Angles — GCSE Maths

Why do angles matter?

• Angles appear throughout geometry — in triangles, polygons, parallel lines and more.
• At GCSE, you must know and apply angle rules to find missing angles.
• You will often need to give reasons for your answers — this is important.
• Angles are measured in degrees (°).

Key Words

• Acute angle — Less than 90°
• Obtuse angle — Between 90° and 180°
• Reflex angle — Greater than 180°
• Vertically opposite — Angles formed at the same point when two lines cross — they are equal
• Co-interior angles — Angles inside parallel lines on the same side — they add up to 180°

Basic angle rules

• Angles on a straight line add up to 180°.
• Angles around a point add up to 360°.
• Vertically opposite angles are equal.
• Example: two angles on a straight line are x and 115°. So x = 180 − 115 = 65°.
• Always write the rule as your reason.

Angles in triangles and polygons

• Angles in a triangle add up to 180°.
• Angles in a quadrilateral add up to 360°.
• Sum of interior angles in any polygon = (n − 2) × 180°, where n is the number of sides.
• Each interior angle of a regular polygon = sum ÷ number of sides.
• Example: regular hexagon interior angle = (6−2)×180÷6 = 120°.

Parallel line angle rules

• Alternate angles (Z angles) are equal.
• Corresponding angles (F angles) are equal.
• Co-interior angles (C angles) add up to 180°.
• You must identify which rule applies from the shape of the angle pair.
• Always state the reason: "alternate angles are equal" etc.

Worked Examples

Practice Questions

• Angles on a straight line. One angle is 73°. Find the other.
  Answer: 107°
  180 − 73 = 107°. Angles on a straight line sum to 180°.
• A triangle has angles 55° and 80°. Find the third angle.
  Answer: 45°
  180 − 55 − 80 = 45°.
• Vertically opposite angles are:
  Answer: Equal
  Vertically opposite angles are always equal.
• Interior angle sum of a pentagon?
  Answer: 540°
  (5−2) × 180 = 3 × 180 = 540°.
• Alternate angles are:
  Answer: Equal
  Alternate angles (Z angles) between parallel lines are equal.