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Glossary
Chapter 1: Number systems and arithmetic
- Overview
- 1.1 The number systems ℕ, ℤ, ℚ and ℝ
- 1.2 Irrational numbers and surds
- 1.3 Geometric construction of √2 and √3
- 1.4 Absolute value
- 1.5 Indices and the rules of indices
- 1.6 Solving problems with indices
- 1.7 Scientific notation and accuracy
- Review
Chapter 2: Arithmetic in context
- Overview
- 2.1 Percentages and percentage change
- 2.2 Profit, loss and mark-up
- 2.3 Income tax, USC and net pay
- 2.4 Simple and compound interest
- 2.5 Present value
- 2.6 Checking whether a result is reasonable
- Review
Chapter 3: Length, area and volume
- Overview
- 3.1 Perimeter and area of plane figures
- 3.2 The circle: circumference and area
- 3.3 Volume of prisms and cylinders
- 3.4 Nets of prisms, cylinders and spheres
- 3.5 Surface area and curved surface area
- 3.6 Similar figures: length, area and volume ratios
- 3.7 The volume and surface area of a sphere
- Review
Chapter 4: Algebra: expressions and equations
- Overview
- 4.1 Generating expressions from patterns
- 4.2 Representing relations in tables and graphs
- 4.3 Finding a formula from a pattern
- 4.4 Features of linear relationships
- 4.5 Evaluating and simplifying expressions
- 4.6 Solving linear equations
- 4.7 Simultaneous equations in two variables
- 4.8 Solving quadratic equations
- 4.9 Linear inequalities
- 4.10 Inequalities and the modulus
- Review
Chapter 5: Complex numbers
- Overview
- 5.1 The imaginary unit and complex numbers
- 5.2 Arithmetic with complex numbers
- 5.3 The complex conjugate and division
- 5.4 The Argand diagram and modulus
- 5.5 Cubic equations and the conjugate root theorem
- Review
Chapter 6: Functions
- Overview
- 6.1 Relations and functions
- 6.2 Domain, range and notation
- 6.3 Composite and inverse functions
- 6.4 Injective, surjective and bijective functions
- 6.5 Graphing functions and transformations
- 6.6 Exponential and logarithmic functions
- Review
Chapter 7: Differentiation
- Overview
- 7.1 The derivative from first principles
- 7.2 Differentiating powers and polynomials
- 7.3 The product rule
- 7.4 The quotient rule
- 7.5 The chain rule
- 7.6 The second derivative and concavity
- 7.7 Rates of change in context
- 7.8 Stationary points, maxima and minima
- 7.9 Curve sketching
- 7.10 Differentiating exponential and trigonometric functions
- Review
Chapter 8: Integration
- Overview
- 8.1 Antiderivatives
- 8.2 The definite integral and area
- 8.3 The fundamental theorem of calculus
- 8.4 Area between a curve and the x-axis
- 8.5 Integration by substitution
- 8.6 Average value and accumulated change
- Review
Chapter 9: Sequences and series
- Overview
- 9.1 Arithmetic sequences
- 9.2 Arithmetic series
- 9.3 Geometric sequences and series
- 9.4 Infinite geometric series
- 9.5 Recurrence relations
- 9.6 Proof by induction
- Review
Chapter 10: Counting and probability
- Overview
- 10.1 Counting outcomes and the fundamental principle
- 10.2 Permutations
- 10.3 Combinations
- 10.4 Sample spaces and set theory
- 10.5 Axioms of probability
- 10.6 Conditional probability
- 10.7 Independent events
- 10.8 Bernoulli trials and the binomial distribution
- Review
Chapter 11: Statistics
- Overview
- 11.1 Populations and samples
- 11.2 Sampling methods and bias
- 11.3 Representing data: charts and histograms
- 11.4 Measures of centre and spread
- 11.5 The standard deviation
- 11.6 The normal distribution
- 11.7 Sampling variability and the margin of error
- 11.8 The central limit theorem
- 11.9 Hypothesis testing and confidence intervals
- 11.10 Correlation and regression
- 11.11 Statistical reasoning and misused statistics
- Review
Chapter 12: Trigonometry
- Overview
- 12.1 The trigonometric ratios in a right triangle
- 12.2 Pythagoras' theorem
- 12.3 Radians and degrees
- 12.4 The unit circle and trigonometric functions of any angle
- 12.5 Graphs of the trigonometric functions
- 12.6 The sine and cosine rules
- 12.7 Area of a triangle
- 12.8 Trigonometric identities
- 12.9 Solving trigonometric equations
- 12.10 Bearings and three-dimensional problems
- Review
Chapter 13: Coordinate geometry
- Overview
- 13.1 Distance, midpoint and slope
- 13.2 The equation of a line
- 13.3 Parallel and perpendicular lines
- 13.4 The equation of a circle
- 13.5 Tangents to a circle
- 13.6 Intersection of a line and a circle
- 13.7 Dividing a line segment in a given ratio
- Review
Chapter 14: Transformation geometry
- Overview
- 14.1 Translations
- 14.2 Axial symmetry and reflection
- 14.3 Central symmetry and rotation
- 14.4 Enlargements and the scale factor
- 14.5 Transformations and symmetry in the plane
- 14.6 Composing transformations
- Review
Chapter 15: Synthetic geometry: foundations
- Overview
- 15.1 Axioms and the rules of deduction
- 15.2 The first theorems on angles
- 15.3 Congruent triangles
- 15.4 Parallel lines
- 15.5 The isosceles triangle and its converse
- 15.6 Perpendicular lines and right angles
- 15.7 Constructions 1–6
- 15.8 Writing a formal proof
- Review
Chapter 16: Synthetic geometry: quadrilaterals, similarity and area
- Overview
- 16.1 Quadrilaterals and parallelograms
- 16.2 Corollaries of the parallelogram theorems
- 16.3 Similar triangles
- 16.4 Proportionality and ratios in triangles
- 16.5 Pythagoras' theorem proved
- 16.6 Area and the area of a triangle
- 16.7 Constructions 7–15
- Review
Chapter 17: Synthetic geometry: circles
- Overview
- 17.1 Chords and the centre
- 17.2 The angle in a semicircle and related theorems
- 17.3 Cyclic quadrilaterals
- 17.4 Tangents to a circle
- 17.5 The special points of a triangle
- 17.6 Constructions 16–22
- 17.7 Proving a circle theorem
- 17.8 Enlargements and similar triangles in the circle
- Review
Chapter 18: Vectors
- Overview
- 18.1 Vectors and their representation
- 18.2 Vector arithmetic
- 18.3 The dot product
- 18.4 Perpendicularity and the angle between vectors
- 18.5 Vectors in three dimensions
- 18.6 Geometric applications of vectors
- Review
About this book

Chapter 15: Synthetic geometry: foundations

The axioms, the first theorems, angles, congruent triangles and parallels — the beginning of the shared geometry course every paper draws on.

The chapter problem

A proof you can build on. You will prove that the angles of a triangle add to 180°, using only what comes before it. Every later theorem in this course will rest on results like this one, and by the end of the chapter you can say exactly what it rests on.

Where it is used: Every geometrical fact in the course, and the reasoning behind it.

Your turn to connect it: Which geometrical fact do you rely on most without knowing why it is true?

Sections

15.1 Axioms and the rules of deduction — tier O · refs 2.1
15.2 The first theorems on angles — tier O · refs 2.1
15.3 Congruent triangles — tier O · refs 2.1
15.4 Parallel lines — tier O · refs 2.1
15.5 The isosceles triangle and its converse — tier O · refs 2.1
15.6 Perpendicular lines and right angles — tier O · refs 2.1
15.7 Constructions 1–6 — tier O · refs 2.1
15.8 Writing a formal proof — tier O · refs 2.1