Similarity and congruence

Congruent figures have the same shape and size. Similar figures have the same shape but may differ in size: if the length scale factor is k, the area scale factor is k². Congruent figures are the special case k = 1. Name the correspondence first; if triangle ABC corresponds to triangle DEF, then AB pairs with DE, BC with EF and AC with DF, regardless of orientation.

What this guide covers

Recognising congruent and similar figures, triangle conditions, corresponding parts and length, area and volume ratios. These examples focus on reasoning rather than compass constructions.

Syllabus reference: 2026 Mathematics 4052, G2, page 9. These are Mentora search groupings, not new official syllabus categories.

The content in this guide was also checked against 2027 G3 Mathematics K310, G2, page 9 on 2026-09-12. This is subject-content alignment, not a claim of complete syllabus coverage.

What information establishes the claim?

Similarity: Equal corresponding angles (AA for triangles), proportional corresponding sides (SSS), or two proportional sides with equal included angle (SAS).
Congruence: Use SSS, SAS, ASA/AAS, or RHS for right triangles. AAA establishes shape, not equal size; two sides and a non-included angle alone are not a general congruence test.
Scaling: For similar figures with length multiplier k, area multiplies by k². For similar solids, volume multiplies by k³.

Worked example

Right triangles ABC and DEF correspond in that order. AB = 3 cm, BC = 4 cm, AC = 5 cm and DE = 6 cm, EF = 8 cm, DF = 10 cm. Compare their lengths and areas.

Every large-to-small corresponding side ratio equals 2, so the triangles are similar by SSS similarity, but not congruent. Their areas are ½ × 3 × 4 = 6 cm² and ½ × 6 × 8 = 24 cm². The area ratio is 4, the square of the length scale factor, not 2.

Corresponding right triangles ABC and DEF have sides 3, 4, 5 cm and 6, 8, 10 cm. A↔D, B↔E, C↔F. The large triangle is an enlargement by factor 2.

Try three practice checks

Attempt each on paper before revealing the worked answer. These sample particular skills; they do not diagnose the whole topic or establish mastery.

Check 1

Two similar triangles have small-to-large corresponding lengths 4 : 10. A second small side is 6 cm. Find its large counterpart.

Show worked answer

The multiplier is 10/4 = 2.5, so the corresponding side is 6 × 2.5 = 15 cm.

Check 2

Similar solids have surface areas in the ratio 9 : 16. Find their volume ratio.

Show worked answer

The length ratio is √9 : √16 = 3 : 4. Cube this ratio to obtain 27 : 64. Do not cube the area ratio directly.

Check 3

Two triangles each have sides 5 cm and 7 cm with the included angle 50°. Are they congruent? What if only their three angles matched?

Show worked answer

The stated sides and included angle establish SAS congruence. Matching three angles alone establishes similarity, not congruence: one triangle could be an enlargement.

Common mistakes to watch for

Match vertices in the stated order rather than pairing sides by their position on the page.
Use squared or cubed scale factors for areas or volumes, not the length factor.
State the conditions that justify similarity or congruence; appearance in a drawing is insufficient.

If one of these occurs, compare the relevant step with the worked example, then try a different question before drawing conclusions.

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