Properties of geometric figures - advanced

This lesson comprises two (2) master classes focusing on:

• Using congruence in proofs
• Constructing formal proofs involving congruent and similar triangles
• Applying logical reasoning to proofs involving plane shapes

Content:

MA5-GEO-P-01

Identify and explain congruence

• Identify figures as congruent figures if translations, reflections and rotations can move one figure exactly on top of another
• Match the sides and angles of 2 congruent polygons
• Indicate 2 polygons are congruent using the symbol \( (\equiv) \) and name the vertices in matching order
• Determine that having equal radii is the condition for 2 circles to be congruent

Develop and use the conditions for congruent triangles

• Establish the 4 congruence tests for triangles (SSS, SAS, AAS and RHS)
• Use the congruence tests to identify a pair of congruent triangles from a selection of 2 or more triangles

Develop and apply the minimum conditions for triangles to be similar

• Examine the minimum conditions needed and establish the 4 tests for 2 triangles to be similar
• Apply the minimum conditions needed and determine whether 2 triangles are similar using an appropriate test

Establish and apply properties of similar shapes and solids

• Establish for 2 similar figures with similarity ratio \( a:b \) that their areas are in the ratio \( a^2:b^2 \) and their volumes are in the ratio \( a^3:b^3 \)
• Solve problems involving areas and volumes of similar shapes and solids

Apply logical reasoning to numerical problems involving plane shapes

• Apply geometrical facts, properties and relationships to find the sizes of unknown sides and angles of plane shapes in diagrams, providing appropriate reasons
• Define the exterior angle of a convex polygon
• Establish the sum of the exterior angles of any convex polygon is \( 360^\circ \) and verify this result
• Apply the result for the sum of the interior angles of a triangle to find, by dissection, the sum of the interior angles of a polygon with more than 3 sides
• Apply the results for the interior and exterior angles of polygons to solve problems involving polygons

MA5-GEO-P-02

Construct formal proofs involving congruent and similar triangles

• Construct formal proofs of the congruence of triangles, preserving the matching order of vertices and drawing relevant conclusions
• Construct formal proofs of the similarity of triangles, preserving the matching order of vertices and drawing relevant conclusions

Apply logical reasoning to proofs involving plane shapes

• Recognise that a definition is the minimum amount of information needed to identify a particular figure
• Prove the properties of isosceles and equilateral triangles and special quadrilaterals from the formal definition of the shapes
• Prove and apply theorems and properties related to triangles and quadrilaterals
• Prove and apply tests for quadrilaterals
• Solve numerical and non-numerical problems in Euclidean geometry based on known assumptions and proven theorems