Angles in Polygons & Parallel LinesCambridge IGCSE Maths: Revision notes

Several fundamental angle facts are used throughout geometry:

• Angles at a point sum to $360°$
• Angles on a straight line sum to $180°$
• Vertically opposite angles (formed where two lines cross) are equal
• Angle sum of a triangle is $180°$
• Angle sum of a quadrilateral is $360°$

Always state which fact you are using as your geometric reason — Cambridge mark schemes require the correct terminology, not just the right numerical answer.

When a straight line (a transversal) crosses a pair of parallel lines, three special angle relationships appear:

• Corresponding angles are equal (same position at each intersection, often described as an "F" pattern)
• Alternate angles are equal (on opposite sides of the transversal, between the parallel lines, a "Z" pattern)
• Co-interior angles (also called supplementary or "C" pattern angles) sum to $180°$

Since questions must be described in words without diagrams, expect angle relationships to be stated in terms like "angle ABC and angle DEF are corresponding angles on parallel lines $AB$ and $DE$".

For any polygon with $n$ sides: $\text{sum of interior angles} = (n-2) \times 180°$

For a regular polygon (all sides and angles equal): $\text{each interior angle} = \frac{(n-2)\times180°}{n}$

At each vertex, the interior and exterior angle lie on a straight line, so: $\text{interior angle} + \text{exterior angle} = 180°$

The exterior angles of any polygon always sum to $360°$, so for a regular polygon: $\text{each exterior angle} = \frac{360°}{n}$

Example: For a regular hexagon ($n=6$): exterior angle $=\frac{360}{6}=60°$; interior angle $=180-60=120°$.

Angles are described using three-letter notation, e.g. angle $ABC$, where $B$ is the vertex (the middle letter is always the point where the angle is measured).

When answering structured questions:

1. State the numerical value of the angle
2. State the geometric reason using correct terminology (e.g. "alternate angles are equal", "angle sum of a triangle is $180°$")

Both the value and the reason are usually needed to earn full marks, since Cambridge mark schemes test AO2 (communicating a method clearly) alongside AO1 (calculation).