Outcomes
A student:

 MA41WM
communicates and connects mathematical ideas using appropriate terminology, diagrams and symbols

 MA42WM
applies appropriate mathematical techniques to solve problems

 MA43WM
recognises and explains mathematical relationships using reasoning

 MA418MG
identifies and uses angle relationships, including those related to transversals on sets of parallel lines
Content
 Students:
 Use the language, notation and conventions of geometry
 define, label and name points, lines and intervals using capital letters
 label the vertex and arms of an angle with capital letters
 label and name angles using \( \angle P\) or \(\angle QPR\) notation
 use the common conventions to indicate right angles and equal angles on diagrams
 Recognise the geometrical properties of angles at a point
 use the terms 'complementary' and 'supplementary' for angles adding to 90° and 180°, respectively, and the associated terms 'complement' and 'supplement'
 use the term 'adjacent angles' to describe a pair of angles with a common arm and a common vertex, and lie on opposite sides of the common arm
 identify and name right angles, straight angles, angles of complete revolution and vertically opposite angles embedded in diagrams
 recognise that adjacent angles can form right angles, straight angles and angles of revolution (Communicating, Reasoning)
 Identify corresponding, alternate and cointerior angles when two straight lines are crossed by a transversal (ACMMG163)
 identify and name perpendicular lines using the symbol for 'is perpendicular to' (\(\bot\)), eg \(AB \bot CD\)
 use the common conventions to indicate parallel lines on diagrams
 identify and name pairs of parallel lines using the symbol for 'is parallel to' \(\left( \,\parallel\, \right)\), eg \(PQ\parallel RS\)
 define and identify 'transversals', including transversals of parallel lines
 identify, name and measure alternate angle pairs, corresponding angle pairs and cointerior angle pairs for two lines cut by a transversal
 use dynamic geometry software to investigate angle relationships formed by parallel lines and a transversal (Problem Solving, Reasoning)
 recognise the equal and supplementary angles formed when a pair of parallel lines is cut by a transversal
 Investigate conditions for two lines to be parallel (ACMMG164)
 use angle properties to identify parallel lines
 explain why two lines are either parallel or not parallel, giving a reason (Communicating, Reasoning)
 Solve simple numerical problems using reasoning (ACMMG164)
 find the sizes of unknown angles embedded in diagrams using angle relationships, including angles at a point and angles associated with parallel lines, giving reasons
 explain how the size of an unknown angle was calculated (Communicating, Reasoning)
Background Information
Dynamic geometry software and prepared applets are useful tools for investigating angle relationships; angles and lines can be dragged to new positions while angle measurements update automatically.
Students could explore the results relating to angles associated with parallel lines cut by a transversal by starting with corresponding angles and moving one vertex and all four angles to the other vertex by a translation. The other two results then follow, using vertically opposite angles and angles on a straight line. Alternatively, the equality of the alternate angles can be seen by rotation about the midpoint of the transversal.
Students should give reasons when finding the sizes of unknown angles. For some students, formal setting out could be introduced. For example,
\(\angle ABQ = 70^{\circ} \) (corresponding angles, \(AC \parallel PR\)).
In his calculation of the circumference of the Earth, the Greek mathematician, geographer and astronomer Eratosthenes (c276–c194 BC) used parallel line results.
Purpose/Relevance of Substrand
The development of knowledge and understanding of angle relationships, including the associated terminology, notation and conventions, is of fundamental importance in developing an appropriate level of knowledge, skills and understanding in geometry. Angle relationships and their application play an integral role in students learning to analyse geometry problems and developing geometric and deductive reasoning skills, as well as problemsolving skills. Angle relationships are key to the geometry that is important in the work of architects, engineers, designers, builders, physicists, land surveyors, etc, as well as the geometry that is common and important in everyday situations, such as in nature, sports, buildings, astronomy, art, etc.
Language
Students in Stage 4 should write geometrical reasons without the use of abbreviations to assist them in learning new terminology, and in understanding and retaining geometrical concepts, eg 'When a transversal cuts parallel lines, the cointerior angles formed are supplementary'.
Some students may find the use of the terms 'complementary' and 'supplementary' (adjectives) and 'complement' and 'supplement' (nouns) difficult. Teachers should model the use of these terms in sentences, both verbally and in written form, eg, '50° and 40° are complementary angles', 'The complement of 50° is 40°'.
Students should be aware that complementary and supplementary angles may or may not be adjacent.