Angles (Copy)
Cheat Sheet: Calculating Angles and Geometrical Reasoning (IGCSE Mathematics 0580 – CORE)
Topic: Angles at a Point, Straight Lines, Triangles, Quadrilaterals, Parallel Lines, and Polygons
1. Basic Angle Properties
- Sum of angles at a point = 360°
- Sum of angles on a straight line = 180°
- Vertically opposite angles are equal
- Angle sum of a triangle = 180°
- Angle sum of a quadrilateral = 360°
2. Angles in Parallel Lines
| Property | Explanation |
|---|---|
| Corresponding angles are equal | Same side of transversal, same position (e.g., both above parallel lines) |
| Alternate angles are equal | “Z” shape angles are equal |
| Co-interior angles are supplementary | “C” shape angles add up to 180° |
3. Properties of Triangles
| Type of Triangle | Properties |
|---|---|
| Equilateral Triangle | All angles = 60° |
| Isosceles Triangle | Two equal sides and two equal angles |
| Scalene Triangle | No equal sides, no equal angles |
| Right-angled Triangle | One angle = 90° |
4. Properties of Quadrilaterals
| Shape | Properties |
|---|---|
| Square | 4 right angles, all sides equal |
| Rectangle | 4 right angles, opposite sides equal |
| Rhombus | Opposite sides equal, opposite angles equal |
| Parallelogram | Opposite sides parallel and equal, opposite angles equal |
| Trapezium | One pair of parallel sides |
5. Angles in Polygons
- Sum of interior angles of an n-sided polygon = (n – 2) × 180°
- Exterior angles:
- The sum of exterior angles of any polygon = 360°.
- Each exterior angle of a regular polygon = 360° ÷ number of sides.
Key Formulas:
- Interior angle of regular polygon = (n – 2) × 180° ÷ n
- Exterior angle of regular polygon = 360° ÷ n
6. Three-Letter Notation for Angles
- Example: angle ABC means the angle at point B between points A and C.
7. Common Geometric Reasons
| Situation | Reason |
|---|---|
| Two angles vertically opposite | Vertically opposite angles are equal |
| Angles on a straight line | Sum to 180° |
| Angles around a point | Sum to 360° |
| Corresponding angles | Equal because of parallel lines |
| Alternate angles | Equal because of parallel lines |
| Co-interior angles | Add up to 180° because of parallel lines |
| Triangle angle sum | 180° |
| Quadrilateral angle sum | 360° |
| Exterior angles of polygon | Add up to 360° |
This cheat sheet covers all CORE-level facts you need to calculate unknown angles and give proper geometrical explanations.
Cheat Sheet: Calculating Angles and Geometrical Reasoning (IGCSE Mathematics 0580 – CORE)
Topic: Angles at a Point, Straight Lines, Triangles, Quadrilaterals, Parallel Lines, and Polygons
1. Basic Angle Properties
- Sum of angles at a point = 360°
- Sum of angles on a straight line = 180°
- Vertically opposite angles are equal
- Angle sum of a triangle = 180°
- Angle sum of a quadrilateral = 360°
2. Angles in Parallel Lines
| Property | Explanation |
|---|---|
| Corresponding angles are equal | Same side of transversal, same position (e.g., both above parallel lines) |
| Alternate angles are equal | “Z” shape angles are equal |
| Co-interior angles are supplementary | “C” shape angles add up to 180° |
3. Properties of Triangles
| Type of Triangle | Properties |
|---|---|
| Equilateral Triangle | All angles = 60° |
| Isosceles Triangle | Two equal sides and two equal angles |
| Scalene Triangle | No equal sides, no equal angles |
| Right-angled Triangle | One angle = 90° |
4. Properties of Quadrilaterals
| Shape | Properties |
|---|---|
| Square | 4 right angles, all sides equal |
| Rectangle | 4 right angles, opposite sides equal |
| Rhombus | Opposite sides equal, opposite angles equal |
| Parallelogram | Opposite sides parallel and equal, opposite angles equal |
| Trapezium | One pair of parallel sides |
5. Angles in Polygons
- Sum of interior angles of an n-sided polygon = (n – 2) × 180°
- Exterior angles:
- The sum of exterior angles of any polygon = 360°.
- Each exterior angle of a regular polygon = 360° ÷ number of sides.
Key Formulas:
- Interior angle of regular polygon = (n – 2) × 180° ÷ n
- Exterior angle of regular polygon = 360° ÷ n
6. Three-Letter Notation for Angles
- Example: angle ABC means the angle at point B between points A and C.
7. Common Geometric Reasons
| Situation | Reason |
|---|---|
| Two angles vertically opposite | Vertically opposite angles are equal |
| Angles on a straight line | Sum to 180° |
| Angles around a point | Sum to 360° |
| Corresponding angles | Equal because of parallel lines |
| Alternate angles | Equal because of parallel lines |
| Co-interior angles | Add up to 180° because of parallel lines |
| Triangle angle sum | 180° |
| Quadrilateral angle sum | 360° |
| Exterior angles of polygon | Add up to 360° |
This cheat sheet covers all CORE-level facts you need to calculate unknown angles and give proper geometrical explanations.