CaseNaming the parts — points, lines and the circle's vocabulary
Marks are lost every year to vocabulary alone, so learn the words exactly. A vertex is a corner (plural vertices); an edge is a line segment where two faces meet; parallel lines never meet and are marked with matching arrows; perpendicular lines meet at 90°, marked with a small square. AQA will hand you a written description — 'ABCD is a quadrilateral with AB parallel to DC and angle A a right angle' — and expect you to draw it accurately, labelling the vertices in order around the shape.
The circle has its own dictionary, and every one of these terms appears in later mensuration work. The centre is the middle point; the radius runs from centre to edge; the diameter runs right across through the centre and is twice the radius; the circumference is the perimeter. A chord is any straight line joining two points on the circle without passing through the centre. A tangent touches the circle at exactly one point and sits at right angles to the radius drawn to that point. An arc is a piece of the circumference; a sector is a 'pizza slice' bounded by two radii and an arc; a segment is the region cut off by a chord.
MechanismConstructions and loci — the ruler-and-compass toolkit
These are the questions where students throw away marks by rubbing out their working. The construction arcs are the answer — leave them showing. Set your compasses, keep the same radius throughout each construction, and use a sharp pencil.
The perpendicular bisector of a line segment: open the compasses to more than half its length, draw arcs from each end above and below the line, and join the two crossing points. That line is every point equidistant from the two ends. The angle bisector: from the corner, draw an arc crossing both arms; from those two crossing points draw two more equal arcs; join their intersection back to the corner. That line is every point equidistant from the two arms.
A locus is the set of all points obeying a rule, and each construction is a locus in disguise. Points a fixed distance from a single point form a circle. Points a fixed distance from a straight line form a 'racetrack' — two parallel lines capped by semicircles. Points equidistant from two points are the perpendicular bisector; points equidistant from two lines are the angle bisector. Exam loci problems — 'shade the region closer to the fence than to the tree and within 5 m of the gate' — are just two or three of these rules combined, then a shaded overlap.
To find the point equidistant from three towns A, B and C on a map: construct the perpendicular bisector of AB and the perpendicular bisector of BC. Where those two bisectors cross is equidistant from all three — every point on the first line is equal distance from A and B, every point on the second is equal distance from B and C, so the crossing point is equal distance from all three. Leave all four arcs visible; that is what the method marks are checking.
ModelCongruence and similarity — identical versus scaled
Two shapes are congruent if they are identical — same shape, same size — even if one is rotated or flipped. For triangles you never need to check all six measurements; four shortcuts guarantee congruence. SSS: three pairs of equal sides. SAS: two sides and the angle between them. ASA: two angles and the side between them. RHS: a right angle, the hypotenuse and one other side. Quote the four letters — that label is the mark.
Watch the trap: equal angles alone (AAA) do not prove congruence, only similarity, and 'SSA' (two sides and a non-included angle) is not a valid rule at all, because two different triangles can fit it.
Two shapes are similar if one is a scaled copy of the other — all angles equal, all sides in the same ratio, called the scale factor. This is the retriangulation idea again: fix the angles and one length, and the whole triangle is determined. Similar triangles are the tool for finding a missing length in a scale drawing, a shadow problem or a pair of nested triangles: match up corresponding sides, find the scale factor from a known pair, and multiply.
Prove the base angles of an isosceles triangle are equal. Take triangle ABC with AB = AC, and drop the bisector of the apex angle A to meet BC at D. Compare triangles ABD and ACD: AB = AC (given), angle BAD = angle CAD (AD bisects angle A), and AD = AD (shared side). That is two sides and the included angle — SAS — so the triangles are congruent, which forces angle B = angle C. A four-line proof, and every GCSE congruence proof follows this exact template: name the two triangles, list three matching facts, quote the criterion, state what follows.
MechanismTransformations — slide, flip, spin, resize
Four transformations move shapes around the grid, and each needs a specific, complete description — give the wrong ingredients and you score zero even with the right picture. A translation slides a shape by a column vector: so many across, so many up or down. A reflection flips it in a mirror line, so you must name that line by its equation (y = x, or x = 2). A rotation spins it, so you must give three things: the centre of rotation, the angle, and the direction (clockwise or anticlockwise). An enlargement resizes it, so you must give the centre of enlargement and the scale factor.
Translation, reflection and rotation all keep the shape the same size — the image is congruent to the original. Enlargement changes the size, so the image is similar, not congruent. The one that catches people out: a scale factor between 0 and 1 is still an enlargement, it just makes the shape smaller. A scale factor of ½ halves every length. (Negative scale factors, which flip the shape through the centre, are Higher-tier only, so at Foundation your scale factors are positive.)
A triangle has a vertex at (2, 3). It is enlarged by scale factor 2, centre the origin (0, 0). Where does that vertex go? Measure how far the point is from the centre and double it: from the origin, (2, 3) becomes (4, 6). Every point's distance from the centre doubles, so the image sits twice as far out and is twice as big — congruent in shape, but similar, not congruent, in size. If the scale factor had been ½, the same vertex would map to (1, 1.5), a smaller copy.
DataSolids — faces, edges, vertices, and the flat views
A 3D solid is described by its faces (flat surfaces), edges (where two faces meet) and vertices (corners). A cube and a cuboid each have 6 faces, 12 edges and 8 vertices. A triangular prism has 5 faces, 9 edges and 6 vertices. A square-based pyramid has 5 faces, 8 edges and 5 vertices. Curved solids stretch the language: a cylinder has 2 flat circular faces and a curved surface; a cone has 1 flat circular face, a curved surface and a single apex; a sphere is one smooth curved surface with no edges or vertices at all. For any solid with flat faces, the counts obey Euler's relation, faces + vertices = edges + 2 — a cube gives 6 + 8 = 14 = 12 + 2, a free way to check you have counted correctly.
AQA also asks you to read a solid from its three flat views. The plan is the view looking straight down from above; the front elevation is the view from the front; the side elevation is the view from one side. Drawn on squared paper to scale, these three views together specify the solid completely — the same principle an architect uses on a set of building plans. The classic exam task hands you a shape built from cubes and asks for all three elevations, or hands you the elevations and asks you to build or name the solid.
A solid is made of six centimetre cubes: a 2-by-2 square base of four cubes with two more cubes stacked on the back-left corner. The plan (from above) is a 2-by-2 square. The front elevation (from the front) is an L-shape two squares wide and two tall on the left. The side elevation (from the side) is also two squares tall on the back edge and one tall at the front. Drawing each on squared paper, one square per centimetre, is what earns the marks — label which view is which, because AQA marks them separately.
VocabularyKey terms the mark scheme pays for
TrapsMisconceptions that cost marks
ExamWhat examiners want
This section lives on all three papers, and much of it — angle chasing, polygon angles, naming solids — needs no calculator, so expect it on Paper 1. The single biggest source of dropped marks is missing reasons. On any 'give reasons for your answer' angle question, AQA awards marks for the justification, not just the number: write 'alternate angles are equal', 'angles on a straight line sum to 180°', 'the base angles of an isosceles triangle are equal' at every step. A correct final angle with no reasoning can score zero.
On constructions and loci, bring a working pair of compasses and a sharp pencil, and never rub out your arcs — the construction arcs are the evidence the method marks are looking for. Keep the compass setting fixed within each construction. For congruence proofs, name the two triangles, list exactly three matching facts with their reasons, then quote the criterion (SSS, SAS, ASA or RHS) explicitly — the criterion label is itself a mark.
For transformations, give the full description or lose everything: a rotation needs centre, angle and direction; a reflection needs the mirror line's equation; an enlargement needs centre and scale factor; a translation needs a column vector. Describing a translation as a 'move' or giving a rotation without its direction is a guaranteed lost mark. And when a question says 'describe fully the single transformation', the word 'single' is a warning — one transformation only, fully specified.