Lines, Angles, and Shapes

1. Lines and Angles

Basic Definitions

  • Parallel Lines: Lines in a plane that never meet, no matter how far they extend.
  • Perpendicular Lines: Lines that intersect at a right angle ($90^\circ$).
  • Types of Angles:
    • Acute: $0^\circ < \theta < 90^\circ$
    • Right: $\theta = 90^\circ$
    • Obtuse: $90^\circ < \theta < 180^\circ$
    • Reflex: $180^\circ < \theta < 360^\circ$

Basic Angles Diagram

Angle Relationships

  • Angles on a straight line: Sum to $180^\circ$.
  • Angles around a point: Sum to $360^\circ$.
  • Vertically opposite angles: Are equal.
  • Parallel Line Angles:
    • Corresponding Angles: Equal (F-shape).
    • Alternate Angles: Equal (Z-shape).
    • Co-interior Angles: Sum to $180^\circ$ (C-shape).

Parallel Line Angles Diagram

2. Triangles

Types of Triangles

  • Equilateral: All sides equal, all angles $60^\circ$.
  • Isosceles: Two sides equal, two base angles equal.
  • Scalene: No sides equal, no angles equal.
  • Right-angled: One angle is $90^\circ$.

Triangles Diagram

Properties

  • Angle Sum: The interior angles of any triangle sum to $180^\circ$.
  • Exterior Angle: The exterior angle of a triangle is equal to the sum of the two opposite interior angles.

3. Quadrilaterals

A four-sided polygon. Interior angles sum to $360^\circ$.

Shape Sides Angles Parallel Sides
Square 4 equal 4 right angles Opposite sides parallel
Rectangle Opposite sides equal 4 right angles Opposite sides parallel
Parallelogram Opposite sides equal Opposite angles equal Opposite sides parallel
Rhombus 4 equal Opposite angles equal Opposite sides parallel
Trapezium Varies Varies $\ge 1$ pair parallel
Kite 2 pairs adjacent equal 1 pair opposite equal None required

Quadrilaterals

4. Polygons

  • Regular Polygon: All sides and all interior angles are equal.
  • Irregular Polygon: Sides and angles are not all equal.
  • Interior Angle Sum: $(n - 2) \times 180^\circ$, where $n$ is the number of sides.
  • Interior Angle (Regular): $\frac{(n - 2) \times 180^\circ}{n}$.
  • Exterior Angle Sum: Always $360^\circ$ for any convex polygon.
  • Exterior Angle (Regular): $\frac{360^\circ}{n}$.

5. Circles

Terminology

  • Centre: The fixed point from which all points on the circumference are equidistant.
  • Radius: Distance from centre to circumference.
  • Diameter: Distance across the circle through the centre ($d = 2r$).
  • Circumference: The perimeter of the circle ($C = 2\pi r$).
  • Chord: A line segment joining two points on the circumference.
  • Tangent: A line that touches the circle at exactly one point.
  • Arc: A part of the circumference.
  • Sector: A “pie slice” area bounded by two radii and an arc.
  • Segment: Area bounded by a chord and an arc.

Circle Terminology Diagram

6. Basic Construction

  • Perpendicular Bisector: A line that cuts another line in half at $90^\circ$.
  • Angle Bisector: A line that divides an angle into two equal parts.
  • Triangle Construction: Using a ruler and compass to draw a triangle given three side lengths (SSS).