GCSE Maths Angles & constructions revision

Angles filling a full turn about a point sum to 360°; angles filling half a turn along a straight line sum to 180°. When two straight lines cross, four angles appear in two equal pairs: each pair of vertically opposite angles is equal, and neighbouring pairs sit on a straight line so they sum to 180°. Exam answers expect a reason alongside each number — “angles on a straight line”, “vertically opposite angles are equal”.

Unknowns hide inside these facts. Three angles at a point might be 3x, 2x and 150°, giving 3x + 2x + 150 = 360, so 5x = 210 and x = 42. The routine never changes: collect what you know, write the sum (180°, 360° or a triangle’s 180°), solve the linear equation, then substitute back for the angle itself — the question usually wants the angle, not x.

When a transversal cuts two parallel lines, three rules do all the work. Corresponding angles are equal — the two angles sit in matching corners, an F shape. Alternate angles are equal — they sit on opposite sides of the transversal between the parallels, a Z shape. Co-interior angles sit on the same side between the parallels, a C or U shape, and sum to 180°. Label the shape you see on the diagram; the name is the reason the mark scheme wants.

Chains of two or three steps are standard: find one angle on the transversal, transfer it to the other parallel line, then finish with a triangle or straight-line sum. Arrowheads (» ) mark the parallel lines on diagrams; right-angle boxes mark perpendicularity.

A triangle’s interior angles sum to 180° — so 47° and 86° leave 180 − 133 = 47° for the third. Push one side outwards and the exterior angle it forms equals the two opposite interior angles added (35° + 77° = 112°). In an isosceles triangle the two equal sides face two equal base angles: an apex of 40° leaves 140° to split as 70° and 70°. Equilateral triangles are the special case with all three angles at 60°.

Quadrilaterals have interior sum 360°. Each special one carries properties worth memorising: a parallelogram has equal opposite sides and angles; a rhombus adds equal sides and bisecting diagonals; a kite has two pairs of equal adjacent sides and one pair of equal angles; a trapezium has exactly one pair of parallel sides (isosceles trapezia add equal base angles).

For any n-sided polygon, walking a full turn outside it turns you through 360° — one exterior angle per vertex, so exteriors always total 360° no matter how irregular the shape. Interiors total (n − 2) × 180° (a pentagon: 3 × 180 = 540°). For a regular polygon both sums split equally: a regular polygon with exterior angle x° has n = 360⁄x sides and interior 180 − x. Working backwards is common: interior 150° → exterior 30° → 360 ÷ 30 = 12 sides.

Ruler-and-compass constructions replace estimating with logic. The perpendicular bisector of AB comes from two equal-radius arcs centred at A and B; their crossings both sit the same distance from A and B, so joining them gives the locus of equidistant points. An angle bisector uses arcs of one radius from the vertex, then equal arcs from the two new points. Keep the compass radius locked between arcs — that equality is the whole proof. A locus question converts words into boundaries: a fixed distance from a point draws a circle arc; from a line, a parallel pair; “equidistant from A and B” draws the perpendicular bisector; “equidistant from two lines” draws the angle bisector. Shade where the conditions overlap.