Geometry

Summary: Covers angle properties, triangles, polygons, all seven circle theorems, symmetry, loci and constructions, similarity, and congruence. Geometry is a key visual topic in IGCSE Mathematics 0580, appearing in both Core and Extended papers with a strong emphasis on circle theorems and geometric reasoning. Tags: igcse mathematics geometry Created: 2026-07-16 Last Updated: 2026-07-16

1. Angles

Types of angles:

• Acute: 0° < angle < 90°
• Right angle: exactly 90°
• Obtuse: 90° < angle < 180°
• Reflex: 180° < angle < 360°

Fundamental angle facts:

• Angles on a straight line sum to 180°
• Angles around a point sum to 360°
• Vertically opposite angles are equal (X-shape)

Example: Two angles on a straight line are 3x and 2x. Find x. 3x + 2x = 180° → 5x = 180° → x = 36°.

Parallel lines — when a transversal crosses two parallel lines:

• Alternate angles (Z-shape): equal
• Corresponding angles (F-shape): equal
• Co-interior angles (C-shape / allied angles): sum to 180°

Example: AB ∥ CD, a transversal intersects at A and C. Angle at A = 65°. Alternate angle = 65° (equal). Co-interior angle = 180° − 65° = 115°.

2. Triangles

Angle sum: The three interior angles of any triangle sum to 180°.

Exterior angle theorem: The exterior angle of a triangle equals the sum of the two opposite interior angles.

Example: In triangle ABC, A = 50°, B = 70°. Find C and the exterior angle at C. C = 180° − (50° + 70°) = 60°. Exterior angle at C = 50° + 70° = 120°.

Types of triangles by sides:

• Scalene: all three sides different, all angles different
• Isosceles: two equal sides, two equal base angles
• Equilateral: all sides equal, all angles = 60°

Types of triangles by angles: acute (all angles < 90°), right-angled (one angle = 90°), obtuse (one angle > 90°).

Pythagoras’ theorem: In a right-angled triangle, a² + b² = c², where c is the hypotenuse.

3. Polygons

A polygon is a closed 2D shape with straight sides.

Interior angle sum = (n − 2) × 180° (n = number of sides) Exterior angle sum = 360° (always, for any convex polygon) Interior angle + exterior angle = 180° (at each vertex).

For a regular polygon (all sides and angles equal):

• Each interior angle = (n − 2) × 180° / n
• Each exterior angle = 360° / n

Example: A regular polygon has exterior angles of 40°. How many sides? n = 360 / 40 = 9 sides (a nonagon).

Example: Find the interior angle of a regular hexagon. Exterior = 360°/6 = 60° → Interior = 180° − 60° = 120°.

Example: A polygon has interior angles summing to 1260°. How many sides? (n − 2) × 180° = 1260° → n − 2 = 7 → n = 9 sides.

4. Circle Theorems

IGCSE 0580 requires knowledge of all seven circle theorems, applied with formal reasons.

Theorem 1 — Angle at centre: The angle subtended by an arc at the centre is twice the angle subtended at any point on the circumference.

Theorem 2 — Angle in a semicircle: The angle subtended by a diameter at the circumference is a right angle (90°).

Theorem 3 — Angles in the same segment: Angles subtended by the same chord (or arc) in the same segment are equal.

Theorem 4 — Cyclic quadrilateral: The sum of opposite angles in a cyclic quadrilateral is 180°. (All four vertices lie on the circumference.)

Theorem 5 — Tangent and radius: The angle between a tangent and the radius drawn to the point of contact is 90°.

Theorem 6 — Alternate segment theorem: The angle between a tangent and a chord through the point of contact equals the angle in the alternate segment.

Theorem 7 — Tangents from a point: Two tangents drawn from the same external point to a circle are equal in length.

Example: In a circle, angle at centre = 130°. Find the angle at circumference subtended by the same arc. Angle at circumference = 130° / 2 = 65°.

Example: ABCD is a cyclic quadrilateral. Angle A = 85°, angle B = 110°. Find C and D. Opposite angles sum to 180°: C = 180° − 85° = 95°. D = 180° − 110° = 70°.

Example: A tangent touches a circle at P. Chord PQ makes an angle of 55° with the tangent. Find the angle in the alternate segment. By the alternate segment theorem: 55°.

Example: AB is a diameter, C is a point on the circumference. Angle BAC = 37°. Find angle ACB and angle ABC. Angle ACB = 90° (angle in a semicircle, Theorem 2). Angle ABC = 180° − 90° − 37° = 53° (triangle angle sum).

5. Symmetry

Line symmetry (reflection symmetry): A shape can be folded along a line so both halves match. The fold line is the line of symmetry.

Rotational symmetry: A shape fits onto itself when rotated. The order is the number of times it fits in a full 360° turn.

Plane of symmetry (3D): A flat surface dividing a 3D shape into two mirror-image halves.

For 3D shapes: a cube has 9 planes of symmetry; a cuboid has 3 planes of symmetry.

6. Loci and Constructions

A locus (plural: loci) is the set of all points satisfying a given condition.

Standard loci:

• Points at a fixed distance r from a point → circle (radius r)
• Points equidistant from two points → perpendicular bisector
• Points equidistant from two intersecting lines → angle bisector
• Points at a fixed distance from a line → pair of parallel lines

Standard compass-and-straightedge constructions:

1. Perpendicular bisector of a line segment
2. Angle bisector
3. Perpendicular from a point to a line / through a point on a line
4. Equilateral triangle / 60° angle
5. Triangle given three sides (SSS)

7. Similarity

Two shapes are similar if one is an enlargement of the other — same shape, different size. Corresponding angles are equal; corresponding sides are in the same ratio.

Conditions for similar triangles:

• AAA (or AA): Two pairs of corresponding angles are equal (implies the third).
• SAS ratio: Two pairs of sides in the same ratio and the included angle equal.

Scale factors:

• Length scale factor = k
• Area scale factor = k²
• Volume scale factor = k³

Example: Triangle PQR ~ triangle XYZ (similar). PQ = 4 cm, XY = 10 cm. Area of PQR = 6 cm². Find the area of XYZ. Length SF = 10/4 = 2.5. Area SF = 2.5² = 6.25. Area of XYZ = 6 × 6.25 = 37.5 cm².

8. Congruence

Two shapes are congruent if they are identical in shape and size.

Four criteria for proving triangles congruent:

1. SSS (Side-Side-Side): All three pairs of corresponding sides are equal.
2. SAS (Side-Angle-Side): Two sides and the included angle are equal.
3. ASA / AAS (Angle-Side-Angle / Angle-Angle-Side): Two angles and any corresponding side are equal.
4. RHS (Right angle-Hypotenuse-Side): Right angle, equal hypotenuses, one other pair of equal sides. Only for right-angled triangles.

Important: ASS/SSA is NOT a valid congruence criterion — the angle must be between the two known sides for SAS.

Example: Prove triangles ABC and DEF are congruent, given AB = DE = 8 cm, BC = EF = 6 cm, and angle B = angle E = 90°. Both are right-angled. AB = DE (hypotenuse), BC = EF (side). By RHS, the triangles are congruent.

Sources

• BBC Bitesize GCSE Mathematics — Angles, Triangles, and Polygons, BBC (free educational resource)
• OpenStax Math — Geometry and Trigonometry, Rice University (free, CC BY 4.0)
• Cambridge IGCSE Mathematics 0580 — Geometry, Cambridge Assessment International Education
• CK-12 Mathematics — Geometry Fundamentals, CK-12 Foundation (free, CC BY-NC 3.0)

Related Notes

• Mensuration — Perimeter, area, and volume formulas for 2D and 3D shapes
• Trigonometry — Trigonometry in right triangles, sine/cosine rules
• Vectors and Transformations — Transformations and vector geometry
• Algebra and Graphs — Coordinate geometry, equation of a line
• IGCSE-Maths-Index — Full IGCSE Mathematics index

Common Misconceptions