Similarity and congruence
Congruent figures have the same shape and size; similar figures have the same shape but may differ in size. 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.
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.
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.
Useful foundations to check
These connections suggest what may help with this skill. Selecting a topic says nothing about your current attainment.
Ratios and their simplest form
Compare corresponding sides in the same direction to obtain one length scale factor.
Practise ratio and proportionCongruent figures and conditions
Distinguish the same shape and size from the same shape at a different scale before selecting a triangle criterion.
Practise similarity and congruence
Applications and next topics
Use similarity for scale drawings and indirectly measured lengths. Mensuration then converts a length scale into area, capacity or material requirements.
Choose a useful next resource
To understand a recurring error rather than relearn the whole skill: Check area and volume scaling errors.
Explore the foundations connected to similarity and congruence
Sources and review
Original Mentora public practice, authored for this guide. Worked answers and syllabus mapping checked during editorial QA on 2026-09-12; no independent expert review is claimed. Graph connections are Mentora learning hints, not an official SEAB sequence. Read the source and examination-year notes.