O-Level Maths Similarity: Match the Vertices Before Writing Ratios

A student in a public E-Maths discussion asked whether naming a similarity test was enough or whether the angle reasoning should be written out. The useful habit is to make the argument readable: identify the equal angles, give their reasons and state the triangles in matching order. This guide offers a transparent method, without promising that a particular abbreviation earns marks.

Use the triangle names as a correspondence map

If triangle ABC is similar to triangle DEF, the order indicates A corresponds to D, B to E and C to F. Consequently, AB corresponds to DE, BC to EF and AC to DF. A diagram may rotate or reflect one triangle, so “the left side” is an unreliable way to select a matching pair. Labels and equal angles provide a firmer basis.

The 2026 SEAB Mathematics 4052 syllabus includes determining similarity, proportional corresponding sides and area ratios. It also assesses mathematical reasoning and communication. A short explanation connecting facts is therefore a useful revision target alongside obtaining the numerical answer.

Follow an original worked configuration

In triangle ABC, D lies on AB and E lies on AC, with DE parallel to BC. Let AD = 6 cm, DB = 3 cm and AE = 8 cm. To establish similarity, angle ADE equals angle ABC because DE is parallel to BC and AD lies along AB. Angle DAE equals angle BAC because both use the same two rays from A.

Thus triangle ADE is similar to triangle ABC by the angle-angle test. Write the smaller triangle first in both names. A maps to A, D to B and E to C. The side ratios are AD/AB = AE/AC = DE/BC. Notice that AB is 6 + 3 = 9 cm: DB is only the remaining section of AB.

The smaller-to-larger length ratio is 6/9 = 2/3. Therefore 8/AC = 2/3, giving AC = 12 cm. The requested remaining section EC would be 12 − 8 = 4 cm. Before circling an answer, reread whether the question asks for the whole side or its remaining part.

Keep the scale direction consistent

Starting with smaller/larger is convenient here, but larger/smaller works equally well if used consistently. You could write AB/AD = AC/AE = 3/2. A common error is to put AD/AB on one side of an equation and AC/AE on the other. Those ratios run in opposite directions.

Use a verbal check: “six is two-thirds of nine, so eight must be two-thirds of the whole AC.” This makes AC = 12 plausible. An answer smaller than 8 cm would contradict E lying between A and C. Such checks can catch an inverted ratio before you repeat the algebra.

Square the ratio only when comparing areas

The area ratio of triangle ADE to triangle ABC is (2/3)² = 4/9. If the larger triangle has area 54 cm², the smaller area is 24 cm². The area of the remaining region DBCE is 54 − 24 = 30 cm². Its ratio to the larger triangle is 5/9, not 4/9.

Why square? Both a corresponding base and its perpendicular height scale by the same length factor. Their product therefore scales by that factor twice. Do not apply the squared factor to an ordinary side length. LibreTexts' similar-triangle explanation provides background on matching angles and sides.

Make your written proof easy to check

Write each angle equality with its reason next to it, then give the similarity statement and test. Avoid relying on an unmarked visual impression that two lines are parallel. In the worked example, parallelism is explicitly given; without that information, the corresponding-angle argument would need another justification.

Choose a similarity question from the Secondary E-Maths collection. Cover the solution and build a three-column correspondence table for vertices, sides and direction of ratio. After marking, identify whether any error came from the proof, matching sides, whole-versus-part lengths or area scaling. Rework that specific step, then rotate your sketch and check that the same reasoning still works.