NSW Syllabuses

# Mathematics K–10 - Stage 5.3 - Measurement and Geometry Properties of Geometrical Figures §

## Outcomes

#### A student:

• MA5.3-1WM

uses and interprets formal definitions and generalisations when explaining solutions and/or conjectures

• MA5.3-2WM

generalises mathematical ideas and techniques to analyse and solve problems efficiently

• MA5.3-3WM

uses deductive reasoning in presenting arguments and formal proofs

• MA5.3-16MG

proves triangles are similar, and uses formal geometric reasoning to establish properties of triangles and quadrilaterals

## Content

• construct and write geometrical arguments to prove a general geometrical result, giving reasons at each step of the argument, eg prove that the angle in a semicircle is a right angle
• Apply logical reasoning, including the use of congruence and similarity, to proofs and numerical exercises involving plane shapes (ACMMG244)
• write formal proofs of the similarity of triangles in the standard four- or five-line format, preserving the matching order of vertices, identifying the scale factor when appropriate, and drawing relevant conclusions from this similarity
• prove that the interval joining the midpoints of two sides of a triangle is parallel to the third side and half its length, and the converse (Communicating, Problem Solving)
• establish and apply for two similar figures with similarity ratio $$1\!:\!k$$ the following:
• matching angles have the same size
• matching intervals are in the ratio $$1\!:\!k$$
• matching areas are in the ratio $$1\!:\!k^2$$
• matching volumes are in the ratio $$1\!:\!k^3$$
• solve problems involving similarity ratios and areas and volumes (Problem Solving)
• state a definition as the minimum amount of information needed to identify a particular figure
• prove properties of isosceles and equilateral triangles and special quadrilaterals from the formal definitions of the shapes:
• a scalene triangle is a triangle with no two sides equal in length
• an isosceles triangle is a triangle with two sides equal in length
• an equilateral triangle is a triangle with all sides equal in length
• trapezium is a quadrilateral with one pair of opposite sides parallel
• parallelogram is a quadrilateral with both pairs of opposite sides parallel
• rhombus is a parallelogram with two adjacent sides equal in length
• rectangle is a parallelogram with one angle a right angle
• square is a rectangle with two adjacent sides equal
• use dynamic geometry software to investigate and test conjectures about geometrical figures (Problem Solving, Reasoning)
• prove and apply theorems and properties related to triangles and quadrilaterals:
• the sum of the interior angles of a triangle is 180º
• the exterior angle of a triangle is equal to the sum of the two interior opposite angles
• if two sides of a triangle are equal, then the angles opposite those sides are equal; conversely, if two angles of a triangle are equal, then the sides opposite those angles are equal
• each angle of an equilateral triangle is equal to 60º
• the sum of the interior angles of a quadrilateral is 360º
• the opposite angles of a parallelogram are equal
• the opposite sides of a parallelogram are equal
• the diagonals of a parallelogram bisect each other
• the diagonals of a rhombus bisect each other at right angles
• the diagonals of a rhombus bisect the vertex angles through which they pass
• the diagonals of a rectangle are equal
• recognise that any result proven for a parallelogram would also hold for a rectangle (Reasoning)
• give reasons why a square is a rhombus, but a rhombus is not necessarily a square (Communicating, Reasoning)
• use a flow chart or other diagram to show the relationships between different quadrilaterals
(Communicating)
• prove and apply tests for quadrilaterals:
• if both pairs of opposite angles of a quadrilateral are equal, then the quadrilateral is a parallelogram
• if both pairs of opposite sides of a quadrilateral are equal, then the quadrilateral is a parallelogram
• if all sides of a quadrilateral are equal, then the quadrilateral is a rhombus
• solve numerical and non-numerical problems in Euclidean geometry based on known assumptions and proven theorems
• state possible converses of known results, and examine whether or not they are also true (Communicating, Reasoning)

### Background Information

Similarity of triangles is used in Circle Geometry to prove further theorems on intersecting chords, secants and tangents.

Attention should be given to the logical sequence of theorems and to the types of arguments used. The memorisation of proofs is not intended. Every theorem presented could be preceded by a straight-edge-and-compasses construction to confirm the theorem before it is proven in an appropriate manner, by way of an exercise or an investigation.

In Euclidean geometry, congruence is the method used to construct symmetry arguments. It is often helpful to see exactly what transformation, or sequence of transformations, will map one triangle into a congruent triangle. For example, the proof that the opposite sides of a parallelogram are equal involves constructing a diagonal and proving that the resulting triangles are congruent – these two triangles can be transformed into each other by a rotation of 180º about the midpoint of the diagonal.

In the German-born physicist Albert Einstein's (1879–1955) general theory of relativity, three-dimensional space is curved and, as a result, the sum of the angles of a physical triangle of cosmological proportions is not 180º. Abstract geometries of this nature were developed by other mathematicians, including the German Carl Friedrich Gauss (1777–1855), the Hungarian János Bolyai (1802–1860), the Russian Nikolai Lobachevsky (1792–1856), the German Bernhard Riemann (1826–1866) and others, in the nineteenth century, amid suspicions that Euclidean geometry may not be the correct description of physical space.

The Elements, a treatise written by the Greek mathematician Euclid (c325−265 BC), gives an account of geometry written almost entirely as a sequence of axioms, definitions, theorems and proofs. Its methods have had an enormous influence on mathematics. Students could read some of Book 1 of the Elements for a systematic account of the geometry of triangles and quadrilaterals.

### Language

If students abbreviate geometrical reasons that they use in deductive geometry, they must take care not to abbreviate the reasons to such an extent that the meaning is lost.