NSW Syllabuses

# Mathematics K–10 - Stage 5.2 - Measurement and Geometry Properties of Geometrical Figures

## Outcomes

#### A student:

• MA5.2-1WM

selects appropriate notations and conventions to communicate mathematical ideas and solutions

• MA5.2-2WM

interprets mathematical or real-life situations, systematically applying appropriate strategies to solve problems

• MA5.2-3WM

constructs arguments to prove and justify results

• MA5.2-14MG

calculates the angle sum of any polygon and uses minimum conditions to prove triangles are congruent or similar

## Content

• write formal proofs of the congruence of triangles, preserving matching order of vertices
• apply congruent triangle results to prove properties of isosceles and equilateral triangles:
• if two sides of a triangle are equal in length, then the angles opposite the equal sides are equal
• conversely, if two angles of a triangle are equal, then the sides opposite those angles are equal
• if the three sides of a triangle are equal, then each interior angle is 60°
• use the congruence of triangles to prove properties of the special quadrilaterals, such as:
• the opposite angles of a parallelogram are equal
• the diagonals of a parallelogram bisect each other
• the diagonals of a rectangle are equal
• investigate the minimum conditions needed, and establish the four tests, for two triangles to be similar:
• if the three sides of a triangle are proportional to the three sides of another triangle, then the two triangles are similar
• if two sides of a triangle are proportional to two sides of another triangle, and the included angles are equal, then the two triangles are similar
• if two angles of a triangle are equal to two angles of another triangle, then the two triangles are similar
• if the hypotenuse and a second side of a right-angled triangle are proportional to the hypotenuse and a second side of another right-angled triangle, then the two triangles are similar
• explain why the remaining (third) angles must also be equal if two angles of a triangle are equal to two angles of another triangle (Communicating, Reasoning)
• determine whether two triangles are similar using an appropriate test
• Apply logical reasoning, including the use of congruence and similarity, to proofs and numerical exercises involving plane shapes (ACMMG244)
• apply geometrical facts, properties and relationships to find the sizes of unknown sides and angles of plane shapes in diagrams, providing appropriate reasons
• recognise that more than one method of solution is possible (Reasoning)
• compare different solutions for the same problem to determine the most efficient method (Communicating, Reasoning)
• apply the properties of congruent and similar triangles, justifying the results (Communicating, Reasoning)
• apply simple deductive reasoning to prove results for plane shapes
• define the exterior angle of a convex polygon
• establish that the sum of the exterior angles of any convex polygon is 360º
• use dynamic geometry software to investigate the constancy of the exterior angle sum of polygons for different polygons (Reasoning)
• apply the result for the interior angle sum of a triangle to find, by dissection, the interior angle sum of polygons with more than three sides
• use dynamic geometry software to investigate the interior angle sum of different polygons (Reasoning)
• express in algebraic terms the interior angle sum of a polygon with $$n$$ sides, eg $$\, \mbox{interior angle sum} = \left( n-2 \right) \times 180^{\circ} \,$$ (Communicating)
• apply interior and exterior angle sum results for polygons to find the sizes of unknown angles

### Background Information

Students are expected to give reasons when proving properties of plane shapes using congruent triangle results.

Dynamic geometry software and prepared applets are useful tools for investigating the interior and exterior angle sums of polygons, allowing students a visual representation of a result.

The concept of the exterior angle sum of a convex polygon may be interpreted as the amount of turning required when completing a circuit of the boundary.

Comparing the perimeters of inscribed and circumscribed polygons leads to an approximation for the circumference of a circle. This is the method that the Greek mathematician and scientist Archimedes (c287–c212 BC) used to develop an approximation for the ratio of the circumference to the diameter of a circle, ie $$\pi$$.

### Language

The term 'equiangular' is often used to describe a pair of similar figures (which includes congruent figures), as the angles of one figure are equal to the matching angles of the other figure.

The term 'angle sum' is generally accepted to refer to the interior angle sum of a polygon. When calculating the exterior angle sum of a polygon, students need to refer explicitly to the 'exterior angle sum'.