Every geometry question from Sec 3 to the O-Level is built on a small set of angle facts. This includes angle-chasing, congruent and similar triangles, and circle properties. These notes cover the basic angle facts, parallel lines cut by a transversal, triangle and special quadrilateral properties, interior and exterior angle sums of polygons, and accurate constructions with instruments.
Examiners award marks for correctly cited reasons, not just correct values. A right answer with no reason, or a vague reason such as “angles add up”, routinely loses marks. Learn the exact reason wording for each fact and write it on every line of working.
What these notes cover
- •Types of angles
- •Angles on a straight line, at a point and vertically opposite
- •Parallel lines cut by a transversal
- •Angle sum and exterior angle of a triangle
- •Isosceles and equilateral triangles
- •Properties of special quadrilaterals
- •Naming a quadrilateral from its diagonals
- •Interior and exterior angle sums of polygons
- •Regular polygons
- •Constructions with ruler, compasses, protractor and set squares
- •Common mistakes and exam technique
Key definitions
- Reflex angle
- An angle greater than 180° but less than 360°, measured the long way round the vertex. Reflex ∠AOB = 360° − ∠AOB.
- Transversal
- A straight line that cuts across two or more other lines.
- Interior (co-interior) angles
- Angles between a pair of parallel lines on the same side of the transversal. They add up to 180°.
- Exterior angle of a triangle
- The angle formed outside a triangle when one side is produced, i.e. extended in a straight line.
- Regular polygon
- A polygon with all sides equal and all angles equal.
Key formulas
Angle sum of a triangle
a + b + c = 180°
a, b and c are the three interior angles of any triangle. Reason: (∠ sum of △).
Exterior angle of a triangle
x = a + b
x is the exterior angle formed when a side is produced. a and b are the two interior opposite angles, never the adjacent one. Reason: (ext. ∠ of △).
Interior angle sum of an n-sided polygon
sum of interior angles = (n − 2) × 180°
n is the number of sides. Diagonals from one vertex split the polygon into n − 2 triangles. Valid for any polygon.
Exterior angle sum
sum of exterior angles = 360°
Holds for any convex polygon, whatever the value of n. At each vertex, interior angle + exterior angle = 180°.
Regular polygon — each exterior and interior angle
Regular polygons only. This route uses smaller numbers than (n − 2) × 180° ÷ n and gives the same answer.
Number of sides of a regular polygon
Use this to find n from a given angle. n must be a whole number and at least 3; otherwise reject it.
Worked example
The interior angle of a regular polygon P is in the ratio 7 : 2 to its exterior angle. (a) Find the number of sides of P. (b) An equilateral triangle ABX is drawn on the outside of P, sharing the edge AB with P. Find the angle at A not inside either shape.
- 1Interior and exterior angles at a vertex lie on a straight line, so together they make 180°.
- 2Sharing 180° in the ratio 7 : 2 gives each exterior angle = (2 ÷ 9) × 180° = 40°.
- 3n = 360° ÷ 40° = 9, which is a whole number, so P is a regular 9-sided polygon (a nonagon).
- 4Each interior angle of P = 180° − 40° = 140°.
- 5At A, the interior angle of P, the 60° angle of the equilateral triangle and the required angle add up to 360° (∠s at a pt.).
- 6Required angle = 360° − 140° − 60° = 160°.
Watch out: Always state the check that n is a whole number. A non-integer n means an error, or that no such polygon exists.
Common mistakes
Citing letter shapes instead of angle names
Wrong: “Z angles”, “F angles”, “C angles”.
Right: “alt. ∠s, AB ∥ CD”, “corr. ∠s, AB ∥ CD”, “int. ∠s, AB ∥ CD”. Always name the pair of parallel lines.
Treating interior angles between parallel lines as equal
Wrong: x = 118° (int. ∠s), copying the given angle across.
Right: Interior angles are supplementary: x = 180° − 118° = 62° (int. ∠s, AB ∥ CD). Only alternate and corresponding angles are equal.
Dividing the interior angle sum by n for an irregular polygon
Wrong: For an irregular pentagon, each angle = 540° ÷ 5 = 108°.
Right: Divide by n only when the polygon is stated to be regular. For an irregular polygon, add the given angles, set the total equal to (n − 2) × 180° and solve for the unknown.
Questions students ask
What is the sum of the interior angles of a polygon?
The interior angles of an n-sided polygon add up to (n − 2) × 180°. Drawing every diagonal from one vertex splits the polygon into n − 2 triangles, and each triangle contributes 180°. For example, a hexagon gives 4 × 180° = 720°.
How do you find the number of sides of a regular polygon from its interior angle?
Subtract the interior angle from 180° to get the exterior angle, then use n = 360° ÷ exterior angle. For an interior angle of 156°, the exterior angle is 24°, so n = 360° ÷ 24° = 15. Check that n is a whole number of at least 3; if it is not, no such regular polygon exists.
What is the difference between alternate, corresponding and interior angles?
When a transversal cuts parallel lines, corresponding angles are equal, alternate angles are equal, and interior angles add up to 180°. Write the reasons as corr. ∠s, alt. ∠s or int. ∠s and name the parallel lines, e.g. alt. ∠s, AB ∥ CD. Never write “Z angles” or “F angles”.
How do you identify a special quadrilateral from its diagonals?
If the diagonals bisect each other, the shape is in the parallelogram family. Equal diagonals give a rectangle, perpendicular diagonals give a rhombus, both give a square and neither gives a parallelogram. If the diagonals are perpendicular but only one is bisected, the shape is a kite; always give the most specific name the properties force.
In the full PDF
- •Labelled diagrams for each angle type, the parallel-line transversal, the exterior angle property and polygon triangulation
- •The full properties table for all six special quadrilaterals, including symmetry, and the classification hierarchy
- •Further worked examples: algebraic angles at intersecting lines, a Paper 2–style isosceles angle chase, and both routes for a regular 12-gon
- •Step-by-step SSS, SAS, quadrilateral and parallel-line constructions, with three worked construction examples
- •All six mark-costly misconceptions, the construction mistakes and the exam technique checklist
- •Section summaries and the five-point final summary