NSW Syllabuses

Mathematics K–10 - Stage 4 - Measurement and Geometry Angle Relationships

Outcomes

A student:

• MA4-1WM

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

• MA4-2WM

applies appropriate mathematical techniques to solve problems

• MA4-3WM

recognises and explains mathematical relationships using reasoning

• MA4-18MG

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
• recognise that adjacent angles can form right angles, straight angles and angles of revolution (Communicating, Reasoning)
• 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 co-interior 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 problem-solving 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 co-interior 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.

National Numeracy Learning Progression links to this Mathematics outcome

When working towards the outcome MA4‑18MG the sub-elements (and levels) of Number patterns and algebraic thinking (NPA7-NPA8) and Understanding geometric properties (UGP5-UGP6) describe observable behaviours that can aid teachers in making evidence-based decisions about student development and future learning.

The progression sub-elements and indicators can be viewed by accessing the National Numeracy Learning Progression.