IP EMaths Notes (Upper Sec, Year 3-4): 12) Similarity and Congruency

Study guide08 Nov 2025, 00:00 ZUpdated 30 Nov 2025

Use similarity ratios and congruency criteria to prove geometric relationships and scale figures accurately.

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Q: What does IP EMaths Notes (Upper Sec, Year 3-4): 12) Similarity and Congruency cover?
A: Use similarity ratios and congruency criteria to prove geometric relationships and scale figures accurately.

The core idea is simple: Similar shapes have matching angles and scaled sides; congruent shapes match exactly.

Use it as a working check: Write corresponding vertices in the correct order before using ratios. Similarity scales lengths by $k$ , areas by $k^2$ , and volumes by $k^3$ . Congruency transfers equal sides and angles directly.

Then go one layer deeper: Use the lamppost and proof examples to practise clear geometry writing: state the matching angles or sides, name the test, set up the ratio, and conclude in words.

Similarity compares shapes with proportional sides and equal angles, while congruency proves exact matches. Keep diagrams annotated with tick marks, matching angle symbols, and explicit ratio statements so the reader can follow the logic without guessing.

Reviewed by

Marcus Pang·Managing Director (Maths)

Question target	First evidence to mark	Ratio or test to use	Common trap
Prove two triangles match exactly	Equal sides, equal angles, or a right angle plus hypotenuse and side	$\text{SSS}$ , $\text{SAS}$ , $\text{ASA}$ , $\text{AAS}$ , or $\text{RHS}$	Using similarity language when the answer needs exact equality
Find an unknown length in similar figures	Corresponding vertices and one known side pair	Side ratio $k$	Pairing a side with the wrong corresponding side
Find an area after a length scale changes	Side ratio first	Area ratio $k^2$	Multiplying area by $k$ instead of $k^2$
Find a volume after a length scale changes	Side ratio first	Volume ratio $k^3$	Squaring the scale factor for a 3D solid

Evidence pattern	What to write before the test	Test you may use	Common trap
Three matching sides	$AB = DE$ , $BC = EF$ , $CA = FD$	$\text{SSS}$	Listing side lengths without showing which sides correspond.
Two sides and the included angle	$AB = DE$ , $\angle ABC = \angle DEF$ , $BC = EF$	$\text{SAS}$
Two angles and one matching side	Two equal angles plus one corresponding side	$\text{ASA}$ or $\text{AAS}$	Forgetting that angle order fixes the matching vertices.
Right angle, hypotenuse, and one side	Both triangles are right-angled, hypotenuse pair equal, one side pair equal	$\text{RHS}$	Calling it RHS when the equal side is actually another hypotenuse.

IP EMaths Notes (Upper Sec, Year 3-4): 12) Similarity and Congruency

Sources

Quick reference

Method-choice checkpoint

Similarity fundamentals

Congruency fundamentals

Congruency proof-writing checkpoint

Worked example - Height via similar triangles

Worked example - Congruency proof in an isosceles triangle

Ratio consequences in practice

Worked example - Parallel slice in a triangle

Practice Quiz

Try this

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