Sheet 4 of 10 · Maths
Angles and shapes
Angle facts, polygons, triangles, quadrilaterals and symmetry. 12 rules, each with a worked example.
Angle facts
Angles on a straight line
a + b equals 180°
Examplea = 65°, so b = 180° − 65° = 115°Angles around a point
a + b + c + d equals 360°
Example100° + 80° + 120° + d = 360°, so d = 60°Vertically opposite angles
Vertically opposite angles are equal
- Where two straight lines cross, the angles opposite each other match
ExampleOne angle is 40°: the opposite one is 40°, the other two are 140° eachAngles in a triangle
a + b + c equals 180°
Example50° + 60° + c = 180°, so c = 70°Angles in a quadrilateral
a + b + c + d equals 360°
Example90° + 90° + 110° + d = 360°, so d = 70°
Polygons
Interior angle sum
Sum of interior angles equals (n − 2) × 180°
- n = number of sides
ExamplePentagon: (5 − 2) × 180° = 540°Regular polygon: each interior angle
Each interior angle equals sum ÷ n
ExampleRegular hexagon: (6 − 2) × 180° = 720°, and 720° ÷ 6 = 120°Exterior angles
Exterior angles always add up to 360°
- Interior + exterior = 180° at each corner
ExampleRegular octagon: 360° ÷ 8 = 45° exterior, so 180° − 45° = 135° interior
Types and properties
Types of angle
Acute < 90° · Right equals 90° · Obtuse 90°–180° · Reflex > 180°
- Obtuse: more than 90° but less than 180° · Reflex: more than 180° but less than 360°
Types of triangle
Equilateral: 3 equal sides, all 60° · Isosceles: 2 equal sides, 2 equal angles · Scalene: no equal sides
- Right-angled: one angle is 90°
ExampleIsosceles, top angle 40°: base angles = (180° − 40°) ÷ 2 = 70° eachQuadrilaterals
Square, rectangle, rhombus, parallelogram, trapezium, kite
- Square: 4 equal sides, 4 right angles · Rectangle: opposite sides equal, 4 right angles
- Rhombus: 4 equal sides, opposite angles equal · Parallelogram: opposite sides equal and parallel
- Trapezium: one pair of parallel sides · Kite: two pairs of equal sides next to each other
Lines of symmetry
A regular polygon with n sides has n lines of symmetry
ExampleSquare 4 · Rectangle 2 · Equilateral triangle 3 · Isosceles triangle 1 · Kite 1Tip: A parallelogram that isn't a rectangle or rhombus has no lines of symmetry.
Want the full explanation?
The study guide goes through each rule step by step, with more examples.