Sample Notes: Angles

O Level and IGCSE Mathematics – Detailed Notes
Chapter 4.6: Angles

1. Basic Angle Properties

1.1 Sum of Angles at a Point = 360°

• When multiple angles meet at a single point and form a complete circle, their total is 360°.
• Example:
  If angles at a point are 120°, 90°, and x°, then:
  • 120° + 90° + x° = 360°
  • x = 360° − 210° = 150°

1.2 Sum of Angles on a Straight Line = 180°

• Angles that lie on a straight line and are adjacent (next to each other) form a linear pair, summing up to 180°.
• Example:
  If angle A = 135°, then the angle next to it (angle B) on the straight line:
  • A + B = 180°
  • 135° + B = 180° → B = 45°

1.3 Vertically Opposite Angles are Equal

• When two lines intersect, the angles opposite each other (not adjacent) are equal.
• Example:
  If angle x and angle y are vertically opposite and x = 70°, then:
  • y = 70°

2. Angle Sum in Triangles and Quadrilaterals

2.1 Sum of Angles in a Triangle = 180°

• The three interior angles in any triangle always add up to 180°.
• Example:
  If two angles in a triangle are 40° and 65°, then:
  • x = 180° − 40° − 65° = 75°

2.2 Sum of Angles in a Quadrilateral = 360°

• The four interior angles in any quadrilateral always sum up to 360°.
• Example:
  If three angles in a quadrilateral are 90°, 85°, and 95°, then:
  • x = 360° − (90° + 85° + 95°) = 90°

3. Angles in Parallel Lines

When a pair of parallel lines is intersected by a transversal line, it creates several angle pairs with important relationships.

3.1 Corresponding Angles are Equal

• These are angles in matching corners when a transversal crosses parallel lines.
• Rule: If lines are parallel, corresponding angles are equal.
• Notation: ∠A ≅ ∠E
• Example:
  If one corresponding angle is 112°, then the other is also 112°.

3.2 Alternate Angles are Equal

• These angles are on opposite sides of the transversal but between the two lines.
• Notation: Often called ‘Z angles’.
• Rule: If lines are parallel, alternate angles are equal.
• Example:
  If one alternate angle is 60°, then the other is 60°.

3.3 Co-Interior Angles Sum to 180° (Supplementary)

• Also called consecutive interior angles.
• These are between the two lines and on the same side of the transversal.
• Rule: If lines are parallel, co-interior angles add up to 180°.
• Example:
  If one co-interior angle is 115°, then:
  • x = 180° − 115° = 65°

4. Angle Properties of Polygons

4.1 Interior Angle Sum of a Polygon

• Formula:
  Sum = (n − 2) × 180°, where n = number of sides.
• Example (Hexagon, n = 6):
  (6 − 2) × 180° = 4 × 180° = 720°

4.2 Exterior Angle of a Regular Polygon

• Exterior angles of a regular polygon always add up to 360°.
• Each exterior angle:
  = 360° / n
• Example (Octagon):
  360° / 8 = 45° per exterior angle

4.3 Relationship Between Interior and Exterior Angles

• Interior angle + Exterior angle = 180°
• Example (Regular Pentagon):
  Exterior angle = 360° / 5 = 72°
  Interior angle = 180° − 72° = 108°

5. Naming Angles Using Three-Letter Notation

• Always place the vertex (middle point) of the angle in the center of the three letters.
• Example: If angle is between line AB and line BC, and B is the vertex:
  • Written as ∠ABC or ∠CBA

6. Regular vs Irregular Polygons

6.1 Regular Polygon

• All sides and all angles are equal.
• Each interior and exterior angle can be calculated directly.
• Examples: Square, Equilateral Triangle, Regular Hexagon

6.2 Irregular Polygon

• Sides and/or angles are not equal.
• Must use angle sum and logical reasoning to find unknowns.

7. Examples and Practice Problems

Example 1: Find the missing angle at a point

• Given: ∠A = 110°, ∠B = 90°, ∠C = x°
• Total = 360°
• x = 360° − (110° + 90°) = 160°

Example 2: Triangle Missing Angle

• ∠A = 38°, ∠B = 74°, ∠C = ?
• ∠C = 180° − 38° − 74° = 68°

Example 3: Quadrilateral

• ∠A = 92°, ∠B = 89°, ∠C = 110°, ∠D = ?
• ∠D = 360° − (92° + 89° + 110°) = 69°

Example 4: Parallel Lines

• Corresponding angles: If ∠A = 123°, what is corresponding angle ∠E?
• ∠E = 123°

Example 5: Polygon Interior Angle

• 9-sided polygon:
  • Sum = (9 − 2) × 180° = 1260°
  • Each interior angle = 1260° / 9 = 140°

8. Key Terms and Definitions

9. Summary Checklist

• Know total angle at a point = 360°
• Know straight-line angles = 180°
• Understand triangle and quadrilateral angle sums
• Master parallel line rules: corresponding, alternate, co-interior
• Use polygon formulas for interior and exterior angles
• Apply proper three-letter notation (e.g., ∠ABC)
• Identify regular vs irregular polygons
• Justify answers using geometric properties