Overview
What you’ll learn
- Recognise congruent figures and match their corresponding sides and angles.
- Name the four conditions under which two triangles must be congruent (SSS, SAS, ASA, RHS).
- Recognise similar figures: equal corresponding angles and proportional corresponding sides.
- Find the scale factor of an enlargement and use it to compute missing lengths.
- Interpret map and model scales of the form 1 : n in both directions.
- Solve multi-step problems chaining similar figures.
In 30 seconds
- 1Congruent figures have the same shape and the same size — one slides, flips or spins onto the other.
- 2Two triangles are definitely congruent if they match by SSS, SAS, ASA or RHS — three well-chosen facts beat checking all six.
- 3Similar figures share equal corresponding angles and proportional corresponding sides, but may differ in size.
- 4The scale factor is new length ÷ corresponding old length; it is the same for every pair of matching sides.
- 5A map scale of 1 : n means 1 cm on the map represents n cm in reality — convert units before comparing.
- 6An enlargement with scale factor between 0 and 1 shrinks the figure; the shape still counts as similar.
Notes · 1 of 6
Congruent figures
Identical twins: same shape, same size, any orientation.
Two figures are congruent when one can be placed exactly on top of the other by sliding (translation), turning (rotation) or flipping (reflection). Every pair of corresponding sides is equal, and every pair of corresponding angles is equal. Orientation is irrelevant — a triangle flipped over is still congruent to its mirror partner.
| Transformation | Effect on size | Effect on shape | Image congruent? |
|---|---|---|---|
| Translation (slide) | unchanged | unchanged | yes |
| Rotation (turn) | unchanged | unchanged | yes |
| Reflection (flip) | unchanged | unchanged | yes |
| Enlargement (scale) | changed | unchanged | no — only similar |
Notes · 2 of 6
When are two triangles certainly congruent?
Four combinations of facts lock a triangle down completely.
| Condition | Meaning | Check |
|---|---|---|
| SSS | all three pairs of sides equal | the triangle is rigid once its sides are fixed |
| SAS | two sides and the angle between them | the angle must sit between the two sides |
| ASA | two angles and the side between them | the side joins the two angles |
| RHS | right angle, hypotenuse and one other side | only for right-angled triangles |
Three equal angles alone (AAA) do not force congruence — the triangles could be different sizes, which is exactly similarity. And SSA (two sides with a non-included angle) can match in more than one way. The four conditions above are the safe ones to name.
Notes · 3 of 6
Similar figures
Same shape, different size: angles hold, sides scale.
Two figures are similar when their corresponding angles are equal and their corresponding sides are in the same ratio. A photograph and its enlargement are similar: nothing about the shape changes, only the size.
| Rectangle | Width | Length | Width ÷ Length |
|---|---|---|---|
| Small | 4 cm | 6 cm | 4/6 = 2/3 |
| Large | 10 cm | 15 cm | 10/15 = 2/3 |
The internal ratio width : length stays 2 : 3 in both — that is the “same shape” guarantee. Meanwhile the matching sides 4 → 10 and 6 → 15 both grew by the factor 2.5, the scale factor. Similarity lives on both facts: equal angles and one common scale factor.
Notes · 4 of 6
Scale factors and enlargement
One number rules every matching pair: new ÷ old.
- 11. Pick a matching pair
Choose one side and its clearly corresponding partner.
- 22. Divide
Scale factor = new length ÷ old length.
- 33. Multiply everywhere
Every remaining old length scales by the same factor.
- 44. Sanity-check
SF > 1 enlarges, SF = 1 keeps congruent, 0 < SF < 1 shrinks.
| Side | Old | New |
|---|---|---|
| Shortest | 2 | 6 |
| Middle | 5 | 15 |
| Longest | 7 | 21 |
Notes · 5 of 6
Map and model scales
The ratio 1 : n is a scale factor wearing a costume.
A scale of 1 : 50 000 says 1 cm on the map represents 50 000 cm in the real world — which is 500 m. Convert the ratio into comfortable units once, then work in them: 1 : 25 000 is 1 cm to 250 m; 1 : 100 000 is 1 cm to 1 km.
| Scale | Map to reality | Reality to map |
|---|---|---|
| 1 : 25 000 | 6 cm → 1.5 km | 3.5 km → 14 cm |
| 1 : 100 000 | 8 cm → 8 km | 5 km → 5 cm |
Notes · 6 of 6
Chaining similar figures
Long problems are just several scale factors in a row.
Exam questions love a chain: figure P similar to Q, and Q similar to R. Similarity is transitive — P and R are similar too. Work one link at a time: find the scale factor P → Q, compute Q’s sides, then a new scale factor Q → R for the last figure.
| Figure | Sides | Scale factor to next |
|---|---|---|
| P | 3, 4, 6 | P → Q: 2 (Q’s longest side is 12) |
| Q | 6, 8, 12 | Q → R: 2.5 (R’s shortest side is 15) |
| R | 15, 20, 30 | — |
Revise
Key terms
The words examiners expect you to use precisely.
- Congruent figures
- Figures identical in both shape and size; one maps onto the other by sliding, turning or flipping.
- Corresponding sides
- Matching sides of two congruent or similar figures, as fixed by the letter pairing.
- Corresponding angles
- Matching angles of two figures; equal in both congruent and similar pairs.
- SSS
- A congruence condition: all three pairs of sides equal.
- SAS
- A congruence condition: two pairs of sides and the angle between them equal.
- ASA
- A congruence condition: two pairs of angles and the side between them equal.
Show 6 more key termsShow fewer key terms
- RHS
- A congruence condition for right-angled triangles: right angle, hypotenuse and one side equal.
- Similar figures
- Figures with equal corresponding angles and corresponding sides in a constant ratio.
- Scale factor
- The number new length ÷ corresponding old length, shared by every matching pair.
- Enlargement
- A transformation that multiplies every length by the scale factor; factors below 1 shrink.
- Map scale (1 : n)
- A ratio saying 1 unit on the map represents n of the same units in reality.
- Model scale
- The ratio in which a model is shrunk from its real object, such as 1 : 24.
Revise
Common mistakes
Where students most often lose marks in this chapter — and what to do instead.
Mistake: Calling two shapes congruent because they have equal areas.
A 6 × 4 rectangle and a 12 × 2 rectangle share an area but not a shape.
Instead: Congruence needs every corresponding side and angle equal — area alone proves nothing.
Mistake: Claiming two triangles with equal angles must be congruent.
Equal angles fix the shape but not the size — that is similarity (AAA), not congruence.
Instead: Congruence needs side information too: SSS, SAS, ASA or RHS.
Show 8 more mistakesShow fewer mistakes
Mistake: Dividing old ÷ new for the scale factor.
The scale factor runs from the original to the image: new ÷ old.
Instead: A 5 cm side becoming 15 cm is a scale factor of 15 ÷ 5 = 3, not 1/3.
Mistake: Using the map ratio without converting units, e.g. treating 1 : 50 000 cm as 50 m.
The ratio compares identical units; 50 000 cm is 500 m.
Instead: Convert once: 1 : 50 000 means 1 cm : 500 m, then compute in comfortable units.
Mistake: Scaling only the known sides and leaving the rest unchanged in an enlargement.
Every length of the figure multiplies by the same scale factor.
Instead: Multiply all sides by the factor; angles never change in an enlargement.
Mistake: Matching the wrong sides in a similar pair, e.g. pairing the longest side with the shortest.
Ratios only hold between corresponding sides.
Instead: Order the sides (or follow the letters) and pair shortest with shortest before dividing.
Mistake: Adding the scale factor difference instead of multiplying, e.g. “2 cm longer” instead of “twice as long”.
Similarity is multiplicative; differences do not transfer between figures.
Instead: Compute the factor from a pair, then multiply — never add.
Mistake: Declaring triangles congruent from two equal sides alone (SS).
Two equal side pairs leave the shape free to flex — the third side can differ.
Instead: Use a full condition: SSS, SAS, ASA or RHS. Two sides need the included angle (SAS) or the third side (SSS).
Mistake: Doubling angles during an enlargement, e.g. a 40° angle becoming 80° under scale factor 2.
Enlargements scale lengths only; the shape’s angles are fixed.
Instead: Every angle stays exactly the same; only the side lengths multiply.
Mistake: Reading 1 : 100 000 as “1 cm to 100 m”.
The ratio compares centimetres with centimetres: 100 000 cm is 1 km, not 100 m.
Instead: Convert the map unit once: 1 cm : 100 000 cm = 1 cm : 1 km, then compute.
Check yourself
Quick quiz
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Question 1 of 3
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Define congruence
Two figures are congruent when they have
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Practise
Topical questions
Exam-style practice in three tiers, from recall to data questions.
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15 questions worth 58 marks, each with a model answer, mark scheme and examiner tips.
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Question paper
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10 multiple-choice questions and structured questions with answer lines. Chapter 6.
Mark scheme
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Model answers with a mark for each creditworthy point — for marking at home or in class.
Where this chapter is tested
Each school sets its own scope — check your assessment outline.
- Term 2Coming soonWeighted Assessment 2 (WA2)
- Term 4Coming soonEnd-of-Year Examination (EOY)
Questions
Frequently asked questions
What is Sec 2 Math Chapter 6 about?
Chapter 6 covers congruence and similarity: congruent figures and the transformations that preserve them, the four triangle congruence conditions (SSS, SAS, ASA, RHS), similar figures with proportional sides, scale factors and enlargement, and map and model scales.
What is the difference between congruent and similar?
Congruent figures match in both shape and size — every corresponding side and angle is equal. Similar figures match in shape only: angles stay equal but the sides all scale by one common factor.
How do I find the scale factor?
Pick one clearly corresponding pair of lengths and divide: scale factor = new length ÷ old length. Check the factor on a second pair before using it; both pairs must give the same number.
What does a map scale of 1 : 25 000 mean?
One centimetre on the map represents 25 000 centimetres in reality — 250 m. So 6 cm on the map is 6 × 250 m = 1.5 km on the ground.
Up next
Chapter 7: Pythagoras’ Theorem and Irrational Numbers