Overview
What you’ll learn
- Recognise congruent figures and decide whether two triangles are congruent using SSS, SAS, ASA/AAS or RHS.
- Recognise similar figures and use equal corresponding angles with proportional corresponding sides to test similarity.
- Solve for unknown sides and angles in similar triangles and polygons, including simple algebraic cases.
- Perform enlargements and reductions by a given scale factor and interpret scale drawings.
- Use the ratio of areas of similar plane figures (k²) and the ratio of volumes of similar solids (k³).
- Solve applied similarity problems, including shadow (indirect measurement) and model problems.
In 30 seconds
- 1Congruent figures match in both shape and size; one is an exact copy of the other.
- 2Four congruence tests: SSS, SAS, ASA (or AAS), RHS — three equal angles alone prove nothing about size.
- 3Similar figures share equal corresponding angles and proportional corresponding sides.
- 4For similar figures with scale factor k: lengths ×k, areas ×k², volumes ×k³.
- 5Two triangles are similar if two pairs of angles match (AA) — the third pair is forced.
- 6A scale drawing ratio 1 : n sends drawing lengths ×1/ n and drawing areas ×1/n².
- 7Perimeters of similar figures scale like k; areas never scale like k.
- 8Shadow problems work because the sun’s rays create similar triangles: height ÷ shadow stays constant.
Notes · 1 of 5
Congruent figures and the four tests
Exact copies — and the shortcuts that prove it without measuring everything.
Two figures are congruent when one fits exactly on top of the other: same shape, same size. For triangles, matching all six parts (three sides, three angles) is unnecessary — four standard tests guarantee congruence. In each test the pairs refer to corresponding sides and angles, matched in order.
| Test | What must match | Memory hook |
|---|---|---|
| SSS | three pairs of equal sides | rigid frame |
| SAS | two equal sides with the included angle between them | hinge |
| ASA / AAS | two equal angles plus any corresponding equal side | shape locked |
| RHS | right angle, hypotenuse, one other side | right triangles only |
The classic trap: three equal angles (AAA) make triangles similar, not congruent — a photo and its enlargement share every angle but not a single length. No test uses "three angles" or "two sides and a non-included angle".
Notes · 2 of 5
Similar figures
Same shape, scaled size: equal angles, proportional sides.
Two figures are similar when corresponding angles are equal and corresponding sides grow by the same scale factor k. For triangles the shortcut is decisive: two pairs of equal angles (AA) force similarity, because the third pair is left over from the 180° sum. Rectangles are similar only when their side ratios match — having four right angles is not enough.
- 1Match the vertices in order
ΔABC ~ ΔPQR means A ↔ P, B ↔ Q, C ↔ R.
- 2Write the ratio once
AB/PQ = BC/QR = CA/RP = k.
- 3Solve for the unknown
If AB = 4, PQ = 6 (so k = 2/3) and BC = 5, then QR = 5 ÷ (2/3) = 7.5.
- 4Check plausibility
The bigger triangle must get the bigger side — a reversed ratio shows instantly.
Notes · 3 of 5
Enlargement, reduction and scale drawings
Scaling shapes up and down, and reading the 1 : n language of plans and maps.
An enlargement with scale factor k multiplies every length by k; a reduction is the same idea with 0 < k < 1. The shape stays similar: angles never change. A scale drawing records real objects at a stated ratio, such as 1 : 200, meaning 1 cm on paper equals 200 cm in reality. Lengths divide by n; areas divide by n squared — the detail people forget.
- Scale 1 : 200, drawing 5 cm → real length 5 × 200 = 1000 cm = 10 m.
- Scale 1 : 200, drawing area 6 cm² → real area 6 × 200² = 240 000 cm² = 24 m².
- Scale 1 cm : 2 m, room area on plan 15 cm² → real area 15 × 2² = 60 m².
Notes · 4 of 5
Ratios of areas and volumes
The k, k², k³ ladder for every similar pair of figures.
For similar figures with scale factor k (big ÷ small): perimeters and any other lengths scale by k, areas by k², and volumes by k³. The reason is structural: area is a product of two lengths (k × k), volume of three (k × k × k). This ladder runs both ways — given a volume ratio you can claw back k by cube-rooting.
| Quantity | Small | Big | Ratio (big : small) |
|---|---|---|---|
| Side | 6 cm | 9 cm | 3 : 2 |
| Perimeter | 18 cm | 27 cm | 3 : 2 |
| Area | 24 cm² | 54 cm² | 9 : 4 |
Watch the direction of the ratio: "area ratio 9 : 4" means k² = 9/4, so k = 3/2 (not 9/2, and not 9 ÷ 4 ÷ 2). Write k = big ÷ small at the top of your working and keep every ratio consistent with it.
Notes · 5 of 5
Applied similarity problems
Shadows, models and nested triangles — similar figures doing real work.
Similarity solves measurement problems that are too big or awkward to measure directly. In sunshine, every vertical object and its shadow form similar right triangles, because the sun’s rays hit the ground at the same angle: height ÷ shadow length is the same constant for all objects at that moment.
- 1Set the constant
A 1.6 m person casts a 2 m shadow: height/shadow = 1.6/2 = 0.8.
- 2Apply to the unknown
A flagpole casts a 7.5 m shadow: height = 0.8 × 7.5 = 6 m.
- 3Justify the similarity
Both triangles share the sun-angle (AA), so the proportionality is valid.
Models work the same way. A 1 : 20 car model with volume 120 cm³ represents 120 × 20³ = 960 000 cm³ = 0.96 m³ of real car. State the direction of your scale factor, label the similar triangles, and keep the k / k² / k³ ladder straight — that is the entire topic in three lines.
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 exactly.
- SSS
- Congruence test: three pairs of equal corresponding sides.
- SAS
- Congruence test: two pairs of equal sides with equal included angles.
- ASA / AAS
- Congruence test: two pairs of equal angles with one equal corresponding side.
- RHS
- Congruence test for right triangles: right angle, hypotenuse and one side equal.
- Similar figures
- Figures with equal corresponding angles and proportional corresponding sides.
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- Scale factor (k)
- The constant multiplying corresponding lengths between similar figures.
- AA
- Similarity test: two pairs of equal angles force the triangles to be similar.
- Scale drawing
- A drawing at a stated ratio 1 : n; lengths divide by n and areas by n².
- Area ratio
- For similar figures, k² where k is the length scale factor.
- Volume ratio
- For similar solids, k³ where k is the length scale factor.
- Enlargement
- A transformation scaling a figure by factor k > 1; a reduction uses 0 < k < 1.
Revise
Common mistakes
Where students most often lose marks in this chapter — and what to do instead.
Mistake: Claiming two triangles with three equal angles are congruent.
Equal angles fix shape, not size.
Instead: AAA gives similarity only; congruence needs SSS, SAS, ASA/AAS or RHS.
Mistake: Using SAS with a non-included angle: AB = PQ, BC = QR and angle A = angle P.
The angle must sit between the two sides.
Instead: For SAS the angle is at the shared vertex of the two known sides (angle B here); otherwise no test applies.
Show 10 more mistakesShow fewer mistakes
Mistake: Assuming any two rectangles are similar.
Equal angles are not enough for polygons with more than three sides.
Instead: Rectangles are similar only when length : width ratios match.
Mistake: Writing the proportionality upside down: matching the short side of one triangle to the long side of the other.
Corresponding vertices pair big with big and small with small.
Instead: Match vertices in the similarity order: ΔABC ~ ΔPQR pairs A with P throughout.
Mistake: Scaling areas by k: "scale factor 2 doubles the area".
Area is two-dimensional.
Instead: Scale factor 2 quadruples area (k² = 4).
Mistake: Scaling areas by k² when volumes are asked.
Volume is three-dimensional.
Instead: Volumes scale by k³: scale factor 2 multiplies volume by 8.
Mistake: Reading the volume ratio 27 : 8 and quoting k = 27/8.
The cube root is missing.
Instead: k³ = 27/8 gives k = 3/2.
Mistake: Converting map area 6 cm² at 1 : 200 as 6 × 200 = 1200 cm².
Each direction scales, so area scales twice.
Instead: 6 × 200² = 240 000 cm² = 24 m².
Mistake: In shadow problems, equating shadow/height instead of height/shadow between the two objects.
The constant ratio flips halfway through.
Instead: Pick one form — height ÷ shadow — and use it on both objects.
Mistake: Quoting the congruence test "SSA" (two sides and a non-included angle).
No such test exists; the data can fit two different triangles.
Instead: Use SAS (angle included) or move to RHS for right triangles.
Mistake: For a scale 1 : n plan, computing the real area as plan area × n.
n is a length ratio.
Instead: Real area = plan area × n².
Mistake: In a nested triangle (DE ∥ BC), writing AD/DB = AE/EC as the similarity ratio.
That is the intercept ratio (often true), but the similarity proportion pairs AD with AB, the whole side.
Instead: ΔADE ~ ΔABC gives AD/AB = AE/AC = DE/BC.
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Chapter 10 Practice Paper
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7 multiple-choice questions and structured questions with answer lines. Chapter 10 (G2 Congruence and Similarity: congruence tests, similar triangles, scale drawings, area and volume ratios).
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Frequently asked questions
What does Sec 3 E-Math Chapter 10 cover?
It covers syllabus sub-topic G2 of O-Level 4052 / SEC G3 K310 in full: the four congruence tests (SSS, SAS, ASA/AAS, RHS), similarity via equal angles and proportional sides, enlargements and scale drawings, area ratios (k²) and volume ratios (k³), and applied problems including shadows and models.
Why is AAA not a congruence test?
Equal angles fix the shape but not the scale — a triangle and its enlargement share all three angles. Congruence needs at least one length to pin the size down, which is why every congruence test (SSS, SAS, ASA/AAS, RHS) includes a side.
How do I remember which power of k to use?
Count dimensions. Lengths are one-dimensional: ×k. Areas come from two multiplied lengths: ×k². Volumes from three: ×k³. Perimeters behave like lengths; paint behaves like area; capacity behaves like volume.
How do I set up a shadow problem?
Sunlight gives every vertical object the same triangle shape, so height ÷ shadow is constant at that moment. Compute the constant from the object you know both numbers for, then multiply the unknown object’s shadow by it. Quote AA similarity as the reason.
When is this chapter usually tested in school?
Congruence and similarity appear in Weighted Assessment 2 and End-of-Year papers, usually as a proportion problem with a nested triangle or a scale-drawing conversion, plus the k²/k³ rules in a mensuration-flavoured part. Marks hang on stating correspondences and reasons clearly.
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Chapter 11: Properties of Circles