Lines and Angles
- Complementary angles add to 90°; supplementary angles add to 180°. Angles on a straight line add to 180° and angles round a point to 360°.
- Vertically opposite angles are equal.
- With parallel lines and a transversal: corresponding angles are equal, alternate interior angles are equal, and co-interior angles add to 180°.
Triangles
- The angles of a triangle add to 180°. An exterior angle equals the sum of the two opposite interior angles.
- Any two sides together are longer than the third side. The largest side is opposite the largest angle.
- Pythagoras: in a right triangle, hypotenuse² = base² + height². Common triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25.
- Congruence tests: SSS, SAS, ASA (or AAS) and RHS. Similar triangles have equal angles and proportional sides; the ratio of their areas is the square of the ratio of their sides.
- Basic proportionality (Thales): a line parallel to one side divides the other two sides in the same ratio. The line joining the midpoints of two sides is parallel to the third and half its length.
| Centre | Meeting point of | Note |
|---|---|---|
| Centroid | Medians | Divides each median 2 : 1 from the vertex |
| Circumcentre | Perpendicular bisectors | In a right triangle it is the midpoint of the hypotenuse |
| Incentre | Angle bisectors | Centre of the inscribed circle |
| Orthocentre | Altitudes | In a right triangle it is the right-angle vertex |
Quadrilaterals and Polygons
- The angles of a quadrilateral add to 360°. A parallelogram has opposite sides and angles equal and its diagonals bisect each other. A rhombus's diagonals bisect at right angles; a rectangle's diagonals are equal.
- A polygon with n sides: interior angles add to (n − 2) × 180°; exterior angles always add to 360°. Each exterior angle of a regular polygon = 360°/n, so a regular hexagon has exterior angles of 60° and interior angles of 120°.
- Number of diagonals of an n-sided polygon = n(n − 3)/2. A hexagon has 9.
Circles
- The angle an arc subtends at the centre is twice the angle it subtends anywhere on the remaining circle. The angle in a semicircle is 90°. Angles in the same segment are equal.
- Opposite angles of a cyclic quadrilateral add to 180°.
- A tangent is perpendicular to the radius at the point of contact. The two tangents from an outside point are equal in length.
- A perpendicular from the centre bisects a chord. A chord of length 16 cm at 6 cm from the centre: radius = √(8² + 6²) = 10 cm.
Coordinate Geometry
| Quantity | Formula |
|---|---|
| Distance between (x₁, y₁) and (x₂, y₂) | √[(x₂ − x₁)² + (y₂ − y₁)²] |
| Midpoint | ((x₁ + x₂)/2, (y₁ + y₂)/2) |
| Point dividing in ratio m : n | ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)) |
| Centroid of a triangle | ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3) |
| Area of a triangle | ½ |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)| |
Common Mistakes
Taking the interior angle sum of a polygon as n × 180°; confusing the centroid (medians) with the circumcentre; and forgetting that area scales with the square of the side ratio for similar figures.