How do you recognise congruent and similar shapes, prove congruence and use scale factors for lengths, areas and volumes?
Recognise and prove congruence (SSS, SAS, ASA, RHS) and similarity, use scale factors to find missing lengths in similar shapes, and apply area and volume scale factors (Higher tier).
A focused answer to the WJEC GCSE Mathematics geometry content on similarity and congruence, covering the congruence conditions SSS, SAS, ASA and RHS, recognising similar shapes, using linear scale factors and applying area and volume scale factors at Higher tier.
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What this dot point is asking
WJEC asks you to recognise congruent and similar shapes, to prove congruence using the four conditions (SSS, SAS, ASA, RHS), to use scale factors to find missing lengths in similar shapes, and at Higher tier to apply area and volume scale factors. The central ideas are that congruent shapes are identical in size and shape, while similar shapes have the same shape but a different size linked by a scale factor. The proof-style congruence questions reward precise reasoning (AO3), and the scale-factor questions reward knowing that areas square and volumes cube the linear factor.
Congruence
Congruent shapes can be placed exactly on top of one another.
To prove congruence, state the equal sides and angles with reasons, then name the condition (for example "congruent by SAS"). WJEC marks the named condition, so always finish by quoting it.
Similarity
Similar shapes have the same shape but a different size.
To find a missing length, work out from a known pair of corresponding sides, then multiply (going to the larger shape) or divide (going to the smaller). Matching the correct corresponding sides is essential, so it helps to redraw the two shapes the same way up before pairing the sides.
A common WJEC configuration is two triangles that share an angle, where a line parallel to one side creates a smaller similar triangle inside the larger one. Equal angles (from the parallel lines and the shared vertex) prove the triangles similar, and then a pair of corresponding sides gives the scale factor. Recognising this "triangle within a triangle" pattern, and identifying which sides correspond, is the skill these questions reward.
Length, area and volume scale factors (Higher)
Areas and volumes scale differently from lengths.
This is the most error-prone idea in the topic: candidates often multiply areas or volumes by the linear factor instead of by or . Always find the linear scale factor first, then raise it to the right power.
Worked scale-factor problem
A typical Higher question chains the area scale factor.
Why this matters
Similarity and congruence test two complementary skills: the reasoning of a congruence proof and the calculation of a scale-factor problem. Congruence proofs are pure AO3 communication, rewarding a clear, named chain of equal sides and angles, while similarity links to enlargement in transformations and to mensuration through the area and volume factors. The squared and cubed scale factors are a classic Higher discriminator, and getting the direction of the scale factor right (multiplying up, dividing down) secures the routine marks.
Exam-style practice questions
Practice questions written in the style of WJEC exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.
WJEC 20193 marksTwo triangles are similar. The smaller has a side of cm; the corresponding side of the larger is cm. A second side of the smaller triangle is cm. Work out the corresponding side of the larger triangle. (Foundation and Higher, Unit 2, calculator.)Show worked answer →
Find the linear scale factor from the matching pair: .
Multiply the known side of the smaller triangle by the scale factor: cm.
Markers award a mark for the scale factor , a mark for multiplying and a mark for the answer cm. Dividing instead of multiplying (going the wrong way between the shapes) is the common error.
WJEC 20224 marksTwo similar solids have heights cm and cm. The smaller has volume cm. Work out the volume of the larger. (Higher, Unit 2, calculator.)Show worked answer →
The linear scale factor is . The volume scale factor is the cube: .
Multiply the smaller volume: cm.
Markers give marks for the linear scale factor, for cubing it to get the volume scale factor, and for the final volume. Using the linear or area scale factor instead of cubing is the usual mistake.
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Sources & how we know this
- WJEC GCSE Mathematics specification (3300) — WJEC (2015)
- WJEC GCSE Mathematics specification PDF (3300) — WJEC (2015)