Outcomes
A student:

 MA5.21WM
selects appropriate notations and conventions to communicate mathematical ideas and solutions

 MA5.22WM
interprets mathematical or reallife situations, systematically applying appropriate strategies to solve problems

 MA5.23WM
constructs arguments to prove and justify results

 MA5.214MG
calculates the angle sum of any polygon and uses minimum conditions to prove triangles are congruent or similar
Content
 Students:
 Formulate proofs involving congruent triangles and angle properties (ACMMG243)
 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
 Use the enlargement transformations to explain similarity and to develop the conditions for triangles to be similar (ACMMG220)
 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 rightangled triangle are proportional to the hypotenuse and a second side of another rightangled 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 \( \, \mbox{interior angle sum} = \left( n2 \right) \times 180^{\circ} \,\) (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'.