1 Angle Relationships
Basic Angle Pairs
| Pair | Definition | Sum |
|---|---|---|
| Complementary | Two angles that together form a right angle | 90° |
| Supplementary | Two angles that together form a straight line | 180° |
| Vertical angles | Opposite angles formed when two lines intersect | Equal to each other |
| Angles on a line | All angles on one side of a straight line | 180° |
| Angles at a point | All angles around a single point | 360° |
Parallel Lines with a Transversal
When a transversal crosses two parallel lines, special angle pairs are formed:
| Pair name | Position | Relationship |
|---|---|---|
| Corresponding angles | Same position at each intersection (F-shape) | Equal |
| Alternate interior angles | Between the lines, on opposite sides (Z-shape) | Equal |
| Co-interior (same-side interior) | Between the lines, on the same side (C-shape) | Sum = 180° |
🔑Remember: F = equal, Z = equal, C = 180°
2 Triangles
Triangle Angle Sum
Interior angles
angle A + angle B + angle C = 180°
Types of Triangles by Sides
| Type | Sides | Angles |
|---|---|---|
| Equilateral | All 3 sides equal | All 3 angles = 60° |
| Isosceles | 2 sides equal | 2 base angles equal |
| Scalene | No sides equal | No angles equal |
Types of Triangles by Angles
| Type | Largest angle |
|---|---|
| Acute | All angles < 90° |
| Right | One angle = 90° exactly |
| Obtuse | One angle > 90° |
Exterior Angle Theorem
Exterior angle
exterior angle = sum of the two non-adjacent interior angles
✏️Triangle with angles 40° and 65°. Exterior angle at 3rd vertex = 40° + 65° = 105°
3 Quadrilaterals
Sum of angles
sum of interior angles of a quadrilateral = 360°
| Shape | Key properties |
|---|---|
| Square | 4 equal sides · 4 right angles · diagonals bisect at 90° |
| Rectangle | Opposite sides equal · 4 right angles · diagonals equal |
| Parallelogram | Opposite sides parallel & equal · opposite angles equal |
| Rhombus | 4 equal sides · opposite angles equal · diagonals bisect at 90° |
| Trapezoid | Exactly 1 pair of parallel sides |
4 Perimeter & Area
Perimeter
| Shape | Formula |
|---|---|
| Square | P = 4s |
| Rectangle | P = 2l + 2w = 2(l + w) |
| Triangle | P = a + b + c |
| Circle (circumference) | C = 2πr = πd |
Area
Square
A = s²
Rectangle
A = l × w
Triangle
A = (base × height) / 2
Parallelogram
A = base × height
Trapezoid
A = ((b₁ + b₂) / 2) × height
Circle
A = πr²
⚠️The height of a triangle or parallelogram must be perpendicular to the base — not the slant side.
5 Volume & Surface Area
Volume
Rectangular prism
V = l × w × h
Cylinder
V = πr²h
Triangular prism
V = (base area) × height
Pyramid / cone
V = (1/3) × base area × height
Sphere
V = (4/3)πr³
Surface Area
Rectangular prism
SA = 2(lw + lh + wh)
Cylinder
SA = 2πr² + 2πrh
Sphere
SA = 4πr²
💡Surface area = sum of the areas of all faces. Draw a net (unfolded shape) to help visualise all faces.
6 Pythagorean Theorem
In a right-angled triangle, the square of the hypotenuse (longest side, opposite the right angle) equals the sum of squares of the other two sides.
Pythagorean theorem
a² + b² = c² (c = hypotenuse)
✏️
Find the hypotenuse: legs = 6 cm and 8 cm
c² = 6² + 8² = 36 + 64 = 100
c = √100 = 10 cm
c² = 6² + 8² = 36 + 64 = 100
c = √100 = 10 cm
✏️
Find a leg: hypotenuse = 13 cm, one leg = 5 cm
5² + b² = 13²
25 + b² = 169
b² = 144
b = √144 = 12 cm
5² + b² = 13²
25 + b² = 169
b² = 144
b = √144 = 12 cm
Pythagorean Triples (memorise these)
| Triple | Scaled versions |
|---|---|
| 3, 4, 5 | 6–8–10, 9–12–15, 15–20–25 |
| 5, 12, 13 | 10–24–26 |
| 8, 15, 17 | — |
🔑Only applies to right triangles. If a² + b² = c² holds, then the triangle has a right angle at C.
7 Common Mistakes to Avoid
| Mistake | What to do instead |
|---|---|
| Angles in a triangle sum to 360° | Triangle angles sum to 180°. Quadrilateral angles sum to 360°. |
| Using slant height as the height in area | Height must be perpendicular to the base. Draw the altitude. |
| Adding sides to find area | Area uses base × height (or formula). Perimeter is adding sides. |
| Pythagorean theorem on non-right triangles | a² + b² = c² only works when there is a 90° angle. Check first. |
| Identifying the hypotenuse wrong | Hypotenuse is always opposite the right angle and is the longest side. |
| Forgetting π in circle formulas | Circumference = 2πr and Area = πr². Always include π ≈ 3.14159. |
| Surface area vs volume confusion | Surface area is in square units (cm²). Volume is in cubic units (cm³). |