NSW Syllabuses

Mathematics K–10 - Stage 4 - Measurement and Geometry Properties of Geometrical Figures

Properties of Geometrical Figures 1

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

• MA4-17MG

classifies, describes and uses the properties of triangles and quadrilaterals, and determines congruent triangles to find unknown side lengths and angles

Related Life Skills outcomes: MALS-30MG, MALS-31MG

Content

• Students:
• Classify triangles according to their side and angle properties and describe quadrilaterals (ACMMG165)
• label and name triangles (eg triangle $$\textit{ABC}$$ or $$\Delta \textit{ABC}$$) and quadrilaterals (eg $$\textit{ABCD}$$) in text and on diagrams
• use the common conventions to mark equal intervals on diagrams
• recognise and classify types of triangles on the basis of their properties (acute-angled triangles, right-angled triangles, obtuse-angled triangles, equilateral triangles, isosceles triangles and scalene triangles)
• recognise that a given triangle may belong to more than one class (Reasoning)
• explain why the longest side of a triangle is always opposite the largest angle (Reasoning)
• explain why the sum of the lengths of two sides of a triangle must be greater than the length of the third side (Communicating, Reasoning)
• sketch and label triangles from a worded or verbal description (Communicating)
• the opposite sides are parallel
• the opposite sides are equal
• the adjacent sides are perpendicular
• the opposite angles are equal
• the diagonals are equal
• the diagonals bisect each other
• the diagonals bisect each other at right angles
• the diagonals bisect the angles of the quadrilateral
• use techniques such as paper folding or measurement, or dynamic geometry software, to investigate the properties of quadrilaterals (Problem Solving, Reasoning)
• sketch and label quadrilaterals from a worded or verbal description (Communicating)
• classify special quadrilaterals on the basis of their properties
• describe a quadrilateral in sufficient detail for it to be sketched (Communicating)
• investigate and determine lines (axes) of symmetry and the order of rotational symmetry of polygons, including the special quadrilaterals
• determine if particular triangles and quadrilaterals have line and/or rotational symmetry (Problem Solving)
• investigate the line and rotational symmetries of circles and of diagrams involving circles, such as a sector or a circle with a marked chord or tangent
• identify line and rotational symmetries in pictures and diagrams, eg artistic and cultural designs
• Demonstrate that the angle sum of a triangle is 180° and use this to find the angle sum of a quadrilateral (ACMMG166)
• justify informally that the interior angle sum of a triangle is 180°, and that any exterior angle equals the sum of the two interior opposite angles
• use dynamic geometry software or other methods to investigate the interior angle sum of a triangle, and the relationship between any exterior angle and the sum of the two interior opposite angles (Reasoning)
• use the angle sum of a triangle to establish that the angle sum of a quadrilateral is 360°
• use the angle sum results for triangles and quadrilaterals to determine unknown angles in triangles and quadrilaterals, giving reasons
• Use the properties of special triangles and quadrilaterals to solve simple numerical problems with appropriate reasoning
• find unknown sides and angles embedded in diagrams, using the properties of special triangles and quadrilaterals, giving reasons
• recognise special types of triangles and quadrilaterals embedded in composite figures or drawn in various orientations (Reasoning)

Background Information

The properties of special quadrilaterals are important in the Measurement and Geometry strand. For example, the perpendicularity of the diagonals of a rhombus and a kite allows a rectangle of twice the size to be constructed around them, leading to formulas for finding area.

In Stage 4, the treatment of triangles and quadrilaterals is still informal, with students consolidating their understanding of different triangles and quadrilaterals and being able to identify them from their properties.

Students who recognise class inclusivity and minimum requirements for definitions may address this Stage 4 content concurrently with content in Stage 5 Properties of Geometrical Figures, where properties of triangles and quadrilaterals are deduced from formal definitions.

Students should give reasons orally and in written form for their findings and answers. For some students, formal setting out could be introduced.

A range of examples of the various triangles and quadrilaterals should be given, including quadrilaterals containing a reflex angle and figures presented in different orientations.

Dynamic geometry software and prepared applets are useful tools for investigating properties of geometrical figures.

When using examples of Aboriginal rock carvings and other Aboriginal art, it is recommended, wherever possible, that local examples be used. Consult with local Aboriginal communities and education consultants for such examples.

Purpose/Relevance of Substrand

In geometry, students study two-dimensional shapes, three-dimensional objects, and position, before moving on to the study and application of angle relationships and the properties of geometrical figures. As the focus moves to relationships and properties, students learn to analyse geometry problems. They develop geometric and deductive reasoning skills, as well as problem-solving skills. Students also develop an understanding that geometry is linked to measurement and is very important in the work of architects, engineers, designers, builders, physicists, land surveyors, etc. However, they also learn that geometry is common and important in everyday situations, including in nature, sports, buildings, astronomy, art, etc.

Language

In Stage 4, students should use full sentences to describe the properties of plane shapes, eg 'The diagonals of a parallelogram bisect each other'. Students may not realise that in this context, the word 'the' implies 'all' and so this should be made explicit. Using the full name of the quadrilateral when describing its properties should assist students in remembering the geometrical properties of each particular shape.

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.

This syllabus uses the phrase 'line(s) of symmetry', although 'axis/axes of symmetry' may also be used.

'Scalene' is derived from the Greek word skalenos, meaning 'uneven'. 'Isosceles' is derived from the Greek words isos, meaning 'equals', and skelos, meaning 'leg'. 'Equilateral' is derived from the Latin words aequus, meaning 'equal', and latus, meaning 'side'. 'Equiangular' is derived from aequus and another Latin word, angulus, meaning 'corner'.

Properties of Geometrical Figures 2

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

• MA4-17MG

classifies, describes and uses the properties of triangles and quadrilaterals, and determines congruent triangles to find unknown side lengths and angles

Related Life Skills outcomes: MALS-30MG, MALS-31MG

Content

• recognise congruent figures in tessellations, art and design work, eg mosaics (Reasoning)
• recognise that area, lengths of matching sides, and angle sizes are preserved in congruent figures (Reasoning)
• match sides and angles of two congruent polygons
• name vertices in matching order when using the symbol $$\equiv$$ in statements regarding congruence
• determine the condition for two circles to be congruent (equal radii)
• investigate the minimum conditions needed, and establish the four tests, for two triangles to be congruent:
• if the three sides of a triangle are respectively equal to the three sides of another triangle, then the two triangles are congruent (SSS test)
• if two sides and the included angle of a triangle are respectively equal to two sides and the included angle of another triangle, then the two triangles are congruent (SAS test)
• if two angles and one side of a triangle are respectively equal to two angles and the matching side of another triangle, then the two triangles are congruent (AAS test)
• if the hypotenuse and a second side of a right-angled triangle are respectively equal to the hypotenuse and a second side of another right-angled triangle, then the two triangles are congruent (RHS test)
• use dynamic geometry software and/or geometrical instruments to investigate what information is needed to show that two triangles are congruent (Problem Solving)
• explain why the angle in the SAS test must be the included angle (Communicating, Reasoning)
• demonstrate that three pairs of equal matching angles is not a sufficient condition for triangles to be congruent (Communicating, Reasoning)
• use the congruency tests to identify a pair of congruent triangles from a selection of three or more triangles or from triangles embedded in a diagram
• Establish properties of quadrilaterals using congruent triangles and angle properties, and solve related numerical problems using reasoning (ACMMG202)
• apply the properties of congruent triangles to find an unknown side and/or angle in a diagram, giving a reason
• use transformations of congruent triangles to verify some of the properties of special quadrilaterals, including properties of the diagonals, eg the diagonals of a parallelogram bisect each other

Background Information

For some students, formal setting out of proofs of congruent triangles could be introduced.

Dynamic geometry software and prepared applets are useful tools for investigating properties of congruent figures through transformations.

Congruent figures are embedded in a variety of designs, eg tapa cloth, Aboriginal designs, Indonesian ikat designs, Islamic designs, designs used in ancient Egypt and Persia, window lattice, woven mats and baskets. Computer drawing programs enable students to prepare tessellation designs and to compare these with other designs, such as those of the Dutch artist M C Escher (1898–1972).

Language

The meaning of the term 'included angle' should be taught explicitly. Similarly, the use of the adjective 'matching' when referring to the sides and angles of congruent shapes should be made explicit.

The term 'corresponding' is often used in relation to congruent and similar figures to refer to angles or sides in the same position, but it also has a specific meaning when used to describe a pair of angles in relation to lines cut by a transversal. This syllabus has used 'matching' to describe angles and sides in the same position; however, the use of the word 'corresponding' is not incorrect.

The term 'superimpose' is used to describe the placement of one figure upon another in such a way that the parts of one coincide with the parts of the other.