Properties of geometric figures - intermediate

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

• Using congruence in proof
• Proving similar triangles
• Applying similar triangles 

Content:

MA5-GEO-C-01

Identify and describe the properties of similar figures

• Describe similar figures as having the same shape but not necessarily the same size
• Verify and explain that in similar polygons, the corresponding angles are equal and the corresponding side lengths are in the same proportion
• Name the vertices in matching order when using the similar symbol  \( (\sim) \) in a similarity statement
• Match the corresponding sides and angles of similar polygons

Solve problems using ratio and scale factors in similar figures

• Apply an appropriate scale to enlarge or reduce a diagram
• Determine the scale factor for pairs of similar polygons and circles
• Apply knowledge of scale factor to find unknown sides in similar polygons
• Solve problems involving unknown lengths and scale factors of similar figures and related practical problems
• Solve problems involving scale drawings, with or without digital tools

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