SECONDARY 1 MATHEMATICS LEARNING GUIDE · GUIDE 22
Congruence asks whether two figures are exactly the same size and shape. Similarity asks whether they have the same shape even when the size changes. The distinction matters because corresponding lengths, angles, areas and volumes behave differently under scaling.
This guide develops corresponding parts, triangle congruence, similarity, scale factors, perimeter and area scaling, map and model reasoning, indirect measurement and common diagram traps. Exact sequencing varies across subject levels and schools, so use extension sections only when they match the learner’s present course.
Useful prior guides: Geometry, Angles and Polygons, Ratio and Proportion and Geometrical Construction, Scale Drawings and Loci.
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1. Correspondence comes before comparison
If triangle ABC corresponds to triangle PQR, the order tells us A↔P, B↔Q and C↔R.
Therefore side AB corresponds to PQ, BC to QR, and AC to PR.
Why order matters
Correct ratios depend on matching corresponding parts. A wrong pairing can produce a neat calculation with no geometric meaning.
2. Congruent figures have equal corresponding lengths and equal corresponding angles
Congruent figures can be translated, rotated or reflected and still remain congruent.
Orientation does not determine congruence. Size and shape do.
Worked example
If ΔABC ≅ ΔPQR and AB = 7 cm, then PQ = 7 cm.
3. Similar figures preserve shape but not necessarily size
Similar figures have equal corresponding angles and proportional corresponding lengths.
If one figure is an enlarged version of another by scale factor 3, every corresponding length is multiplied by 3.
Example
A 4 cm side corresponds to a 12 cm side. The scale factor from the first figure to the second is 12/4 = 3.
4. A scale factor has a direction
If figure A has side 5 cm and corresponding side in figure B is 8 cm, the scale factor from A to B is 8/5.
The scale factor from B back to A is 5/8.
Do not say “the scale factor is 8/5” without identifying direction when the context could be ambiguous.
5. Congruence can be established from sufficient triangle information
Common congruence tests include SSS, SAS, ASA/AAS and RHS in appropriate school treatments.
These tests are useful because they identify enough information to force one triangle shape and size.
SSS example
If three sides of one triangle are 5, 7 and 9 cm and the other has corresponding sides 5, 7 and 9 cm, the triangles are congruent by SSS.
6. AAA establishes similarity, not congruence
If two triangles have equal corresponding angles, their shape is fixed but their size may differ.
Therefore AAA is a similarity condition, not a congruence condition.
Example
A 3-4-5 triangle and a 6-8-10 triangle have the same angles and are similar, but they are not congruent.
7. SAS similarity uses proportional sides around an equal angle
If two pairs of corresponding sides are proportional and the included angle is equal, the triangles are similar.
Worked example
Triangle A has sides 6 and 9 around a 50° angle. Triangle B has corresponding sides 10 and 15 around a 50° angle.
6/10 = 9/15 = 3/5, so the sides are proportional and the included angle matches.
8. SSS similarity compares all three side ratios
Triangles with side lengths 4, 6, 8 and 6, 9, 12 are similar because:
4/6 = 6/9 = 8/12 = 2/3.
The common ratio shows a constant scale factor.
9. Find missing sides using one consistent scale factor
Suppose two similar triangles have corresponding sides 5 cm and 8 cm. Another side of the first triangle is 12 cm. Find its corresponding side in the second.
Scale factor = 8/5.
Required side = 12×8/5 = 19.2 cm.
10. Equivalent ratio routes should agree
Instead of multiplying by the scale factor, one may write:
5/8 = 12/x.
Then 5x = 96 and x = 19.2.
Both methods express the same proportional relationship.
11. Perimeters scale like lengths
If the linear scale factor is k, the perimeter scale factor is also k.
Worked example
A polygon has perimeter 24 cm. A similar polygon has linear scale factor 1.5 from the first to the second.
New perimeter = 24×1.5 = 36 cm.
12. Areas scale by the square of the linear factor
If all lengths are multiplied by k, area is multiplied by k².
Worked example
A rectangle is enlarged by scale factor 3. If the original area is 20 cm², the new area is 20×9 = 180 cm².
Why square?
Area involves two perpendicular dimensions. Each contributes a factor k.
13. Volumes scale by the cube of the linear factor
If similar solids have linear scale factor k, corresponding volumes scale by k³.
Use this as an extension if the learner’s current course has not yet formalised similar solids.
Example
Scale factor 2 gives volume factor 8.
14. Area ratio can reveal the linear scale factor
Suppose two similar figures have area ratio 25:49.
The linear scale factor is √(25/49) = 5/7 in the corresponding direction.
This reverse reasoning matters because area and length do not share the same ratio.
15. Congruence and similarity interact with transformations
Translation, rotation and reflection preserve lengths and angles, so they produce congruent images.
Enlargement with scale factor not equal to ±1 changes size but preserves shape, producing similar figures.
Negative scale factors
Negative enlargement scale factors are an extension topic in many courses. Treat them only when introduced.
16. Scale drawings are similarity models
A map at scale 1:50,000 is similar to the real region under an idealised scaling model.
1 cm on the map represents 50,000 cm = 500 m in reality.
The entire drawing uses one consistent linear scale factor.
17. Similarity can support indirect measurement
Suppose a 1.5 m stick casts a 2 m shadow at the same time a tree casts an 8 m shadow.
Under the ideal assumption that sun rays create similar triangles:
tree height/8 = 1.5/2.
Tree height = 8×1.5/2 = 6 m.
18. Parallel lines often create similar triangles
When a line is drawn parallel to one side of a triangle, angle relationships can establish similarity between the smaller and larger triangles.
This supports proportional side reasoning.
Example structure
If DE ∥ BC in triangle ABC, then corresponding angles can show ΔADE ~ ΔABC.
19. Diagram size is not evidence of similarity
Two shapes can look similar but fail because corresponding angles differ or side ratios are inconsistent.
Conversely, two valid similar triangles may be drawn at awkward orientations.
Rule
Use properties and ratios, not appearance.
20. Common congruence and similarity errors
| Error | Likely issue | Repair prompt |
|---|---|---|
| Uses AAA for congruence | Shape confused with size | Can equal angles still allow different sizes? |
| Pairs non-corresponding sides | Correspondence not established | Which vertices match? |
| Uses area ratio as length ratio | Dimensional scaling missed | Is this a one-dimensional or two-dimensional quantity? |
| Scale factor inverted | Direction omitted | From which figure to which figure? |
| Assumes sketch is to scale | Visual guess replacing geometry | What stated property proves the comparison? |
21. Practice laboratory
- State whether two 3-4-5 triangles are congruent if both have those exact side lengths.
- State whether a 3-4-5 triangle and a 6-8-10 triangle are congruent or similar.
- Find the scale factor from a 5 cm side to its 12 cm corresponding side.
- A corresponding side is 9 cm in the small figure and 15 cm in the large. A second small side is 12 cm. Find the corresponding large side.
- Two similar polygons have linear scale factor 4. If the smaller perimeter is 18 cm, find the larger perimeter.
- With the same scale factor 4, if the smaller area is 7 cm², find the larger area.
- Two similar figures have area ratio 36:81. Find the corresponding linear ratio.
- Explain why AAA proves similarity but not congruence.
- Triangles have corresponding side triples 5,7,8 and 10,14,16. Are they similar?
- A map scale is 1:25,000. What real distance is represented by 6 cm?
- A 1.2 m pole casts a 0.8 m shadow. At the same moment a structure casts a 10 m shadow. Find its height under a similar-triangle model.
- If two figures are congruent, what is their linear scale factor?
- If linear scale factor is 1/3, what is the area scale factor?
- Explain why rotating a triangle does not change congruence.
22. Explained answers
1. Yes, by SSS.
2. Similar, not congruent.
3. 12/5 = 2.4.
4. Scale factor = 15/9 = 5/3; required side = 12×5/3 = 20 cm.
5. 18×4 = 72 cm.
6. Area factor = 16, so 7×16 = 112 cm².
7. √(36/81) = 2/3.
8. Equal angles fix shape but not size.
9. Yes; each larger side is twice the smaller corresponding side.
10. 6×25,000 = 150,000 cm = 1.5 km.
11. h/10 = 1.2/0.8 = 1.5, so h = 15 m.
12. 1.
13. 1/9.
14. Rotation preserves lengths and angles.
23. Complete mixed problem
Two similar triangular signs have corresponding bases 18 cm and 30 cm. The smaller sign has area 72 cm². Find the larger area.
Linear scale factor from smaller to larger = 30/18 = 5/3.
Area scale factor = (5/3)² = 25/9.
Larger area = 72×25/9 = 8×25 = 200 cm².
A common mistake would be 72×5/3 = 120 cm², incorrectly using the length scale factor for area.
24. Teaching similarity through correspondence
Before calculating ratios, ask learners to mark corresponding vertices with matching symbols. Then write the side pairs explicitly.
This prevents later arithmetic from being built on the wrong pairing.
Changed-case test
If the linear scale factor doubles, ask how perimeter, area and volume factors change. The different dimensional responses reveal whether scaling is understood structurally.
25. Questions students often ask
Can congruent figures face different directions?
Yes. Translation, rotation and reflection can change position or orientation without changing size and shape.
Are all congruent figures similar?
Yes, with scale factor 1.
Are all similar figures congruent?
No. They are congruent only when the scale factor is 1.
Why does area use k²?
Because two independent length dimensions each scale by k.
How do I know which side corresponds?
Use equal angles, vertex order, stated relationships and structural position—not visual closeness.
26. Return path and sources
Congruence and similarity connect geometry to ratio. Revisit Geometry, Angles and Polygons for angle constraints and Ratio and Proportion for multiplicative comparison.
Official curriculum reference: MOE Secondary Syllabus Directory. Exact congruence, similarity and scale-factor depth varies by subject level and school.
Editorial approach: Wintour House V1.0 · CivDJ · eduKate Publishing. Establish correspondence, identify the invariant shape, compute one consistent scale factor, respect dimensional scaling and verify against the diagram constraints.
