Welcome to Geometry: Understanding the Language of Shapes!

Hi there! Before we start calculating areas or solving complicated angle problems, we need to master the basic language of Geometry. Think of this chapter as learning the alphabet for a new language. If you know these terms, reading diagrams and answering geometry questions becomes so much easier.

Don't worry if geometry feels tricky; we will break down every term so you can build a strong foundation. Let's get started!

1. The Absolute Basics: Points, Lines, and Angles

The foundation of all shapes starts with these simple concepts.

Key Definitions of Basic Elements
  • Point: A specific location in space. It has no size or dimensions—we usually represent it with a dot.
    Analogy: The tiny dot marking your current location on a map.
  • Line: A straight path extending infinitely in both directions. In IGCSE diagrams, when we refer to a "line," we often mean a line segment (a straight section between two points).
  • Vertex (plural: Vertices): A corner where two or more lines or edges meet.
Understanding Angles (The Size of a Turn)

Angles are measured in degrees (°). You must be able to recognize and define the following types of angles:

  • Right Angle: Exactly \(90^\circ\). Often shown with a small square symbol in the corner.
  • Acute Angle: Less than \(90^\circ\) (\(0^\circ < \text{angle} < 90^\circ\)). Think: A cute, small angle.
  • Obtuse Angle: Greater than \(90^\circ\) but less than \(180^\circ\) (\(90^\circ < \text{angle} < 180^\circ\)). Think: It's obese, or wide.
  • Reflex Angle: Greater than \(180^\circ\) but less than \(360^\circ\) (\(180^\circ < \text{angle} < 360^\circ\)). This is the large angle around the outside of an acute or obtuse angle.

Did You Know?
The sum of angles at a point is \(360^\circ\). The sum of angles on a straight line is \(180^\circ\).

Key Takeaway: Points, lines, and angles are the building blocks. Always check the type of angle, as this dictates which angle rules you should apply.

2. Geometrical Relationships

These terms describe how lines and shapes interact with each other.

Parallel and Perpendicular Lines
  • Parallel: Lines that are always the same distance apart and will never meet. They often have arrows marked on them in diagrams.
    Analogy: Train tracks or the opposite edges of a ruler.
  • Perpendicular: Lines that intersect (cross) to form a right angle (\(90^\circ\)).

*Extended Content Term (Good to Know):* A Perpendicular Bisector is a line that cuts another line segment into two equal halves AND crosses it at \(90^\circ\).

Similarity and Congruence

These terms describe the relationship between two different shapes.

  1. Congruent: Shapes that are exactly the same size and shape. If you can pick one up and place it perfectly on top of the other, they are congruent.
    Analogy: Identical twins.
  2. Similar: Shapes that have the same shape, but different sizes. Their corresponding angles are equal, but their sides are in proportion.
    Analogy: A photograph and an enlargement of the same photo.

The relationship between the side lengths of similar shapes is described by the Scale Factor. If a shape is similar to Shape A and its sides are twice as long, the scale factor is 2.

3. Triangles, Quadrilaterals, and Polygons (2D Shapes)

A Polygon is any closed 2D shape made up of three or more straight sides.

Vocabulary of Triangles (3 Sides)

The sum of the interior angles in any triangle is always \(180^\circ\).

  • Equilateral: All three sides are equal in length, and all three angles are equal (\(60^\circ\)).
  • Isosceles: Two sides are equal in length, and the two angles opposite those sides (base angles) are equal.
  • Scalene: No sides are equal, and no angles are equal.
  • Right-angled: Contains one right angle (\(90^\circ\)).
Vocabulary of Special Quadrilaterals (4 Sides)

The sum of the interior angles in any quadrilateral is always \(360^\circ\).

Quick Review Box: The 6 Special Quadrilaterals

  1. Square: 4 equal sides, 4 right angles.
  2. Rectangle: Opposite sides equal, 4 right angles.
  3. Rhombus: 4 equal sides (like a squashed square), opposite angles equal.
  4. Parallelogram: Opposite sides are parallel and equal, opposite angles are equal.
  5. Trapezium: Has exactly one pair of parallel sides.
  6. Kite: Two distinct pairs of adjacent sides are equal.
General Polygons

When solving problems, you must know the names of polygons based on their number of sides:

  • 5 sides: Pentagon
  • 6 sides: Hexagon
  • 8 sides: Octagon
  • 10 sides: Decagon

A polygon is Regular if all its sides are equal and all its interior angles are equal. Otherwise, it is Irregular.

Key Takeaway: Knowing the properties (like "all sides are equal" for a rhombus) is essential, especially when calculating interior or exterior angles.

4. Introduction to 3D Shapes (Simple Solids)

When we move from 2D (flat shapes) to 3D, we use new terms to describe the parts.

  • Face (or Surface): Any single flat side of a solid object. (Think of the sides of a box).
  • Edge: The line segment where two faces meet.
  • Vertex: The corner where three or more edges meet.
Common Solids You Need to Know
  • Cube: A solid with 6 square faces, 12 edges, and 8 vertices.
  • Cuboid: A box shape with 6 rectangular faces.
  • Prism: A solid shape with the same cross-section running all the way through (e.g., a triangular prism has a triangle at both ends).
  • Cylinder: A solid shape with two circular ends and one curved surface.
  • Pyramid: A shape with a polygon base and triangular faces that meet at a single point (apex).
  • Cone: A shape with a circular base leading up to a single point (apex).
  • Sphere: A perfectly round three-dimensional object (like a ball).

Key Takeaway: When working with 3D shapes, be clear whether the question asks for the volume (space inside), the surface area (total area of all faces), or dimensions (length of an edge).

5. Vocabulary of a Circle

The circle has its own set of unique terms you need to memorize.

  • Centre: The exact middle point of the circle.
  • Radius (r): The distance from the centre to any point on the edge. (Plural: Radii).
  • Diameter (d): A straight line passing through the centre, connecting two points on the edge. It is always twice the radius: \(d = 2r\).
  • Circumference: The perimeter or distance around the edge of the circle.
  • Semicircle: Exactly half of a circle.
  • Chord: A straight line connecting two points on the circumference (it does NOT have to pass through the centre).
  • Tangent: A straight line that touches the circle at only one point.
  • Arc: A curved part of the circumference. (Can be minor arc or major arc.)
  • Sector: The area bounded by two radii and an arc. (Think: A slice of pizza or pie).
  • Segment: The area bounded by a chord and an arc. (Think: The crusty part left when you slice the chord off the pizza).

Common Mistake to Avoid: Confusing a Sector (radii) with a Segment (chord).

6. Bearings and Direction

Bearing is a crucial term used in navigation and direction problems. It measures direction, and there are three golden rules:

  1. It must be measured from North.
  2. It must be measured in a clockwise direction.
  3. It must always be written using three figures (000° to 360°).

Example: If the bearing is \(45^\circ\), you write it as \(045^\circ\).

You also need familiarity with cardinal directions:

  • North (N): \(000^\circ\) or \(360^\circ\)
  • East (E): \(090^\circ\)
  • South (S): \(180^\circ\)
  • West (W): \(270^\circ\)

Memory Aid: Always start at North and turn Clockwise (N.C.T. - North, Clockwise, Three figures).

Key Takeaway: Bearings are just angles used in a very specific context. They rely heavily on your knowledge of parallel lines (since the North lines at two different locations are parallel).