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Mathematics 05804.1

Angles and polygons

Angle rules, parallel lines, and the angles in polygons.

Learning objectives

What you need to be able to do

Teacher-mapped phrasing — check against the official Cambridge syllabus for exact wording.

  • 4.1.1Use angle properties of lines, triangles, quadrilaterals and parallel lines.
  • 4.1.2Calculate interior and exterior angles of regular and irregular polygons.

6 minute read

Angles and polygons

Angle questions are marked on reasons, not just numbers. Learn the exact phrases.

Basic rules

  • Angles on a straight line add to 180°.
  • Angles at a point add to 360°.
  • Vertically opposite angles are equal.
  • Angles in a triangle add to 180°; in a quadrilateral, 360°.
  • An exterior angle of a triangle equals the sum of the two opposite interior angles.

Parallel lines

  • Corresponding angles are equal (F-shape).
  • Alternate angles are equal (Z-shape).
  • Co-interior (allied) angles add to 180° (C-shape).

Use the proper names in your reasons — "F angles" is not accepted wording in a mark scheme.

Polygons

  • Sum of interior angles = (n − 2) × 180°, because the polygon splits into n − 2 triangles.
  • Sum of exterior angles = 360°, for every polygon regardless of the number of sides.
  • For a regular polygon: each exterior angle = 360/n, and each interior angle = 180 − 360/n.
  • Interior + exterior at the same vertex = 180°, since they lie on a straight line.

Choosing the quickest route

For regular polygons, going via the exterior angle is almost always faster. To find the interior angle of a regular decagon: exterior = 360/10 = 36°, so interior = 180 − 36 = 144°.

Think of it like this

The exterior angles summing to 360° is just walking once around the shape: at every corner you turn by the exterior angle, and by the time you are back where you started, facing the same way, you have turned a full circle.

Worked examples

Method, step by step

A regular polygon has an interior angle of 156°. How many sides does it have?

  1. 1Interior and exterior angles at a vertex sum to 180°, so the exterior angle is 180 − 156 = 24°.
  2. 2For a regular polygon, exterior angle = 360/n.
  3. 3So 24 = 360/n, giving n = 360/24 = 15.

15 sides

Common misconceptions

  • Thinking the exterior angle sum depends on the number of sides. It is always 360°.
  • Using `(n − 2) × 180` and forgetting to divide by n when a **single** interior angle of a regular polygon is wanted.
  • Giving a numerical answer with no reason. Angle questions almost always reserve a mark for the correct named rule.

In the exam

  • Write the reason next to every step: "alternate angles are equal", "angles on a straight line sum to 180°". Those phrases are the marks.
  • For regular polygons, find the exterior angle first — it is one division, and the interior angle follows immediately.